CBSE Class 7 Science Advanced Physics Kinematics Notes Set 01

Kinematics

What This Chapter Covers

Kinematics describes how bodies move without asking what makes them move. Here we meet the kinds of motion, the quantities used to describe motion (distance, displacement, speed, velocity and acceleration), the way motion is drawn as a distance-time graph, and the simple pendulum, which is used to keep time.

Rest and Motion

A body is in motion if its position keeps changing with respect to the objects around it, and it is at rest if its position does not change. A running athlete, a flowing river and a moving car are in motion; a lamp post, a table and a book lying on it are at rest.

Rest and motion are relative. Passengers sitting in a moving bus are at rest with respect to one another and to the bus, but they are moving with respect to a person standing on the road. So we must always say with respect to what a body is at rest or in motion.

Kinds of Motion

1. Translatory motion. The whole body moves from one place to another. It is of two types.
(a) Rectilinear motion: the path is a straight line, for example soldiers marching on a straight road, a car on a straight road or a carrom coin sliding straight.
(b) Curvilinear motion: the path is curved, for example a car taking a turn, a bus on a flyover or a football kicked into the air.

2. Rotatory motion. The body turns about a fixed axis, for example the blades of a ceiling fan, a spinning top, a turbine or the earth moving around the sun. A rolling bicycle wheel shows rotatory and translatory motion together.

3. Oscillatory motion. The body moves to and fro about a fixed position, for example a child on a swing, a clock pendulum, the needle of a sewing machine or the piston of an engine.

Vibratory motion is a very fast to-and-fro motion. A plucked guitar or violin string and the skin of a drum vibrate, and the vibrating body changes slightly in shape or size.

Periodic motion repeats itself after equal intervals of time. All oscillatory motions are periodic, and some rotatory motions are too (the earth spinning on its axis, the earth going round the sun). Non-periodic motion repeats, if at all, at unequal intervals, for example a ceiling fan slowing down after it is switched off, or a drum rolling down a slope.

Distance and Displacement

Both are lengths and are measured in the same units (cm in the CGS system, m in the SI system), but they mean different things.

Distance is the total length of the path actually travelled. Displacement is the straight line from the starting point to the end point, drawn with a direction from start to end. It is a vector quantity: it has magnitude and direction.

Distance depends on the path chosen; displacement depends only on where the body starts and where it ends.

A particle goes from A to B (6 m), then B to C (2 m), then C to D (5 m). Distance = 6 + 2 + 5 = 13 m. The displacement is the arrow \(\overrightarrow{AD}\), whose size is the length of the straight segment AD.

Why direction matters: if a policeman is told only that a thief is 5 km away, he cannot find him, because the distance alone does not say in which direction to go.

Points to Note

1. Distance is always greater than or equal to the magnitude of displacement.
2. They are equal only when the body moves along a straight line without changing direction.
3. \(\overrightarrow{AB}\) and \(\overrightarrow{BA}\) are not the same, because the start and end points are swapped.
4. The displacement is measured in cm (CGS) or m (SI).

Worked Example: A Horse Tied to a Pole

Illustration 1. A horse is tied to a pole by a 5 m rope and moves along a circle of radius 5 m. Find the distance and the displacement for (i) half a revolution, (ii) one full revolution and (iii) three-fourths of a revolution.

Solution.

(i) Half a revolution

Distance = half the circumference = \(\pi r = 5\pi\) m. Displacement = the diameter \(H_1H_2\) = 10 m, directed from H₁ to H₂.

(ii) One full revolution

Distance = the whole circumference = \(2\pi r = 10\pi\) m. The horse returns to the starting point, so the displacement = 0.

(iii) Three-fourths of a revolution

Distance = \(\frac{3}{4} \times 2\pi \times 5 = 7.5\pi\) m. The start H₁ and the end H₂ are joined by the hypotenuse of a right triangle with two sides of 5 m, so the displacement = \(\sqrt{5^2 + 5^2} = 5\sqrt{2}\) m, directed from H₁ to H₂.

