The equation of a straight line is the linear equation. It could be in one variable or two variables.
Linear Equation in One Variable
The equation with one variable in it is known as a Linear Equation in One Variable. (Scroll down to continue …)
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The general form for Linear Equation in One Variable is px + q = s, where p, q and s are real numbers and p ≠ 0.
Example:
x + 5 = 10
y – 3 = 19
These are called Linear Equations in One Variable because the highest degree of the variable is one.
Graph of the Linear Equation in One Variable
We can mark the point of the linear equation in one variable on the number line.
x = 2 can be marked on the number line as follows –
Graph of the Linear Equation in One Variable
Linear Equation in Two Variables
An equation with two variables is known as a Linear Equation in Two Variables. The general form of the linear equation in two variables is
ax + by + c = 0
where a and b are coefficients and c is the constant. a ≠ 0 and b ≠ 0.
Example
6x + 2y + 5 = 0, etc.
Slope Intercept form
Generally, the linear equation in two variables is written in the slope-intercept form as this is the easiest way to find the slope of the straight line while drawing the graph of it.
The slope-intercept form is y = mx+c
Where m represents the slope of the line.
and c tells the point of intersection of the line with the y-axis.
Remark: If b = 0 i.e. if the equation is y = mx then the line will pass through the origin as the y-intercept is zero.
Solution of a Linear Equation
There is only one solution in the linear equation in one variable but there are infinitely many solutions in the linear equation in two variables.
As there are two variables, the solution will be in the form of an ordered pair, i.e. (x, y).
The pair which satisfies the equation is the solution to that particular equation.
Example:
Find the solution for the equation 2x + y = 7.
Solution:
To calculate the solution of the given equation we will take x = 0
2(0) + y = 7
y = 7
Hence, one solution is (0, 7).
To find another solution we will take y = 0
2x + 0 = 7
x = 3.5
So another solution is (3.5, 0).
Graph of a Linear Equation in Two Variables
To draw the graph of a linear equation in two variables, we need to draw a table to write the solutions of the given equation, and then plot them on the Cartesian plane.
By joining these coordinates, we get the line of that equation.
The coordinates which satisfy the given Equation lie on the line of the equation.
Every point (x, y) on the line is the solution x = a, y = b of the given Equation.
Any point, which does not lie on the line AB, is not a solution of Equation.
Example:
Draw the graph of the equation 3x + 4y = 12.
Solution:
To draw the graph of the equation 3x + 4y = 12, we need to find the solutions of the equation.
Let x = 0
3(0) + 4y = 12
y = 3
Let y = 0
3x + 4(0) = 12
x = 4
Now draw a table to write the solutions.
x 0 4
y 3 0
Now we can draw the graph easily by plotting these points on the Cartesian plane.
Linear Equation in Two Variables
Equations of Lines Parallel to the x-axis and y-axis
When we draw the graph of the linear equation in one variable then it will be a point on the number line.
x – 5 = 0
x = 5
This shows that it has only one solution i.e. x = 5, so it can be plotted on the number line.
But if we treat this equation as the linear equation in two variables then it will have infinitely many solutions and the graph will be a straight line.
x – 5 = 0 or x + (0) y – 5 = 0
This shows that this is the linear equation in two variables where the value of y is always zero. So the line will not touch the y-axis at any point.
x = 5, x = number, then the graph will be the vertical line parallel to the y-axis.
All the points on the line will be the solution of the given equation.
Equations of Lines Parallel to the x-axis and y-axis
Similarly if y = – 3, y = number then the graph will be the horizontal line parallel to the x-axis.
There are many objects in our life which are round in shape. A few examples are the clock, dart board, cartwheel, ring, Vehicle wheel, Coins, etc. (Scroll down to continue …)
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Circles
Any closed shape with all points connected at equidistant from the centre forms a Circle.
Any point which is equidistant from anywhere from its boundary is known as the Centre of the Circle.
Radius is a Latin word which means ‘ray’ but in the circle it is the line segment from the centre of the circle to its edge. So any line starting or ending at the centre of the circle and joining anywhere on the border of the circle is known as the Radius of Circle.
Interior and Exterior of a Circle
In a flat surface, the interior of a circle is the line whose distance from the centre is less than the radius.
The exterior of a circle is the line in the plane whose distance from the centre is larger than the radius.
Terms related to circle
Chord: Any straight line segment that’s both endpoints falls on the boundary of the circle is known as Chord. In Latin, it means ‘bowstring’.
Diameter: Any straight line segment or Chord which passes through the centre of the Circle and its endpoints connects on the boundary of the Circle is known as the Diameter of Circle. So in a circle Diameter is the longest chord possible in a circle.
Arc: Any smooth curve joining two points is known as Arc. So in Circle, we can have two possible Arcs, the bigger one is known as Major Arc and the smaller one is known as Minor Arc.
Circumference: It is the length of the circle if we open and straighten it out to make a line segment.
Segment and Sector of the Circle
A segment of the circle is the region between either of its arcs and a chord. It could be a major or minor segment.
Sector of the circle is the area covered by an arc and two radii joining the centre of the circle. It could be the major or minor sector.
Angle Subtended by a Chord at a Point
If in a circle AB is the chord and is making ∠ACB at any point of the circle then this is the angle subtended by the chord AB at a point C.
Likewise, ∠AOB is the angle subtended by chord AB at point O i.e. at the centre and ∠ADB is also the angle subtended by AB at point D on the circle.
Theorem 1: Any two equal chords of a circle subtend equal angles at the centre.
Here in the circle, the two chords are given and PQ = RS with centre O.
So OP = OS = OQ = OR (all are radii of the circle)
∆POQ ≅ ∆SOR
∠POQ = ∠SOR
This shows that the angles subtended by equal chords to the centre are also equal.
Theorem 2: If the angles made by the chords of a circle at the centre are equal, then the chords must be equal.
This theorem is the reverse of the above Theorem 1.
Perpendicular from the Centre to a Chord
Theorem 3: If we draw a perpendicular from the centre of a circle to any chord then it bisects the chord.
If we draw a perpendicular from the centre to the chord of the circle then it will bisect the chord. And the bisector will make a 90° angle to the chord.
Theorem 4: The line which is drawn from the centre of a circle to bisect a chord must be perpendicular to the chord.
If we draw a line OB from the centre of the circle O to the midpoint of the chord AC i.e. B, then OB is the perpendicular to the chord AB.
If we join OA and OC, then
In ∆OBA and ∆OBC,
AB = BC (B is the midpoint of AC)
OA = OC (Both are the radii of the same circle)
OB = OB (same side)
Hence, ΔOBA ≅ ΔOBC (both are congruent by SSS congruence rule)
⇒ ∠OBA = ∠OBC (respective angles of congruent triangles)
∠OBA + ∠OBC = ∠ABC = 180° [Linear pair]
∠OBC + ∠OBC = 180° [Since ∠OBA = ∠OBC]
2 x ∠OBC = 180°
∠OBC = 90o
∠OBC = ∠OBA = 90°
∴ OB ⊥ AC
Circle through Three Points
Theorem 5: There is one and only one circle passing through three given non-collinear points.
In this figure, we have three non-collinear points A, B and C. Let us join AB and BC and then make the perpendicular bisector of both so that RS and PQ the perpendicular bisector of AB and BC respectively meet each other at Point O.
Now take the O as centre and OA as the radius to draw the circle which passes through the three points A, B and C.
This circle is known as Circumcircle. Its centre and radius are known as the Circumcenter and Circumradius.
Equal Chords and Their Distances from the Centre
Theorem 6: Two equal chords of a circle are at equal distance from the centre.
AB and CD are the two equal chords in the circle. If we draw the perpendicular bisector of these chords then the line segment from the centre to the chord is the distance of the chord from the centre.
If the chords are of equal size then their distance from the centre will also be equal.
Theorem 7: Chords at equal distance from the centre of a circle are also equal in length. This is the reverse of the above theorem which says that if the distance between the centre and the chords are equal then they must be of equal length.
Angle Subtended by an Arc of a Circle
The angle made by two different equal arcs to the centre of the circle will also be equal.
There are two arcs in the circle AB and CD which are equal in length.
So ∠AOB = ∠COD.
Theorem 8: The angle subtended by an arc at the centre is twice the angle subtended by the same arc at some other point on the remaining part of the circle.
In the above figure ∠POQ = 2∠PRQ.
Theorem 9: Angles from a common chord which are on the same segment of a circle are always equal.
If there are two angles subtended from a chord to any point on the circle which are on the same segment of the circle then they will be equal.
∠a = (1/2) ∠c (By theorem 8)
∠b = (1/2) ∠c
∠a = ∠b
Cyclic Quadrilaterals
If all the vertices of the quadrilateral come in a circle then it is said to be a cyclic quadrilateral.
Theorem 10: Any pair of opposite angles of a cyclic quadrilateral has the sum of 180º.
∠A + ∠B + ∠C + ∠D = 360º (angle sum property of a quadrilateral)
∠A + ∠C = 180°
∠B + ∠D = 180º
Theorem 11: If the pair of opposite angles of a quadrilateral has a sum of 180º, then the quadrilateral will be cyclic.
Two geometric figures which are the same in shape, such that one is simply a copy of the other on a smaller scale or a larger scale, are called similar geometric figures.
Two geometric figures are said to be similar if and only if they have the same shape but not necessarily the same size. Two congruent geometric figures are always similar but converse may or may not be true. (Scroll down to read more …)
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Similar Polygons: Two polygons of the same number of sides are similar, if
(i) their corresponding angles are equal and
(ii) their corresponding sides are in proportion or their corresponding sides are in the same ratio.
The same ratio of the corresponding sides is referred to as the representative fraction or the scale factor for the polygons.
Similar Triangles :
Two triangles are said to be similar,
if (i) their corresponding angles are equal and
ii) their corresponding sides are in proportion (are in the same ratio).
If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio. Or If a line is drawn parallel to one side of a triangle, intersecting the other two sides in distinct points, the other two sides are divided in the same ratio i .e.. If in ∆ABC, l∥ BC, intersecting in D and E. then
Converse of Basic Proportionality Theorem :
If a line divides any two sides of a triangle in the sameratio, the line is parallel to the third side i.e.
