Force and Motion

Every object in the universe is either at rest or in motion. To decide which, we compare it with a reference point — a point, place or object that we treat as fixed. A force is needed to make a body at rest start moving, or to stop a moving body. In this unit we measure motion (speed, velocity, relative velocity, average velocity and acceleration), read motion graphs, study forces and Newton's laws, learn how a lever makes work easy, and study pressure in solids, liquids and gases, including upthrust and floating.

6.1 Motion: distance, displacement, speed, velocity and acceleration

An object is in motion if it changes its position with respect to a reference point or its surroundings (a flying aeroplane, a rolling ball). It is at rest if it does not change its position (houses, trees, a parked bus). Rest and motion are relative: a passenger sitting in a moving bus is at rest with respect to the bus but in motion with respect to the trees outside.

Uniform motion: equal distances in equal intervals of time. Non-uniform motion: unequal distances in equal intervals of time.

Distance	Displacement
total length of the path travelled	shortest distance between the start and end points in a particular direction
has magnitude only (scalar)	has magnitude and direction (vector)
never zero once a body has moved	can be zero (a runner who returns to the start)

6.1.1 Speed and velocity

A speed of 20 m/s means 20 m is covered every second. The speed of sound in air is about 332 m/s. To change km/h into m/s, multiply by \frac{1000}{3600}=\frac{5}{18}.

6.1.2 Relative velocity

The relative velocity of a body is its velocity with respect to another body (or reference point).

Case	Relative velocity of A with respect to B	What passengers feel
same direction, same speed	v_{AB}=v_A-v_B=0	the other bus seems to be at rest
same direction, different speeds	v_{AB}=v_A-v_B	the other bus seems slow
opposite directions	v_{AB}=v_A+v_B	the other bus seems very fast

6.1.3 Average velocity

In non-uniform motion the velocity keeps changing, so we use the average velocity: the total displacement divided by the total time,

If the velocity changes steadily from u (initial) to v (final), \bar{v}=\frac{u+v}{2}.

6.1.4 Acceleration and retardation

A body starting from rest has u=0; a body that stops has v=0.

Acceleration	Retardation
velocity increases (v>u)	velocity decreases (v<u)
positive value	negative acceleration
e.g. a bus starting from the bus park	e.g. a bus braking at a stop

6.2 Distance-time and velocity-time graphs

On a distance-time graph	Meaning
horizontal line	body at rest
straight sloping line	uniform speed; steeper line = greater speed; slope = speed
curve becoming steeper	accelerating

On a velocity-time graph	Meaning
horizontal line	uniform velocity; area under the line = distance
line sloping up	acceleration; slope = acceleration
line sloping down	retardation

6.3 Force and its effects; balanced and unbalanced forces

A force is a push or pull that changes or tends to change the state of rest or motion of a body, or its shape. Its SI unit is the newton (N). On the earth, the weight of a 1 kg mass is about 10 N (more exactly 9.8 N).

Effects of force: it can (1) move a body at rest, (2) stop a moving body, (3) change its speed, (4) change its direction, (5) change its shape or size.

Balanced forces are forces on a body whose resultant is zero: the body stays at rest or keeps moving uniformly (e.g. equal teams in a tug of war, a book on a table). Unbalanced forces have a non-zero resultant and change the motion (one team pulls harder; a kicked football).

6.4 Newton's laws of motion in everyday examples

First law (law of inertia). A body at rest remains at rest, and a body in uniform motion keeps moving in a straight line, unless an external unbalanced force acts on it. The property of a body to resist a change in its state is inertia; more mass means more inertia. Examples: passengers fall forward when a bus stops suddenly and backward when it starts suddenly; dust leaves a carpet when it is beaten.

Second law. The acceleration of a body is directly proportional to the applied force and inversely proportional to its mass: F=ma. Example: the same push gives an empty cart more acceleration than a loaded one. A force of 1 N gives a 1 kg mass an acceleration of 1 m/s².

Third law. To every action there is an equal and opposite reaction. Examples: a swimmer pushes water back and moves forward; a rocket pushes hot gases down and goes up; a gun recoils when fired.

6.5 Lever: classes, principle of moments and mechanical advantage

Archimedes is said to have claimed: "Give me a lever and a place to stand, and I can move the world."

6.5.1 Classes of lever

Class	Arrangement	Examples
First	fulcrum between load and effort	dhiki, scissors, beam balance, crowbar, see-saw
Second	load between fulcrum and effort	nutcracker, lemon squeezer, wheelbarrow
Third	effort between fulcrum and load	fire tongs, shovel, hammer

6.5.2 Principle of lever (principle of moments)

Hang masses on both sides of a metre scale balanced at its middle (Activity 6.3). The scale balances whenever

So the longer the effort distance, the smaller the effort needed.