Slow and Fast; Speed

A cheetah is faster than a snail and an aeroplane is faster than a car. When objects move for the same time, the faster one covers more distance; when they cover the same distance, the faster one needs less time. This fastness is measured by speed, the distance covered in one unit of time.

A bus rarely keeps a steady speed: it starts from rest, picks up speed, runs steadily for a while and slows down at stops. So we use the average speed:

\[ \text{average speed} = \frac{\text{total distance covered}}{\text{total time taken}} \]

Wherever this book says speed without any other word, average speed is meant. Motion at constant speed along a straight line is uniform motion; motion at changing speed is non-uniform motion.

Uniform and Non-uniform Speed

Uniform speed: the body covers equal distances in equal intervals of time, however small the intervals are. Example: a car that covers 10 m in every second.

Non-uniform speed: the body covers unequal distances in equal intervals of time, or equal distances in unequal intervals. Example: a bus after the brakes are applied.

Measuring Time

Early people judged the time of day from the length and direction of shadows. A day, a month and a year come from the periodic motion of the earth (around its axis and around the sun), and a lunar month comes from the moon going round the earth. Every clock uses some periodic motion; old wall clocks used a swinging pendulum, while modern quartz clocks use an electric circuit and are more accurate than pendulum or winding clocks.

A simple pendulum is a small heavy metal ball (the bob) hung from a fixed support by a thread. The bob at rest hangs at the mean position. Pull it aside and let go, and it swings to and fro.

Figure 2.1 A simple pendulum

Figure 2.2 Positions of the bob: O is the mean position, A and B are the extreme positions

In Fig. 2.2 the bob going O → B → O → A → O makes one oscillation; so does A → B → A. A and B are the extreme positions. The time for one oscillation is the time period. It depends on the length of the pendulum but not on the mass, size or shape of the bob (the bob must not be extremely light, like paper).

To find the time period accurately, time about 20 oscillations with a stop watch and divide the time by 20.

Units of Time and Speed

The basic unit of time is the second (s); minute and hour are other units. Because speed is distance divided by time, its SI unit is m s⁻¹, and m min⁻¹ and km h⁻¹ are also used. The unit of time we choose depends on the situation: years for a person's age, but minutes and hours for an aeroplane journey.

Useful conversion: \(1\ \text{km h}^{-1} = \frac{5}{18}\ \text{m s}^{-1}\) and \(1\ \text{m s}^{-1} = \frac{18}{5}\ \text{km h}^{-1}\).

Finding Speed, Distance and Time

Speed of a rolling ball: draw a chalk line, start a stop clock when the ball crosses the line, stop it when the ball comes to rest, and measure the distance from the line to the stopping point. Distance ÷ time gives the speed.

\[ \text{speed} = \frac{\text{distance}}{\text{time}}, \qquad \text{distance} = \text{speed} \times \text{time}, \qquad \text{time} = \frac{\text{distance}}{\text{speed}} \]

A speedometer in a vehicle shows its speed (in km h⁻¹); an odometer shows the distance it has travelled.

Distance-Time Graph

A graph shows information in picture form. For a moving body we draw the distance-time graph, which is a line graph. Take graph paper and draw two perpendicular lines XOX′ and YOY′ meeting at the origin O. Time is marked along the X-axis and distance along the Y-axis.

Figure 2.3 The axes on graph paper

Example: the motion of a motor bike is given in this table.

Steps to draw the graph

1. Draw the perpendicular axes OX and OY meeting at O.
2. Put time on the X-axis and distance on the Y-axis.
3. Choose a scale. Here 1 s = 1 cm on the X-axis and 5 m = 1 cm on the Y-axis.
4. Mark the values on the axes using this scale (Fig. 2.4).
5. Treat each row of the table as a pair (time, distance): (0, 0), (1, 5), (2, 10), (3, 15), (4, 20), (5, 25), (6, 30).
6. To plot (1, 5): go up from 1 s on the X-axis along a line parallel to the Y-axis, go across from 5 m on the Y-axis along a line parallel to the X-axis, and mark the point where they meet. Plot all the other pairs the same way (Fig. 2.5).
7. Join the points. They lie on a straight line, which is the distance-time graph of the bike.