In ∆ABC, if l intersects AB in D and AC in E, such that:
Criteria for Similarity of Triangles:
Two triangles are said to be similar, if
(i) their corresponding angles are equal and (ii) their corresponding sides are in proportion (or are in the same ratio).
2 (i) AA or AAA Similarity Criterion : If two angles of one triangle are equal to two corresponding angles of another triangle, then the triangles are similar. If two angles of one triangle are respectively equal to the two angles of another triangle, then the third angles of the two triangles are necessarily equal, because the sum of three angles of a triangle is always 180 0 .
(ii) SAS Similarity Criterion : If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio, then the two triangles are similar. Or If two sides of a triangle are proportional to two corresponding sides of another triangle and the angles included between them are equal, then the triangles are similar.
iii) SSS Similarity Criterion : If in two triangles, sides of one triangle are proportional (or are in the same ratio) to the sides of the other triangle, then the triangles are similar. If ∆ABC~ ∆PQR by any one similarity criterion, then ∠A=∠P, ∠B=∠Q, ∠C=∠R and
i.e., A and P, B and Q, C and R are the corresponding vertices, also AB and PQ. BC and QR. CA and RP are the corresponding sides. 3 Areas of Similar Triangles: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. – The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding medians. – The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding altitudes. – The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding angle bisectors. Pythagoras Theorem : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Converse of Pythagoras Theorem : In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and similar to each other i .e ..
If in ∆ABC, ∠B=90^0 and BD ⊥ AC, then (i) ∆ADB ~ ∆ABC (ii) ∆BDC ~ ∆ABC (iii) ∆ADB ~ ∆BDC
Statistics is the study of collection, organization, analysis and interpretation of data.
Data
Data is a distinct piece of information in the form of fact or figures collected or represented for any specific purpose. The word data is derived from the Latin word Datum.
Collection of Data
In general, data is of two types. They are:
Primary Data
Secondary Data
Primary Data
Primary data is the data collected from any firsthand experience for an explicit use or purpose.
Secondary data
Secondary data is the data collected by any third party for a different purpose other than the user.
Presentation of Data
After collecting data it is important to present it meaningfully. There are many ways to present data.
1. Raw Data or Ungrouped Data
a. Raw Data or Ungrouped Data is the collected data without any change in its form.
Example
The marks obtained by 10 students in a Mathematics test are:
55 36 95 73 60 42 25 78 75 62
Range
Range is the difference between the highest and the lowest values of data.
For Above data: Range = 95 – 36 = 59
b. Frequency Distribution Table – Frequency Distribution Table is the data of a large number of items converted into tabular form.
Frequency is the number of times the item comes to the table.
2. Grouped Data
To present the very large number of items in the data we use a grouped distribution table.
Grouped Distribution Table
a. Class Interval – The group used to classify the data is called the class interval i.e. 20 – 30, 30 – 40.
b. Upper Limit – In each class interval, the greatest number is the upper-class limit.
c. Lower Limit – In each class interval, the smallest number is the lower class limit.
d. Class Size – It is the difference between the upper limit and the lower limit i.e. 10.
Class Size = Upper Limit – Lower Limit
e. Class Mark – The midpoint of each class interval is the class mark.
Grouped data could be of two types as below:-
Inclusive or discontinuous Frequency Distribution.
Exclusive or continuous Frequency Distribution
Inclusive or discontinuous Frequency Distribution – If the upper limit of a class is different from the lower limit of its next class then it is said to be an Inclusive or discontinuous Frequency Distribution.
Example
Draw the histogram of the following frequency distribution.
Daily earnings (in Rs)
700 – 749
750 – 799
800 – 849
850 – 899
900 – 949
950 – 999
No. of stores
6
9
2
7
11
5
Exclusive or continuous Frequency Distribution – If the upper limit of a class is the same as the lower limit of its next class then it is said to be exclusive or continuous Frequency Distribution
Example
Draw the histogram of the following frequency distribution.
Daily earnings (in Rs)
700 – 750
750 – 800
800 – 850
850 – 900
900 – 950
950 – 1000
No. of stores
6
9
2
7
11
5
Graphical Representation of Data
Since a picture represents better than a thousand words, The data is presented graphically. Some of the methods of representing the data graphically are:
1. Bar Graph
2. Histogram
3. Frequency Polygon
1. Bar Graph
It is the easiest way to represent the data in the form of rectangular bars so it is called Bar graph.
The thickness of each bar should be the same.
The space between the bars should also be the same.
The height of the bar should be according to the numerical data to be represented.
Example
Represent the average monthly rainfall of Nepal for the first six months in the year 2014.
Month
Jan
Feb
Mar
Apr
May
Jun
Average rainfall
45
65
40
60
75
30
Solution
On the x-axis mark the name of the months.
On the y-axis mark the class interval which we have chosen.
Then mark the average rainfall respective to the name of the month by the vertical bars.
The bars could be of any width but should be the same.
This is the required bar graph.
2. Histogram
It is similar to Bar graph, but it is used in case of a continuous class interval.
The class intervals are to be taken along an x-axis.
The height represents the frequencies of the respective class intervals.
Example
Draw the histogram of the following frequency distribution.
Daily earnings (in Rs)
700 – 750
750 – 800
800 – 850
850 – 900
900 – 950
950 – 1000
No. of stores
6
9
2
7
11
5
Solution:
Mark the daily earnings on the x-axis.
Mark the no. of stores on the y-axis.
As the scale is starting from 700 so we will mark the zigzag to show the break.
Mark the daily earnings through the vertical bars.
3. Frequency Polygon
Procedure to draw the frequency polygon
First, we need to draw a histogram.
Then join the midpoint of the top of the bars to a line segment and the figure so obtained is the required frequency polygon.
The midpoint of the first bar is to be joined with the midpoint of the imaginary interval of the x-axis
The midpoint of the last bar is to be joined with the midpoint of the next interval of the x-axis.
If we need to draw the frequency polygon without drawing the histogram then first we need to calculate the class mark of each interval and these points will make the frequency polygon.
Example
Draw the frequency polygon of a city in which the following weekly observations were made in a study on the cost of living index without histogram.
Step 1: First of all we need to calculate the class mark of each class interval.
Step 2: Take the suitable scale and represent the class marks on the x-axis.
Step 3: Take the suitable scale and represent the frequency distribution on the y-axis.
Step 4: To complete the frequency polygon we will join it with the x-axis before the first class and after the last interval.
Step 5: Now plot the respective points and join to make the frequency polygon.
Measures of Central Tendency
To make all the study of data useful, we need to use measures of central tendencies. Some of the tendencies are
1. Mean
2. Median
3. Mode
1. Mean (Average)
The mean is the average of the number of observations. It is calculated by dividing the sum of the values of the observations by the total number of observations.
It is represented by x bar or.
The meanof n values x1, x2, x3, …… xn is given by
Mean of Grouped Data (Without Class Interval)
If the data is organized in such a way that the frequency is given but there is no class interval then we can calculate the mean by
where, x1, x2, x3,…… xn are the observations
f1, f2, f3, …… fn are the respective frequencies of the given observations.
Example
Here, x1, x2, x3, x4, and x5 are 20, 40, 60, 80,100 respectively.
and f1 , f2 , f3 , f4, f5 are 40, 60, 30, 50, 20 respectively.
2. Median
The median is the middle value of the given number of the observation which divides into exactly two parts.
For median of ungrouped data, we arrange it in ascending order and then calculated as follows
If the number of the observations is odd then the median will be Term
As in the above figure the no. of observations is 7 i.e. odd, so the median will beterm.
= 4th term.
The fourth term is 44.
If the number of observations is even then the median is the average of n/2 and (n/2) +1 term.
Example
Find the median of the following data.
6, 7, 10, 12, 13, 4, 8, 12
1. First, we need to arrange it in ascending order.
4, 6, 7,8,10,12,12,13
2. The no. of observation is 8. As the no. of observation is even the median is the average of n/2 and (n/2)+1.
3.
4. 4th term is 8 and the 5th term is 10.
5. So the median
3. Mode
The mode is the value of the observation which shows the number that occurs frequently in data i.e. the number of observations which has the maximum frequency is known as the Mode.
Example
Find the Mode of the following data:
15, 20, 22, 25, 30, 20,15, 20,12, 20
Solution
Here the number 20 appears the maximum number of times so
Mode = 20.
Remark: The empirical relation between the three measures of central tendency is: 3 Median = Mode + 2 Mean
The figures which we can be drawn on a flat surface or that lie on a plane are called Plane Figure.
Example – Circle, Square, Rectangle etc.
Solid figures
The 3D shapes which occupy some space are called Solid Figures.
Example – Cube, Cuboid, Sphere etc. (Scroll down to continue …)
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Volume
Space occupied by any solid shape is the capacity or volume of that figure. The unit of volume is a cubic unit.
Surface Area
The area of all the faces of the solid shape is its total surface area. The unit of surface area is a square unit.
Lateral or Curved Surface Area
The surface area of the solid shape after leaving the top and bottom face of the figure is called the lateral surface of the shape. The unit of lateral surface area is a square unit.
Surface Area and Volume of a Cube
Cube is a solid shape having 6 equal square faces.
Lateral surface area of a cube
4s2
Total surface area of a cube
6s2
The volume of a cube
s3
Diagonal
√3 s, s = edge of the cube = side length of face of cube
Surface Area and Volume of a Cube
Example
What is the capacity of a cubical vessel having each side of 8 cm?
Solution
Given side = 8 cm So, Volume of the cubical vessel = l3 = (8)3 = 256 cm3.
Surface Area and volume of a Cuboid
Cuboid is a solid shape having 6 rectangular faces at a right angle.
Lateral surface area of a cuboid
2h(l + b)
Total surface area of a cuboid
2(lb + bh + lh)
Volume of a cuboid
lbh
Diagonal
l = length, b = breadth, h = height
Surface Area and volume of a Cuboid
Example
What is the surface area of a cereal box whose length, breadth and height is 20 cm, 8 cm and 30 cm respectively?
Solution
Given, length = 20 cm, breadth = 8 cm, Height = 30 cm
Total surface area of the cereal box = 2(lb + bh + lh)
= 2(20 × 8 + 8 × 30 + 20 × 30)
= 2(160 + 240 + 600)
= 2(1000) = 2000 cm2.
Surface Area and Volume of a Right Circular Cylinder
If we fold a rectangular sheet with one side as its axis then it forms a cylinder. It is the curved surface of the cylinder. And if this curved surface is covered by two parallel circular bases then it forms a right circular cylinder.
Curved surface area of a Right circular cylinder
2πrh
Total surface area of a Right circular cylinder
2πr2 + 2πrh = 2πr(r + h)
The volume of a Right circular cylinder
πr2h
r = radius, h = height
Surface Area and Volume of a Right Circular Cylinder
Surface Area and Volume of a Hollow Right Circular Cylinder
If a right circular cylinder is hollow from inside then it has different curved surface and volume.
Curved surface area of a Right circular cylinder
2πh (R + r)
Total surface area of a Right circular cylinder
2πh (R + r) + 2π(R2 – r2)
R = outer radius, r = inner radius, h = height
Surface Area and Volume of a Hollow Right Circular Cylinder
Example
Find the Total surface area of a hollow cylinder whose length is 22 cm and the external radius is 7 cm with 1 cm thickness. (π = 22/7)
Solution
Given, h = 22 cm, R = 7 cm, r = 6 cm (thickness of the wall is 1 cm).
Total surface area of a hollow cylinder = 2πh(R + r) + 2π(R2 – r2)
= 2(π) (22) (7+6) + 2(π)(72 – 62)
= 572 π + 26 π = 598 π
= 1878.67 cm2
Surface Area and Volume of a Right Circular Cone
If we revolve a right-angled triangle about one of its sides by taking other as its axis then the solid shape formed is known as a Right Circular Cone.
Curved surface area of a Right Circular Cone
πrl = πr[√(h2 + r2)]
Total surface area of a Right Circular Cone
πr2 + πrl = πr(r + l)
The volume of Right Circular Cone
(1/3) πr2h
r = radius, h = height, l = slant height
Surface Area and Volume of a Right Circular Cone
Surface Area and Volume of a Sphere
A sphere is a solid shape which is completely round like a ball. It has the same curved and total surface area.
Curved or Lateral surface area of a Sphere
4πr2
Total surface area of a Sphere
4πr2
Volume of a Sphere
(4/3) πr3
R = radius
Surface Area and Volume of a Sphere
Surface Area and Volume of a Hemisphere
If we cut the sphere in two parts then is said to be a hemisphere.
Curved or Lateral surface area of a Sphere
2πr2
Total surface area of a Sphere
3πr2
Volume of a Sphere
(2/3) πr3
r = radius
Surface Area and Volume of a Hemisphere
Example
If we have a metal piece of cone shape with volume 523.33 cm3 and we mould it in a sphere then what will be the surface area of that sphere?
Anything (Physical Material not emotions, feelings etc.) which has mass and volume (occupy space) is called matter.
We feel the presence of matter by one or more of our five sense organs.
Matter is made up of particles. (Scroll down to ontinue …)
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Particles:
Particles are very small in size. Therefore we cannot see particles with our naked eye.
Characteristics of the particles of matter:
(1) All matter (elements or compounds) consists of very small particles which can exist independently and are called particles.
(ii) The particles of matter are in a state of continuous motion and possess kinetic energy.
(iii) There are intermolecular spaces in between the particles (molecules) of matter.
(iv) The particles (molecules) of matter attract each other with a force called intermolecular force.
Intermolecular force is maximum in solids and least in the gases.
These material particles can be touched, moved by changing temperature or attracted by decreasing or increasing forces of attraction or repulsion.
2. States of Matter
Matter exists in three different physical states namely solid, liquid and gas.
One substance such as water can exist in all the three states such as, ice in solid state, water in liquid state and steam or vapours in gaseous state.
The state of matter depends on temperature, forces of attraction between their constituent particles etc.
3. Interconversion of Matter
All these three different states of matter are interconvertible depending upon temperature and pressure.
The state of matter can be changed by changing temperature or pressure.
Due to change in temperature and pressure there will be a change in inter-particle space as well as force between them, resulting in change in physical state.
Examples:
Applying pressure and reducing temperature can liquefy gases.
Solid CO₂ gets converted directly to a gaseous state on decrease of pressure to 1 atmosphere without changing into a liquid state. Due to this fact solid CO₂ is also known as DRY ICE.
4. Plasma: It is the fourth state of matter consisting of super energetic and super excited particles. These particles are in the form of ionised gases.
Examples:
The plasma in stars is formed due to high temperature.
Glowing plasma formed in fluorescent tubes and neon sign bulbs.
These devices contain inert gases which get ionised due to the passage of electric current. The colour of the glowing plasma depends upon the nature of the gas.
5. Sublimation: The process in which a solid state directly changes into a gaseous state on heating or vice-versa on cooling.
6. Melting or Fusion: The process of changing a solid into a liquid state by absorbing heat at a constant temperature is known as Melting or Fusion.
7. Freezing or Solidification: The process of changing a liquid into solid state by losing heat at a constant temperature is known as Freezing or Solidification.
8. Condensation: The process of changing a gas into a liquid state by giving out heat at constant temperature is known as Condensation .
Boiling or Vaporisation : The process of changing a liquid into a gaseous state by absorbing heat at constant temperature is known as Boiling or Vaporisation .
Boiling is a bulk phenomenon. Particles from the bulk (whole) of the liquid change into a vapour state.
Evaporation: The phenomenon of changing the physical state from liquid to vapour, at any temperature is called evaporation.
Evaporation is a surface phenomenon. Particles from the surface gain required energy to overcome the forces of attraction present in the liquid and change into the vapour state.
The rate of evaporation depends upon the surface area exposed to the atmosphere, the temperature, the humidity and the wind speed.
Evaporation causes cooling.
Evaporation takes place at all temperatures, below the boiling point of a liquid
Factors affecting evaporation:
• Rate of evaporation increases with increase in surface area.
• Rate of evaporation increases with increase in temperature.
• Rate of evaporation increases with decrease in Humidity.
• Rate of evaporation increases with increase in wind speed.
Latent heat of boiling or Latent heat of Vaporisation: Latent heat of boiling or Latent heat of Vaporisationis the heat energy required to change 1 kg of a liquid to gas at atmospheric pressure at its boiling point.
Kelvin is the SI unit of temperature.
0°C = 273.16 K.
For convenience, we take 0°C = 273 K after rounding off the decimal.
To change a temperature on the Kelvin scale to the Celsius scale you have to subtract 273 from the given temperature, and to convert a temperature on the Celsius scale to the Kelvin scale you have to add 273 to the given temperature.
Conversion Formula: t°C = (t+273) K
Boiling point or Vaporisation point: Boiling point or Vaporisation point is the fixed temperature at which a liquid converts into a gaseous state at atmospheric pressure.
Melting point or Fusion point: Melting point or Fusion point is the temperature at which a solid starts converting into a liquid state at atmospheric pressure.
Evaporation Causes cooling: During evaporation the particles at the surface of the liquid gain energy from the surroundings and change into vapour.. Therefore Evaporation Causes cooling effect.
Sponge can be compressed although it is solid: Sponge contains minute holes in which air is trapped.So when it is pressed, the air gets expelled and the sponge gets compressed. Also,the material of the sponge is not rigid.
Temperature does not change during change of state: The temperature remains constant at its melting and boiling points (during change of state) until all the substance melts or boils.
Because the heat supplied is continuously used up in changing the state of the substance by overcoming the force of attraction between the particles.
There is no increase in the kinetic energy of the particles and thus, temperature does not change.
This heat energy absorbed without showing any rise in temperature is given the name latent heat of fusion/latent heat of vaporisation.
Effect of pressure on physical state of a substance:
If pressure is applied, melting point decreases and boiling point increases
When pressure is increased, the particles come closer and the force of attraction increases between them and this results in a change of state.
Example: When high pressure is applied to a gas by reducing its temperature, the particles of gas come close and get converted to a liquid. This is also known as liquefaction.
The amount of heat energy required in changing a 1 kg of solid into liquid at atmospheric pressure and its melting point is known as the latent heat of fusion.
[ Lice = 80 cal/g = 3.34 × 105 J/kg].
• The amount of heat which is required to convert 1 kg of the liquid (at its boiling point) to vapours of gas without any change in temperature is known as latent heat of vaporisation.
[Lwater =540 cal/g= 22.5 × 105 J/kg].
• The amount of heat absorbed or liberated , Q = mL.
• The specific heat is the amount of heat per unit mass required to raise the temperature by one degree Celsius.
• Q = m.s. t, where m = mass of the body, s = specific heat of the body and t is temperature difference and m.s is called thermal capacity.
• Change of liquid into vapours at any temperature below the boiling point.
Takes the latent heat from the body. Thus, the body cools when evaporation takes place.
Evaporation:
(1) Evaporation is a slow process.
(ii) Evaporation takes place at the surface mass of the liquid.
(iii) Evaporation takes place at all temperatures.
(iv) The substance becomes cool due to evapora- tion process.
(v) Heat is absorbed from the surroundings due to Evaporation. Absorption of heat from the surroundings causes cooling effect.
Boiling:
(1) Boiling is a rapid process.
(ii) Boiling takes place throughout the mass of a liquid.
(iii) Boiling takes place at a definite temperature called the boil- ing point.
(iv) The substance remains hot during the boiling process.
(v) Heat is required from an external source such as a burner for boiling to take place.
Scales of temperature
• Three scales are commonly used for measuring temperature, namely, the Celsius scale, the Fahrenheit scale and the Kelvin scale.
• The relation between the Celsius and the Kelvin scale can be expressed as:
C + 273 = K
• The relation between the Celsius and the Fahrenheit scale can be expressed as follows.