Figure 2.4 Axes with the scale marked

Figure 2.5 Points plotted and joined

A body moving with constant speed gives a straight-line graph. If the speed is not constant, the graph is not a straight line. The graph also lets us read the distance at any instant, which a table cannot do.

Choosing a Scale

Take a bus journey recorded as follows.

With 1 km = 1 cm the Y-axis would be 80 cm long, which will not fit on a sheet. With 10 km = 1 cm it is only 8 cm, which fits but uses just a small part of the paper. A good scale depends on three things:

1. the gap between the largest and the smallest values;
2. the in-between values, which should be easy to plot with the chosen scale;
3. how much of the graph paper the graph fills.

For a 20 cm × 25 cm sheet, a suitable scale for this table is 5 km = 1 cm for distance and 6 min = 1 cm for time.

Velocity

Speed does not say in which direction a body moves. Velocity is speed in a stated direction, so it has a magnitude and a direction. A car moving at 10 m s⁻¹ towards the east has a velocity of 10 m s⁻¹, east.

1. Uniform velocity: speed and direction both stay the same. Example: an aeroplane flying east at 500 km h⁻¹.

2. Variable velocity: the speed, the direction or both change. Examples: a car on a straight road whose speed keeps changing, and a car taking a turn (the direction changes).

3. Average velocity: for a body with variable velocity,

\[ \text{average velocity} = \frac{\text{total displacement}}{\text{total time taken}} \]

Acceleration

Acceleration is the rate at which velocity changes, that is, the change in velocity in one unit of time. Its SI unit is m s⁻² (cm s⁻² and km h⁻² are also used).

Example: a car's velocity goes from 5 m s⁻¹ to 25 m s⁻¹ in 4 s.

\[ a = \frac{25 - 5}{4} = 5\ \text{m s}^{-2} \]

If the velocity changes from the initial value u to the final value v in time t, then

\[ a = \frac{v - u}{t}, \qquad v = u + at \]

A body let go from a height falls because of gravity, and its velocity keeps increasing; this acceleration is called the acceleration due to gravity (g), whose average value is 9.8 m s⁻². When the velocity decreases, the acceleration is negative and is called deceleration (retardation).

More About the Simple Pendulum

Amplitude: the largest displacement of the bob from the mean position. Length (ℓ): the distance from the point of suspension to the centre of the bob. Frequency (n): the number of oscillations in one second; its SI unit is the hertz (Hz), and n = 1/T. A seconds pendulum has a time period of 2 s.

The time period is given by

\[ T = 2\pi\sqrt{\frac{\ell}{g}} \]

At one place g is fixed, so \(T \propto \sqrt{\ell}\): a longer pendulum has a longer time period and a shorter one has a shorter time period.

Experiment: Time Period of a Simple Pendulum

Figure 2.6 Experimental set-up

1. Find the bob's radius R: hold the bob between two wooden blocks, measure the gap with a scale (this is the diameter) and halve it.
2. Pass the thread through a vertically cut cork and clamp the cork firmly. For a pendulum of length 70 cm, the thread must be (70 − R) cm long, because the length is measured up to the centre of the bob.
3. Pull the bob aside so that the thread makes about 5° with the vertical and release it.
4. Start the stop watch when the bob is at the mean (or an extreme) position and time 20 oscillations.
5. Divide this time by 20 to get T.
6. Repeat with lengths such as 80 cm, 90 cm and 100 cm. You will find that T rises as the length rises.

Points to Remember

1. Rest and motion depend on the surroundings chosen; they are relative.
2. The three kinds of motion are translatory, rotatory and oscillatory.
3. Speed = distance ÷ time; average speed = total distance ÷ total time.
4. Uniform speed means equal distances in equal times; otherwise the speed is non-uniform.
5. Velocity is speed with direction; average velocity = total displacement ÷ total time.
6. Acceleration = change in velocity ÷ time, and v = u + at.
7. For a simple pendulum, \(T = 2\pi\sqrt{\ell/g}\); it depends on the length only.