Property
Solid
Liquid
Gas
Inter particle space
Very less
Larger than solid butlesser than gas
Very large
Inter particle force
Very strong
Weaker than solidbut stronger than gas
Very weak
Nature (Rigidity)
Very hard and rigid
Fluid
Highly fluid
Compressibility
Negligible
Very small
Highly compressible
Shape
Definite shape
Indefiniteshape
Indefinite Shape
Volume
Definite Volume
Indefinite shape
Indefinite volume
Density
high
Less than solid
Very low
Kinetic energy
low
high
Very high
Diffusion
Negligible
Slow
Very high
Specific Heat
11.8 NATURAL PHENOMENA AND CONSEQUENCES OF HIGH SPECIFIC HEAT CAPACITY OF WATER
Some consequences of high specific heat capacity of water are given below.
(i) The climate near the seashore is moderate :
The specific heat capacity of water is very high (= 1000 cal kg-1 °C-1 or 4200 J kg-1 K-¹). It is about five times as high as that of sand. Hence the heat energy required for the same rise in temperature by a certain mass of water will be nearly five times that required by the same mass of sand.
Similarly, a certain mass of water will give out nearly five times more heat energy than that given by sand of the same mass for the same fall in temperature.
As such, sand (or earth) gets heated or cooled more rapidly as compared to water under similar conditions.
Thus, a large difference in temperature is developed between the land and the sea due to which land and sea breezes are formed”. These breezes make the climate near the seashore moderate.
(ii) Hot water bottles are used for fomentation: The reason is that water does not cool quickly due to its large specific heat capacity, so a hot water bottle provides heat energy for fomentation for a long time.
(iii) Water is used as an effective coolant: By allowing water to flow in pipes around the heated parts of a machine, heat energy from such parts is removed (e.g. radiators in car and generator are filled with water). Water in pipes extracts more heat from surroundings without much rise in its temperature because of its large specific heat capacity.
(iv) In cold countries, water is used as a heat reservoir for wine and juice bottles to avoid their freezing: The reason is that water due to its high specific heat capacity can impart a large amount of heat before reaching up to the freezing temperature. Hence bottles kept in water remain warm and they do not freeze even when the surrounding temperature falls considerably.
(v) Farmers fill their fields with water to protect the crops from frost: In the absence of water, if on a cold winter night, the atmospheric temperature falls below 0°C, the water in the fine capillaries of plants will freeze, so the veins will burst due to the increase in volume of water on freezing. As a result, plants will die and the crop will be destroyed. In order to save crop on such cold nights, farmers fill their fields with water because water has a high specific heat capacity, so it does not allow the temperature in the surrounding area of plants to fall up to 0°C.
(vi) All plants and animals have a high content of water in their bodies: All plants and animals have nearly 80% to 90% of water in their bodies so it helps in maintaining the body temperature nearly same in all seasons due to high specific heat capacity of water.
SOME EXAMPLES OF HIGH AND LOW THERMAL CAPACITY
(1) The base of a cooking pan is made thick : By making the base of the cooking pan thick, its thermal capacity becomes large and it imparts sufficient heat energy at a low temperature to the food for its proper cooking. Further it keeps the food warm for a long time, after cooking.
(2) The base of an electric iron is made thick and heavy: By doing so, the thermal capacity of the base becomes large and it remains hot for a long duration even after switching off the current.
(3) The vessel used for measurement of heat (i.e., calorimeter) is made of thin sheet of copper:
The reason is that the specific heat capacity of copper is low and by making the vessel thin, its thermal capacity becomes low so that it takes a negligible amount of heat from its contents to attain the temperature of the contents.
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Matter: Anything that occupies space is called matter.
Example: Air, water, rock etc.,
Matter exists in our surroundings in both pure and impure forms. (Scroll down to continue …).
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Mixture: A mixture is a matter that contains more than one pure substance in any ratio/proportion.
A mixture is an impure form of matter.
Example:
Water in milk, lemon juice, Ginger Garlic paste, etc.,
The mixture may or may not be separated into its constituent particles by physical processes.
Substance: A matter that cannot be separated into its constituent particles by any physical process is known as a substance.
Example:
Solution: A homogeneous mixture of two or more substances is called a solution.
Example:
Tea, sugar, and common salt are dissolved in water.
Alloy: A homogeneous mixture of metals is called an alloy.
Properties of the Solution:
A solution is a homogeneous mixture
Particles are extremely small, not visible to the naked eye
The light path is invisible in solution.
Solute particles cannot be separated by filtration
Concentration of solution: The concentration of a solution is the amount of solute present in a given quantity of the solution.
Unsaturated and Saturated Solutions: a solution in which a larger quantity of solute can be dissolved without raising its temperature, is called an unsaturated solution.
• A solution in which no more solute can be dissolved at a certain temperature, is called a saturated solution.
Solubility: The maximum amount of a solute that can be dissolved in 100 grams of a solvent at a specified temperature is known as the solubility of the solute in that solvent.
Suspension: a heterogeneous mixture of solids and liquids where the solid particles are suspended throughout the medium.
Example: Mixture of chalk powder and water
Properties of Suspension
• Particles are visible to the naked eye
• Light path in a suspension is visible
• Particles settle down
Colloidal Solution: Colloidal Solution Is a heterogeneous mixture, but appears to be homogeneous.
Examples: Milk, soap lather, soda water, pumice stone, rubber, bread, fog, cloud, insecticide spray, butter, etc.
Properties of colloidal solutions
• Heterogeneous mixture
• Particle size is small, not visible to the naked eye
• Light path can be visible;
• Particles do not settle down
• Substances cannot be separated by filtration
Tyndall Effect: Scattering of light beam by suspended particles in the solution.
Physical and Chemical changes:
Physical and change: The changes in which no new substances are formed are called physical changes.
Chemical change: The changes in which new substances are formed are called chemical changes.
SEPARATION OF MIXTURES
The method of separation depends on both the type of mixture and the physical properties of its constituents.
These are :
(i) The physical state of the constituents.
(ii) The differences in the physical properties
of the constituents, such as:
(a) boiling point
(b) melting point
(c) density
(d) magnetic properties
(e) ability to sublime
(f) volatility
(g) solubility in various solvents.
• Evaporation: Used for separating mixtures of volatile solvents and non-volatile solutes.
Working Principle:
One component should be non-volatile. It may or may not be soluble in water.
Example: Separating salt from its solution
• Centrifugation used for separating components based on the difference in their weights.
Working Principle:
Difference in the densities of two liquids.
Example: Separating mixtures of cream from milk
• Separating Funnel: Used for separating two or more immiscible liquids.
Working Principle:
Immiscible liquids with different densities get separated into different layers if they are in the same container.
Example: Separating oil and water
Sublimation:
Sublimation is the process of converting a solid into vapour and returning it to the solid state without passing through the liquid state.
Sublimation is used to separate sublimable solids from their mixtures.
Working Principle:
One of the components can be sublime.
Example: Separating ammonium chloride from a mixture
Chromatography:
The process of separating the different dissolved constituents of a mixture by their adsorption (adsorption refers to the collection of one substance on the surface of another substance.) over an appropriate adsorbing material is called chromatography.
Chromatography is used to separate those solutes that dissolve in the same solvent.
Working Principle:
Adsorption/partition
Example: Separating the components of a dye
Distillation:
Distillation is the process of heating a liquid to convert it into vapours and then condensing the vapours back into a liquid.
Distillation is used to separate two miscible liquids that boil without decomposition.
Working Principle:
One component should be a soluble solid in a liquid.
Example: Separating a mixture of acetone and water
Fractional distillation
Fractional distillation is a process that involves the distillation and collection of fractions or different liquids boiling at different temperatures.
Fractional distillation is used to separate a mixture of liquids when their boiling temperatures differ by less than 25 K.
Example: Separating different components of petroleum
Crystallization: Used to separate pure solids from a solution by forming crystals.
Working Principle:
A solid dissolved in a liquid is separated by evaporating the solvent completely by heating the mixture.
Example: Obtaining pure crystals of copper sulphate from an impure sample.
Differences Between Mixture And Compound
Property
Mixture
Compound
Nature
When two or more elements or compounds or both are mixed together, such that they do not combine chemically, a mixture is formed.
When two or more elements unitechemically, a compound is formed.
Structure
Mixtures are generally heterogeneous. However, some mixtures can be homogeneous.
Compounds are always homogeneous.
Composition
In case of mixtures their constituents can be present in any ratio, i.e., mixtures havevariable composition.
In case of compounds, the constituents arepresent in a fixed ratio by weight.
Properties
The constituents of a mixture retain theirindividual chemical and physical properties.
The properties of a compound are entirelydifferent from the properties of itsconstituents
Separation of constituents
The constituents of a mixture can beseparated by applying physical methods likesolubility, filtration, evaporation, distillation,use of magnet, etc.
The constituents of a compound cannot beseparated by applying physical methods.However, constituents of a compound can beseparated by chemical means.
Energy change
There may or may not be energy changeduring the formation of mixture.
During the formation of a compound eitherthe energy is absorbed or given out.
Different kinds of matter contain different kinds of atoms present in them.
Protons were discovered by Ernest Rutherford, in his famous gold foil experiment.
Electrons were discovered by J.J. Thomson, in his cathode ray tube experiment.
Neutrons were discovered by James Chadwick. (Scroll down to continue …).
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Laws of Chemical Combination: Antoine Laurent Lavoisier, is known as ‘Father of Modern Chemistry.
Lavoisier put forward the law of conservation of mass, which laid the foundation of chemical sciences.
Law of Conservation of Mass:Law of Conservation of Mass states that, “mass is neither created nor destroyed in a chemical reaction.
In other words, the mass of the reactants must be equal to the mass of products.
Law of Constant Proportions or Definite Composition:Law of Constant Proportions or Definite Composition states that, in a pure chemical substance, the elements are always present in definite proportions by mass.
Dalton’s Atomic Theory:
(i) Every element is composed of extremely small particles called atoms.
(ii) Atoms of a given element are identical, both in mass and properties.
(iiii) Different chemical elements have different kinds of atoms; in particular, their atoms have different masses.
(iv) The atoms neither be created nor be destroyed or transformed into atoms of other elements.
(v) Compounds are formed when atoms of different elements combine with each other in small whole number ratios.
(vi) The relative number and kinds of atoms in a given compound are constant.
Drawbacks of Dalton’s Atomic Theory:
(i) According to modern theory, an atom is not the ultimate indivisible particle of matter. Today, we know that atoms are divisible, they are themselves made-up of particles (protons, electrons, neutrons,etc.).
(i) In the case of isotopes of an element, the assumption that the atoms of the same element have the same mass does not hold good.
Atom: It is the smallest particle of an element that maintains its chemical identity throughout all chemical and physical changes.
The smallest unit of a substance which can exist independently is called a molecule.
Atomicity: It is defined as the number of atoms present in a molecule of an element or a compound.
Mono atomic: Molecule having only one atom is called mono atomic,
e.g., He, Ne, Ar.
Diatomic: Molecules made-up of two atoms are called diatomic, e.g., H₂, Cl₂, O₂, N2.
Triatomic: Molecules made-up of three atoms, called triatomic.
e.g., O3, H₂O, NO2.
Tetraatomic : Molecules made-up of four atoms, called tetra atomic.
e.g., P4, NH3, SO3
Polyatomic: Molecules made-up of five or more atoms, called polyatomic/
e.g., CH4.
Polyatomic: Any molecule which is made-up of more than four atoms is called polyatomic,
e.g., Sg.
Relative Atomic Mass: It is defined as the number of times one atom of an element is heavier than
(1/12)th of the mass of an atom of Carbon – 12.
Relative Atomic Mass (RAM) = Mass of an atom of an element/
¹/12 th mass of C-12
Molecular Mass: The molecular mass of a substance is the sum of the atomic masses of all atoms in a molecule of a substance,
e.g., molecular mass of water is 18 u.
The mole (or mol) is the SI unit of the amount of a substance. One mole is equal to the amount of substance that contains as many elementary units as there are atoms in 12 g of the carbon-12 isotope.
The elementary units may be atoms, molecules, ions, radicals, electrons, etc., and must be specified.
This number is called Avogadro’s number (No) or Avogadro’s constant
[NA = 6.0221367 x 1023]. Generally,
Avogadro’s Number is rounded to 6.022 x 1023.
For better understanding we can compare avogadro number with a dozen as:
One dozen oranges contain 12 oranges, similarly, 1 mole of hydrogen atoms contain 6.022 x 1023 H atoms.
H₂O = 2 x H + 1 × O
= 2 x 1+1 x 16 = 2+16
= 18 amu or u.
By : 1 mole of a compound has a mass equal to its relative molecular mass expressed in grams.
1 mole = 6.022 × 1023 number
= Relative mass in grams.
A molecule is the smallest particle of an element or a compound capable of independent existence under ordinary conditions. It shows all the properties of the substance.
A chemical formula of a compound shows its constituent elements and the number of atoms of each combining element.
Clusters of atoms that act as an ion are called polyatomic ions. They carry a fixed charge on them.
The chemical formula of a molecular compound is determined by the valency of each element.
In ionic compounds, the charge on each ion is used to determine the chemical formula of the compound.
Scientists use the relative atomic mass scale to compare the masses of different atoms of elements. Atoms of carbon-12 isotopes are assigned a relative atomic mass of 12 and the relative masses of all other atoms are obtained by comparison with the mass of a carbon-12 atom.
The Avogadro constant 6.022 × 1023 is defined as the number of atoms in exactly 12 g of carbon-12.
The mole is the amount of substance that contains the same number of particles (atoms/ions/ molecules/formula units, etc.) as there are atoms in exactly 12g of carbon-12. Mass of 1 mole of a substance is called its molar mass.
The relative atomic mass of the atom of an element is the average mass of the atom as compared to 1/12th mass of one carbon-12 atom.
Hint: We know that chemical formulas can also be written using a criss-cross method. In the criss-cross method, the numerical value of the ion charge of the two atoms is crossed over, which becomes the subscript of the other ion. Using this technique, we will write the chemical formula of the given compounds.
Complete step by step answer:
Let’s us discuss about the given compound as,
A.Magnesium chloride
We have to remember that the atomic number of Magnesium is 12 and has a valency of 2.
It means it has two electrons in the outermost shell for bonding.
The atomic number of chlorine is 17 and has 7 electrons in the outermost shell.
It means it just needs one more atom for bonding.
Hence, we will use atoms of chlorine to bond with one atom of magnesium.
We can apply the criss-cross method for this compound as,
Therefore, the chemical formula of magnesium chloride is MgCl2
B.Calcium oxide
We have to know that the atomic number of calcium is 20 and has a valency of 2, it means it has 2 two atoms in the outermost shell for bonding.
The atomic number of Oxygen is 8
8 and has a valency of 2, it has 6 atoms in the outermost shell, it needs 2 more to complete the octet.
Hence, we need one calcium atom to bond with one oxygen atom.
We can apply the criss-cross method for this compound as,
Therefore, the chemical formula of magnesium chloride is CaO
C. Copper nitrate
We have to know that the atomic number of copper is 29 and has two atoms in the outermost shell for bonding. While a nitrate molecule has only one valence electron.
We need 2 nitrate molecules to satisfy the valency of 1 copper atom.
We can apply the criss-cross method for this compound as,
Therefore, the chemical formula of magnesium chloride is
Cu(NO3)2
D.Aluminium chloride
We have to know that the atomic number of aluminium is 13 and has a valency of 3 atoms and chlorine atom has a valency of 1. Since it has 7 electrons in the outermost shell.
Thus, we need 3 chlorine atoms to satisfy the valency of 1 aluminium atom.
We can apply the criss-cross method for this compound as,
Therefore, the chemical formula of magnesium chloride is AlCl3.
E.Potassium nitrate
We have to remember that the atomic number of potassium is 19 and has a valency of 1 and nitrate also has a valency of 1, since it needs one more atom to complete its octet. Hence, we need only one molecule of nitrate for one atom of potassium.
We can apply the criss-cross method for this compound as,
Therefore, the chemical formula of magnesium chloride is KNO3.
Note: As we know that the criss-cross method is the most efficient way to write the correct chemical formula of the molecule. It is generally used for finding out the formula of a bonding of a metal with a non-metal to form ionic bonds. Signs of the two ions are dropped, the ion value is crossed which becomes the subscript of the crossed atoms.
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Different kinds of matter contain different kinds of atoms present in them.
Protons were discovered by Ernest Rutherford, in his famous gold foil experiment.
Electrons were discovered by J.J. Thomson, in his cathode ray tube experiment.
Neutrons were discovered by James Chadwick. (Scroll down to continue …)
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Charged Particles in Matter
Whenever we rub two objects together, they become electrically charged.
This is because atoms contain charged particles in them.
Therefore, atoms can be divided further into particles i.e proton, electron and neutron.
Atoms consist of an equal number of protons and electrons.
Protons exist in the interiors of the atom and electrons exist in the exteriors of the atom. Therefore, electrons can be removed from an atom.
Since electrons exist in the exteriors of the atom they can be removed from an atom.
Dalton’s Atomic Theory
The postulates of the atomic theory by John Dalton
The matter is made up of tiny particles called Atoms that cannot be divided.
Atoms are never formed or destroyed during a chemical reaction.
Atoms of an element exhibit the same nature.
Atoms of the same element have equal size, mass and they exhibit similar chemical properties.
Atoms of different elements exhibit variant chemical properties.
Atoms form compounds by combining in a ratio of whole numbers.
A compound contains molecules in which a constant number and types of atoms are present.
Failure of Dalton’s Atomic Theory
Dalton suggested that atoms can neither be created nor destroyed and are indivisible.
But the discovery of electrons and protons in atoms disproved this aspect of Dalton’s theory.
Thomson’s Model of an Atom
According to J.J. Thomson, the structure of an atom can be compared to Christmas pudding.
According to this model the electrons are present inside a positive sphere.
An atom is composed of a positively charged sphere in which electrons are embedded.
Atoms are neutral as the positive and negative charges are equal in number.
Rutherford’s Model of an Atom
Rutherford’s Experiment
Rutherford experimented by passing alpha rays through a thin gold foil.
He expected that the gold atoms would deflect the Alpha particles.
Observations
Inferences
Alpha particles which had high speed moved straight through the gold foil
Atom contains a lot of empty space
Some particles got diverted a by small angles
Positive charges in the atom are not occupying much of its space
Only one out of 12000 particles bounced back
The positive charges are concentrated over a particular area of the atom.
Based on his experiment Rutherford gave the nuclear model of an atom as the following.
Rutherford’s Atomic Model
Rutherford’s Atomic Model is known as Planetary Atomic Model and Nuclear Atomic Model.
According to Rutherford’s Atomic Model:
Atoms contain a lot of unoccupied space
The center of the atom is highly positive , Rutherford named it as nucleus
The atom contains an equal amount of positive and negative charges.
Nucleus of Atom
The nucleus is located at the center of the atom.
All the mass of the atom is because of the nucleus.
The electrons revolve around the nucleus in circular parts which called Orbits
The size of an atomic nucleus is much smaller than its atom.
Drawbacks of the Nuclear Atomic Model
The Rutherford’s Atomic Model failed to explain how an atom remains stable despite having positive and negative charges present in it.
Maxwell’s theory of radiation if any charged particle moves in a circular motion it radiates energy.
So, if electrons move in a circular motion around the nucleus they should radiate some energy as a result this decreases at the speed of the electrons. As a result, they would fall into the nucleus and the nucleus should collapse because of its high positive charge.
But it is not happening because the matter is not collapsing.
Nucleons: The subatomic particles present in the nucleus are collectively called Nucleons. Protons and Neutrons are nucleons.
Bohr’s Model of an Atom
Bohr Atomic Model states as the following:
Electrons revolve around the nucleus in particular circular paths, called orbits.
The electrons do not emit any energy while moving in their orbits.
The orbits are also called Energy Levels.
Energy Levels or Orbits are represented by using letters or numbers as shown in the figure.
Neutron:
J. Chadwick discovered Neutron, a subatomic particle of an atom.
Neutron carries no charge.
Subatomic Particles of Atom
Electrons
Electron carry a negative charge
Protons
Protons carry a positive charge
Neutrons
Neutrons are neutral
Electronic Configuration:The distribution of electrons in different shells or orbits is called Electronic Configuration.
If Orbit number = n
Then number of electrons present in an Orbit = 2n2
So, for n =1
Maximum electrons present in shell – K = 2 * (1)2 = 2
The outermost shell can contain at most 8 electrons.
The shells in an atom are filled in sequence.
Thus, until the inner shells of an atom are filled completely the outer shells cannot contain any electrons.
Valency
Valence Electrons – Electrons existing in the outermost orbit of an atom are called Valence Electrons.
The atoms which have completely filled the outermost shell are not very active chemically.
The valency of an atom or the combining capacity of an atom is given by the number of elements present in the outermost shell.
For Example, Helium contains two electrons in its outermost shell which means its valency is two. In other words, it can share two electrons to form a chemical bond with another element.
What happens when the outermost shell contains a number of electrons that are close to its maximum capacity?
Valency in such cases is generated by subtracting the number of electrons present in the outermost orbit from octet (8). For example, oxygen contains 6 electrons in its outermost shell. Its valency is calculated as: 8 – 6 = 2. This means oxygen needs two electrons to form a bond with another element.
Representation Element:
Atomic Number of an Element
Atomic Number (Z) = Number of protons in an atom
Mass Number of an Element
Mass Number = Number of protons + Number of neutrons
Isotopes
The atoms of an element can exist in several forms having similar atomic numbers but varying mass numbers.
Isotopes are pure substances.
Isotopes have a similar chemical nature.
Isotopes have distinct physical characteristics.
Use of Isotopes:
1. The fuel of Nuclear Reactor – Isotope of Uranium
2. Treatment of Cancer – Isotope of Cobalt
3. Treatment of Goiter – Isotope of Iodine
Example: Consider two atomic species namely U and V. Are they isotopes?
U
V
Protons
5
5
Neutrons
5
6
Mass Number
5 + 5 = 10
5 + 6 = 11
Atomic Number
5
5
From the above example, we can infer that U and V are isotopes because their atomic number is the same.
Isobars
The atoms of several elements can have a similar mass number but distinct atomic masses. Such elements are called Isobars.
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What are Living organisms made up of? All living organisms are made up of cells. Cell is the basic structural and functional unit of complex organisms.
History of cell:
Cells were first discovered by Robert Hooke in 1665 with the help of a primitive microscope. Leeuwenhoek, in 1674, with the improved microscope, discovered free-living cells in pond water for the first time. (Scroll down to continue …)
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Robert Brown in 1831 discovered the nucleus in the cell.
Purkinje in 1839 coined the term ‘protoplasm‘ for the fluid part of the cell.
Schleiden in 1838 and Schwann in 1839 proposed the cell theory which stated that all plants and animals are composed of cells.
Rudolf Virchow in 1855 further expanded the cell theory by suggesting that all cells arise from pre-existing cells.
The invention of magnifying lenses led to the discovery of the microscopic world.
Unicellular organisms are the organisms in which a single cell performs all the functions like nutrition, respiration, excretion and reproduction.
Example: Amoeba, Chlamydomonas, Paramecium and Bacteria possess single cells constituting the whole organism. Multicellular organisms are the organisms which possess many cells to perform different functions.
Multicellular organisms represent themselves as a member of a group of cells or as an individual.
individual.
Example: Fungi, plants and animals have many cells that group together to form tissues.
Every multi cellular organism has come from a single cell. All cells thus come from pre existing cell.
Some organisms can also have cells of different kinds.
The shape and size of cell are related to the specific function they perform.
Some cells change their shapes.
Example: Amoeba. In some cases the cell shape could be more or less fixed and the peculiar for a particular type of cell.
Example: Nerve cells. Each living cell has the capacity to perform certain basic functions that are characteristic of all living forms.
There is a division of labour in multicellular organism such as human beings.
This means that different parts of the human body perform different functions.
Similarly division of labour is also seen within a single cell. In fact each such cell has got certain specific components within it known as cell organelles. Each kind of cell organelle performs a special function.
A cell is able to live and perform all its functions because of these organelles.
These organelles together constitute the basic unit called the cell. What is a cell made up of? What is the structural organization of a cell? Every cell would have three features- plasma membrane, nucleus and cytoplasm.
All activities inside the cell and interactions of the cell with its environment are possible due to these features. Plasma membrane or cell membrane: This is the outermost covering of the cell that separates the contents of the cell from its external environment. It is flexible and made up of organic molecules called lipids and proteins.
The flexibility of the cell membrane also enables the cell to engulf in food and other material from its external environment. Such processes are known as endocytosis.
Example: Amoeba It allows the movement of some substances into and out of the cell.
It also prevents movement of some other materials.
Therefore it is called a selectively permeable membrane. Movement of substances through this semi-permeable membrane can be by the process of diffusion, osmosis etc.
Difference between diffusion and osmosis
If we put an animal cell or a plant cell into a hypotonic solution the cell is likely to swell up.
The cell will stay in the same size if it kept it in isotonic solution.
If the solution is hypertonic then the cell will shrink. Unicellular fresh water organism and most plants tend to gain water through osmosis.
Cell wall: It is present only in plant cells. The cell wall is composed of cellulose and is permeable. It separates the contents of the cell from the surroundings. It gives shape and protection to the cell. Cell walls permit the cells of plants, fungi and bacteria to withstand very dilute external media without bursting.
Plasmolysis: It is the process in which cells lose water in a hypertonic solution.
Nucleus: The nucleus has a double layered covering called nuclear membrane. The nuclear membrane has pores which allow the transfer of material from inside to outside. The nucleus contains chromosomes which are composed of Deoxyribonucleic acid (DNA) and proteins. Nucleus controls all the activities of the cell. As the nucleus carries genetic information in the form of DNA, it plays a major role in cell division and cell development. The functional segments of DNA are called genes. Nucleus plays an important role in protein synthesis and transmission of characters from one generation to another generation. It plays a central role in cellular reproduction. In some organisms nuclear membrane is absent and nuclear region contains only nucleic acids called nucleoid. Such organisms called prokaryotes. Eg. Bacteria. are called eukaryotes. Organisms with cells having a nuclear membrane
Cytoplasm:
The cytoplasm is the fluid content inside the plasma membrane. It is a jelly like viscous substance occupying entire cell except the nucleus. It also contains many specialized cell organelles that perform a specific function for the cell.
Cell organelles: Cell organelles include endoplasmic reticulum, Ribosomes, Golgi apparatus, Mitochondria, Plastids, Lysosomes, and Vacuoles. They are important because they carry out some very crucial functions in cells.
Endoplasmic reticulum (ER): The ER is a large network of membrane bound tubes and sheets. It serves as channels for the transport of materials especially proteins between various organs of the cytoplasm or between the cytoplasm and nucleus. It also functions as a cytoplasmic framework providing a surface for some of the biochemical activities of the cell. There are two types of ER- Rough endoplasmic reticulum and smooth endoplasmic reticulum.
RER: These are rough at surface and are associated with ribosomes. These are responsible for the synthesis of proteins. SER: These are smooth at surface and are not associated with ribosomes. It helps in the manufacture of fat molecules or lipids. It also plays a crucial role in detoxifying many poisons and drugs.
Membrane biogenesis: Some of the proteins and lipids synthesized by EF help in building the cell membrane. This process is known as membrane biogenesis.
Golgi Apparatus: These cell organelles are named after the biologist, Camillo Golgi, who first described it. The Golgi consists of a stack of membrane-bound cisternae. These membranes often have connections with the membranes of ER and therefore constitute another portion of a complex cellular membrane system. Its functions include the storage, modification and packaging of products in vesicles. It is also involved in the formation of lysosomes.
Lysososmes: Lysosomes are membranous sacs filled with enzymes. These enzymes are made by RER. They are a kind of waste disposal system of the cell. They help to keep the cell clean by digesting any foreign material as well as worn out cell organelles. Lysosomes contain hydrolytic enzymes which are capable of digesting cellular macromolecules. When the cell gets damaged, the lysosome may burst and its enzymes may digest thecell itself. Hence, lysosomes are called as ‘suicidal bags’.
Mitochondria: These are cellular organelles termed as ‘power houses of the cells’. These are bounded by a double membrane. The outer membrane is smooth while the inner membrane is thrown into folds called as cristae. The cristae increase the area of cellular respiration. Mitochondria releases energy in the form of ATP molecules. ATP is known as the “energy currency of the cell”. Mitochondria have its own DNA DNA ribosomes and are able to make some of their own proteins.
Plastids:
Plastids are present only in plant cells. These are of two types- chromoplasts (coloured plastids) and leucoplasts (white or colourless plastids). Plastid contains pigment called chlorophyll are known as chloroplasts. These are important for photosynthesis in plants. Chromoplasts are the organelles which provide bright colours to the plant structures like buds, flowers etc.
Leucoplasts: are the organelles which store starch, oils and protein granules. Plastids consist of numerous membrane layers embedded in a material called the stroma. Plastids also have their own DNA and ribosomes.
Vacuoles: Vacuoles are membrane bound compartments present in both plant and animal cells. These are storage sacs for solid or liquid contents. These are small sized in animal cells while bigger in plant cell. In plant cells vacuoles are full of sap and provide turgidity and rigidity to the cell. These organelles store water, waste products, and substances like amino acids, sugars and proteins. In some unicellular organisms specialized vacuoles also play important roles in expelling excess water and some wastes from the cell. Difference between plant cells and animal cells
Difference between Plant cells and Animal cells.
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Are plants and Animals made of same types of tissues? Plants are stationary, and hence are provided with some tissues made up of dead cells, which provide mechanical strength. They have to withstand unfavourable conditions like strong winds, storms, floods etc. Animals on other hand move around in search of food, mates, shelter. (Scroll down to continue …)
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They consume more energy as compared to plants. Most of the tissues they contain are living.
Cell growth in animas is more uniform.
The structural organisation of organs and organ systems is far more specialized and localised in complex animals than even in very complex plants.
Plant tissues:
Meristematic Tissue: The growth of plants occurs only in certain specific regions. This is because the dividing tissue also known as meristematic tissue is the region where they are present, meristematic tissues are classified as apical, lateral and intercalary. Apical meristem is present at the apical or growing tips of stems and roots. Apical meristem increases the length of the plant. Lateral meristem is present in the radial portion of the stem or root. Lateral meristem increases the girth of the plant.
Intercalary meristem occurs at the base of the leaves or at the internodes. Intercalary meristem increases the length of the internode. Permanent Tissue Old meristematic cells lose the capacity to divide and transform into permanent tissues.
This process of taking up a permanent shape, size, and function is called differentiation. These are cells which have lost their capacity to divide but are specified to provide strength, flexibility and elasticity to the plant. These tissues can be further classified into simple permanent, complex permanent and special tissues. Simple permanent can be categorized into parenchyma, collenchyma and sclerenchyma based on their function. Parenchyma- they are live cells. They are usually loosely packed. This tissue provides support to plants and also stores food. In some situations it contains chlorophyll and performs photosynthesis and then it is called chlorenchyma. Parenchyma which contains large air cavities in aquatic plants is called aerenchyma. The aerenchyma helps in buoyancy. Collenchyma – These are elongated living cells with small intercellular spaces. Their cell walls are made up of cellulose and pectin. Collenchyma occurs in the peripheral regions of stems and leaves to provide mechanical support and flexibility in plants. Sclerenchyma – These are long, dead cells with a deposit of lignin in their cell wall. They have no intercellular spaces. Sclerenchyma occurs around the vascular tissues in stems, in the veins of leaves, and in the hard covering of seeds and nuts. They provide strength to the plant.
Epidermis aids in protection against loss of water, mechanical injury and invasion by parasitic fungi. Since it has a protective role to play, cells of epidermal tissue form a continuous layer without intercellular spaces. Epidermis of the leaf contains small pores called stomata. These are necessary for gases exchange and transpiration. Cork – This is the outer protective tissue which replaces the epidermal cells in older roots and stems. Cork cells are dead and lack intercellular spaces. Their cell walls are thickened by suberin which makes them impermeable to water and gas molecules.
Complex permanent tissue: Complex permanent tissue comprises of conducting tissues called xylem and phloem. Xylem is useful in transport of water and soluble substances. Xylem consists of tracheids, vessels, fibres and xylem parenchyma. Transport of minerals and water is unidirectional in xylem. Phloem is useful in transport of food molecules. Phloem comprises of sieve tubes, sieve cells, companion cells, phloem fibres and phloem parenchyma. Phloem is unlike xylem in that materials can move in both directions in it.
Animal Tissues:
These are the tissues present only in animals. Different types of animal tissues are epithelial tissue, connective tissue, muscle tissue and nervous tissue.
Epithelial Tissue: Epithelial tissue forms a lining all over the body of the organism. It protects the inner lying parts.
It is also secretory in function to secrete sebum and excrete wastes along with sweat.
Sometimes it is absorptive in nature. Epithelial tissues act like a barrier to keep the different body systems separate. These are tightly packed and form a continuous sheet without intercellular spaces.
Squamous epithelium has flat and thin cells with no intercellular spaces.
Squamous epithelium provides is found in the outer layer of the skin, lining the cavities of blood vessels, lung alveoli, lining of oesophagus and the lining of the mouth. Stratified epithelium has epithelial cells lined up one over another. It is found in the epidermis of the skin.
It helps to prevent wear and tear of tissue. Columnar epithelium consists of cylindrical cells. It is found in the lining of the stomach and intestines, and facilitates the movement across the epithelial barrier.
Columnar epithelial tissue with cilia is known as ciliated epithelium. These cilia push the mucus forward into the nasal tract to clear it. Cuboidal epithelium consists of cubical cells. It is found in the lining of the kidney tubules, salivary glands and thyroid glands, where it provides mechanical support. Glandular epithelium consists of modified columnar cells, and is found in the sweat glands and tear glands to produce secretions.
Connective tissue : Connective tissues are fibrous in nature.
They include blood, bone, ligament, cartilage, areolar and adipose tissues.
These help in binding other tissues together. They also provide support to other tissues.
Blood has plasma and blood cells.
The blood cells suspended in the plasma include RBC’s, WBC’s and platelets.
Blood flows within blood vessels, and transports gases, digested food, hormones and waste materials to different parts of the body. Bone cells are embedded in a hard matrix composed of calcium and phosphorus compounds.
Bones anchor the muscles and support the main organs of the body. Two bones can be connected to each other by another type of connective tissue called ligament. Ligaments are tough and elastic. They provide strength and flexibility. Tendons connect muscles to bones and are another type of connective tissue. Tendons are tough and non-elastic, and provide great strength and limited flexibility. Cartilage has widely spaced cells suspended in a matrix of proteins and sugars. It is found in the nose, ears, and the rings of the trachea to give flexibility. Areolar connective tissue is found between the skin and muscles, around blood vessels and nerves and in the bone marrow. It helps in repair of tissues. Adipose tissue contains cells filled with fat globules. It is found below the skin and acts as an insulator.
Muscular Tissue: Muscle tissues consists of elongated cells also called muscle fibres.
This tissue is responsible for movement.
Muscles contain special proteins called contractile proteins which contract and relax to cause movement.
These are elastic in nature they have tensile strength.
These muscles can be voluntary or involuntary in function. Muscular tissues are of three kinds namely striated muscles, unstriated muscles and cardiac muscles. Striated muscle cells are long, cylindrical, unbranched and multinucleate.
These are voluntary muscles.
Smooth muscles or involuntary muscles are found in the iris of the eye, in ureters and in the bronchi of the lungs.
These are also called unstriated muscles. The cells are long with pointed ends and uninucleate.
Hear muscles or cardiac muscles are cylindrical, branched and uninucleate.
Nervous Tissue Nervous tissues are found in the brain, spinal cord and nerves.
Nervous tissue is the tissue which works in coordinating the organs of the body by generating impulses.
It is made up of special cells called as neurons.
Each neuron consists of a cell body, which contains a nucleus, cytoplasm, called cyton, from which long thin hair like parts arise.
Usually each neuron has a single long part, called the axon, and many short branched parts called dendrites.
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Motion: An object is said to be in motion when its position changes with time.
Rest: An object is said to be at rest when its position does not change with respect to a reference point with time.
A specific point with respect to which we describe the location of an object is called a reference point.
The terms Rest and Motion are relative. (Scroll down to continue …)
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Distance and Displacement
Distance:The total length of path covered by an object is said to be the distance travelled by it.
Displacement: Gap between the initial and final positions of an object is said to be its displacement. Or
The length of a line segment that joins the initial and final positions of an object is known as the displacement.
Difference Between Displacement and Displacement
Distance
Displacement
Distance is defined as the total length of the path travelled by an object to go from one point to another.
Displacement is defined as the length of the line segment that joins the initial and final positions of an object.
Since distance has only magnitude and its direction cannot be specified always, it is a scalar quantity.
Since displacement has magnitude and it is specified in a direction from initial position to final position, it is a vector quantity.
Distance can only have positive values.
Displacement can have both positive and negative values.
Distance depends on the length of the path travelled.
Displacement depends only on the initial and final point regardless of the path travelled.
Difference Between Displacement and Displacement
Speed And VelocitySpeed
Speed: The distance travelled by an object in unit time is referred to as speed.
Its S.I unit is m/s.
In general speed refers to average speed.
Average speed: For non-uniform motion, the average speed of an object is obtained by dividing the total distance travelled by an object by the total time taken.
For a uniform motion, the average speed of an object is equal to its instantaneous speed throughout the path.
Velocity
Average Velocity or Velocity : For a uniform motion in a straight path, the average velocity is equal to its instantaneous velocity throughout the path.
Velocity of an object is equal to the instantaneous velocity of an object.
Differences Between Speed and Velocity
SPEED
VELOCITY
It is defined as the rate of change of distance.
It is defined as the rate of change of net displacement.
It is a scalar quantity.
It is a vector quantity.
It can never be negative or zero.
It can be negative,zero or positive.
Speed is velocity without direction.
Velocity is directed speed.
Speed may or may not be equal to velocity.
A body may possess different velocities but the same speed.
Speed never decreases with time. For a moving body,
Velocity can decrease with time. For a moving body , it can be zero.
Speed is never zero.
Velocity can be zero.
Speed in SI is measured in ms-1
Velocity in SI, is measured in ms-1
Differences Between Speed and Velocity
Uniform And Non-Uniform motion
Uniform motion or non accelerated motion: When an object covers equal distances in equal intervals of time, it is said to be in uniform motion. Uniform motion is a non-accelerated motion.
Non-uniform motion or accelerated motion: Motions where objects cover unequal distances in equal intervals of time. Uniform motion is an accelerated motion.
Acceleration
Acceleration: Change in the velocity of an object per unit time.
Graphical representation of motions
(i) Distance-time graph
For a distance-time graph, time is taken on x-axis and distance is taken on the y-axis.
[Note: All independent quantities are taken along the x-axis and dependent quantities are taken along the y-axis.]
(ii)Velocity-time graph
Equation of motion by graphical methods
Derivation Of Equations Of Motion
Equations of motion can be derived by two methods. They are (i) Graphical Method. (ii) Algebraic Method
Derivation of The Equations of Motion By Algebraic Method:
(a) Velocity-time relation:
Derivation of S = ut + ½ at2
(ii) The equation for position-time relation:
Derivation of v2 – u2 = 2as
(iii) Equation for position-velocity relation:
Conclusions From a Distance – Time Graph
Uniform Circular Motion
When a body moves in a circular path with uniform speed, its motion is called uniform circular motion.
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We push or pull to open a door. (Scroll down to continue …)
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Effects of Force
Force can change the shape and size of an object.
Force can move a stationary object.
Force can change the speed of a body.
Force can stop a moving body.
Force can change the direction of a moving object.
Net or Resultant Force:
Resultant Force or Net Force acts on a body if two or more forces act on it at the same time. Resultant Force or Net Force on a body is defined as the net effective force due to the multiple forces acting on it simultaneously.
Based on Net force, Forces are classified into two types as:
(A) Balanced forces
(B) Unbalanced forces
(A) Balanced Forces
• If the resultant of applied forces is equal to zero, the forces are called balanced forces.
• Balanced forces do not cause any change in state of an object.
• Balanced forces can change the shape and size of an object.
For example, when forces are applied from both sides over a balloon, the size and shape of the balloon is changed.
(B) Unbalanced Forces
• If the resultant of applied forces are greater than zero, the forces are called unbalanced forces.
• Unbalanced forces can do the following :
* Move a stationary object
* Increase the speed of a moving object
* Decrease the speed of a moving object
* Stop a moving object
* Change the shape and size of an object
Laws of Motion :
Galileo Galilei :
Galileo Galilei was the first to say that objects move with a constant speed when no forces act on them.
That is, if there is no unbalanced force acting on the object, the object moves forever with a constant speed without changing its direction.
In other words, if an object is moving on a frictionless path and no other force is acting upon it, the object moves forever with a constant speed without changing its direction.
Galileo’s Experiment:
Galileo’s thought experiment considered rolling balls on inclined planes in the absence of friction or other resistant forces.
Galileo arranged two inclined planes opposite to each other as shown.
He rolls down the ball from the first inclined plane to climb the second inclined plane.
Galileo observations:
Galileo observed that:
The ball rolling down the first inclined plane comes to rest after climbing a certain height on the second inclined plane.
The speed acquired by the ball moving down a plane from a height is sufficient to enable it to reach the same height when climbing up another plane at a different inclination .
As the angle decreases, the body should travel a greater distance.
From these observations, Galileo hypothesized as:
if the force acting on the ball is only gravitational force, the height reached by the ball must be equal to the height from which it was rolled.
When the inclinations of the two planes are the same, the distance travelled by the sphere while rolling down is equal to the distance travelled by it while climbing up.
Now, if the inclination of the second plane is decreased slowly, then the sphere needs to travel over longer distances to reach the same height.
If the second plane is made horizontal, then the sphere must travel forever trying to reach the required height.
This is the case when there is no unbalanced force acting on it.
From his experiments Galileo proposed that the body could travel indefinitely far as , contrary to the Aristotelian notion of the natural tendency of an object to remain at rest unless acted upon by an external force.
Therefore, Galileo can be credited with introducing the concept of inertia, later exploited by Newton.
However, in reality, frictional forces bring the sphere to rest after it travels over a finite distance.
After further study, Newton, in his first law of motion, stated that all objects resist a change in their natural state of motion.
This tendency of resisting any change in the natural state of motion is called “inertia”.
Newton’s Laws of Motion:
Newton studied the ideas of Galileo and gave the three laws of motion. These laws are popular as Newton’s laws of motion.
Newton’s First Law of Motion (Law of Inertia):
Any object remains in the state of rest or in the state of uniform motion along a straight line, until it is compelled to change its state by applying an external force.
Newton’s First Law of Motion in Everyday Life:
(a) A person standing in a bus falls backward when the bus starts suddenly.
This happens because the person and bus both are at rest while the bus is not moving, but as the bus starts moving, an external force is acted by the bus on the legs of the person. This external force moves legs along with the bus. But the rest of his body has the tendency to remain in rest known as inertia of rest. Because of this, the person falls backward; if he is not alert.
(b) A person standing in a moving bus falls forward if the driver applies brakes suddenly. This happens because when the bus is moving, the person standing in it is also in motion along with the bus. But when the driver applies brakes the speed of the bus decreases suddenly or the bus comes to a state of rest suddenly, in this condition the legs of the person which are in contact with the bus come to rest while the rest of his body have the tendency to remain in motion. Because this person falls forward if he is not alert.
(c) Before hanging the wet clothes over the laundry line, usually many jerks are given to the clothes to get them dried quickly. Because of jerks, droplets of water from the pores of the cloth fall on the ground and the reduced amount of water
in clothes dries them quickly. This happens because when suddenly clothes are made in motion by giving jerks, the water droplets in it have the tendency to remain in rest and they are separated from clothes and fall on the ground.
(d) When the pile of coins on the carrom-board is hit by a striker, the coin only at the bottom moves away leaving the rest of the pile of coins at the same place. This happens because when the pile is struck with a striker, the coin at
the bottom comes in motion while rest of the coin in the pile has the tendency to remain in the rest and they vertically falls the carrom-board
and remain at the same place.
Momentum
Momentum of an object at state of rest is zero :
Let an object with mass ‘m’ be at rest.
Since, object is at rest, its velocity, v = 0
We know that
Momentum, p is equal to the product of mass, m and velocity, v = 0
⇒ p = m × 0 = 0
Thus, the momentum of an object in the rest i.e., non-moving, is equal to zero.
Unit of momentum :
SI unit of mass = kg
SI unit of velocity = meter per second i.e., m/s
We know that Momentum (p) = m × v
⇒ p = kg × m/s
Or ⇒ p = kg m/s
Therefore, SI unit of momentum = kg m/s
Impulse and Impulsive Force
If a cricketer catches a ball he moves his hand back while catching the ball. He does this to reduce the impact, due to the force of the ball on his hand. An object in motion has momentum. Momentum is defined as the product of mass and velocity of an object.
The momentum of the object at the starting of the time interval is called the initial momentum and the momentum of the object at the end of the time interval is called the final momentum. The rate of change of momentum of an object is directly proportional to the applied force.
Newton’s second law quantifies the force on an object. The magnitude of force is given by the equation,
F = ma, where ‘m’ is the mass of the object and ‘a’ is its acceleration. The CGS unit of force is dyne and the SI unit is newton (N).
A large amount of force acting on an object for a short interval of time is called impulse or impulsive force. Numerically impulse is the product of force and time. Impulse of an object is equal to the change in momentum of the object.
Impulse and Impulsive Force
The momentum of an object is the product of its mass and velocity. The force acting on a body causes a change in its momentum. In fact, according to Newton’s second law of motion, the rate of change in the momentum of a body is equal to the net external force acting on it.
Another useful quantity that we come across is “impulse”. “Impulse” is the product of the net external force acting on a body and the time for which the force is acted.
If a force “F” acts on a body for “t” seconds, then Impulse I = Ft.
In fact, this is also equal to the change in the momentum of the body. It means that due to the application of force, if the momentum of a body changes from “P” to “P ‘ ”, then impulse,I = P ‘ – P.
For the same change in momentum, a small force can be made to act for a long period of time, or a large force can be made to act for a short period of time. A fielder in a cricket match uses the first method while catching the ball. He pulls his hand down along with the ball to decrease the impact of the ball on his hands.
In a cricket match, when a batsman hits a ball for a six, he applies a large force on the ball for a very short duration. Such large forces acting for a short time and producing a definite change in momentum are called “impulsive forces”.
Newton’s Second Law of Motion
Newton’s Second Law of Motion states that, the rate of change in momentum of an object is proportional to applied unbalanced force in the direction of force.
Mathematical expression:
State and derive newton’s second law of Motion
Statement: Newton’s second law of motion states that the rate of change of momentum of an object is Proportional to the applied unbalanced force in the direction of force.
Derivation of Newton’s second law of motion:
Suppose an object of mass, m is moving along a straight line with an initial velocity, u.
It is uniformly accelerated to velocity, v in time, t by the application of a constant force, F throughout the time t.
⇒Initial momentum of the object, p1 = mu
⇒Final momentum, p2 = mv
⇒Change in momentum = p2 – p1
⇒The change in momentum = mv – mu
⇒The change in momentum = m × (v – u)
⇒The rate of change of momentum = m(v -u)t
⇒ m (v -u)t
According to Newton’s Second Law of Motion,
Applied force α Rate of change in motion
⇒ F m (v -u)t
F=km (v -u)t = kma —————————- (i)
Here, k is a constant of proportionality and
(v -u)t is the rate of change of velocity, which equals acceleration, a.
The SI units of mass and acceleration are kg and m s-2 respectively.
The unit of force is so chosen that the value of the constant, K becomes one For this.
One unit of force is defined as the amount that produces acceleration
of 1 m s-2 in an object of 1 kg mass.
That is,
1 unit of force = k × (1 kg) × (1 m s-2).
Thus, the value of k becomes 1. From Eq. (iii)
F = ma ————————————-
The unit of force is kg m s-2 or newton, with the symbol N.
Newton’s Third Law of Motion
To every action there is an equal and opposite reaction.
Applications:
(i) Walking is enabled by 3rd law.
(ii) A boat moves back when we deboard it.
(iii) A gun recoils.
Rowing of a boat.
Law of Conservation of Momentum
Law of conservation of momentum states that, if two or more bodies collide, the sum of the initial momentum is equal to the sum of the final momentum.
Or
Law of conservation of momentum states that the sum (total) of the individual momentums of the colliding bodies just before the collision is equal to the sum (total) of the individual momentums of the colliding bodies after the collision
Derivation of Law of Conservation of Momentum From Newton’s Third Law of Motion.
Answer:
For a system of bodies ( two or more bodies ), the total vector sum of momenta of all the bodies due to the mutual action and reaction remain unchanged as long as no external force is acted on the system.
Consider two bodies A and B of the masses m1, m2 moving with the initial velocities u1, u2 respectively.
For a system, let, these masses collide and their velocities after collision are v1, v2 respectively.
If ‘A’ applies a F on B for a time, t;
‘B’ applies a force –F on A for time t [according to Newton’s third law of motion].
Then,
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Therefore, the sum of momentum before impact is equal to the sum of the momenta after the impact represents the law of conservation of momentum.
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.
of n values x1, x2, x3, …… xn is given by
Term
term.