6.5.3 Mechanical advantage, velocity ratio and efficiency

• Mechanical advantage (MA) =\frac{\text{load}}{\text{effort}}=\frac{L}{E}. MA = 4 means the machine lifts a load 4 times the effort. It has no unit and is reduced by friction.
• Velocity ratio (VR) =\frac{\text{distance moved by effort}}{\text{distance moved by load}}=\frac{E.D.}{L.D.}. It has no unit and is not affected by friction.
• Input work = effort × effort distance; output work = load × load distance.
• Efficiency \eta=\frac{\text{output work}}{\text{input work}}\times100\%=\frac{MA}{VR}\times100\%.

Because of friction, some input work is wasted, so MA < VR and the efficiency of a real machine is always less than 100 %. A machine with 100 % efficiency is an imaginary ideal machine.

6.6 Pressure in solids, liquids and gases; atmospheric pressure

A pen pressed on the wrist with its pointed end hurts more than with its blunt end; a brick standing on its narrow face sinks deeper into foam. The same force has a bigger effect on a smaller area.

Increasing pressure: sharp ploughshare, knife, sickle and nail tips (small area). Reducing pressure: wide foundations of houses, broad back wheels of tractors, double tyres on buses and trucks, broad feet of camels, wide straps on school bags.

6.6.1 Pressure in liquids

Water squirts farthest from the lowest hole of a bottle (Activity 6.6): liquid pressure increases with depth. A liquid presses in all directions.

Applications: dams and water tanks are thicker at the bottom; deep-sea divers wear special suits; water tanks are placed high on the roof; mercury bottles have thick bottoms. A bucket fills faster on the ground floor than upstairs because the depth of water below the tank — and so the pressure — is greater there. Blood pressure is greater in the feet than in the head for the same reason.

6.6.2 Air pressure, compressed air and atmospheric pressure

Air can be compressed into tyres, balls and cylinders; compressed air has more pressure than the atmosphere and heats up when compressed. It drives air hammers, drills and wrenches, and works the air brakes of trains, buses and trucks. Its pressure is measured with a pressure gauge.

The earth is covered by air, which has weight. The pressure exerted by the weight of the atmosphere on unit area of the earth's surface is atmospheric pressure. It is greatest at sea level, about 760 mmHg ≈ 1.01 × 10⁵ Pa, and decreases with height. It is measured with a barometer. Aeroplanes keep the air pressure inside the cabin suitable for breathing.

Activities. A glass full of water covered with cardboard can be turned upside down without the water falling — the air pressure from below holds the card. A can heated with a little water, closed tightly and cooled gets crushed — the steam inside condenses, the pressure inside falls, and the outside air pressure crushes the can.

Uses of atmospheric pressure: drinking with a straw, filling a fountain pen, filling a syringe, pumping air into a bicycle tyre. When the piston is pulled back, the pressure inside falls and the outside air pushes the liquid in.

A manometer (U-tube with coloured liquid) measures the pressure of a gas or liquid: the extra pressure pushes the liquid down in one limb and up in the other; the difference in levels h measures the pressure.

6.7 Archimedes' principle, upthrust, floatation and density

A body in a liquid feels an upward push called upthrust (buoyant force), because the pressure on its lower surface (deeper) is greater than on its upper surface. A stone feels lighter in water for this reason.

Density is mass per unit volume, d=\frac{m}{V} (kg/m³ or g/cm³). Water has a density of 1000 kg/m³ (1 g/cm³).

Condition	Result	Example
weight > upthrust (body denser than liquid)	sinks	stone, iron nail
weight = upthrust	floats fully immersed	fish, submarine at a fixed depth
weight < maximum upthrust (body less dense)	floats partly out	wood, ice, a ship

A floating body displaces liquid equal to its own weight (law of floatation). An iron ship floats because its hollow shape displaces a large volume of water. A submarine sinks or rises by filling or emptying its tanks with water.

Chapter summary

• Rest and motion are relative to a reference point. Speed =\frac{d}{t}, velocity =\frac{s}{t} (m/s); velocity has direction.
• Relative velocity: v_A-v_B (same direction), v_A+v_B (opposite).
• Average velocity = total displacement ÷ total time =\frac{u+v}{2} for steady change.
• Acceleration a=\frac{v-u}{t} (m/s²); negative acceleration is retardation.
• Slope of d–t graph = speed; slope of v–t graph = acceleration; area under v–t = distance.
• Newton: inertia; F=ma; action = reaction.
• Lever: E\times E.D.=L\times L.D.; MA=\frac{L}{E}, VR=\frac{E.D.}{L.D.}, \eta=\frac{MA}{VR}\times100\% (< 100 % because of friction).
• Pressure P=\frac{F}{A} (Pa); liquid pressure P=hdg; atmospheric pressure at sea level = 760 mmHg; barometer, manometer, pressure gauge.
• Upthrust = weight of fluid displaced (Archimedes); a body floats if it is less dense than the liquid.

Practice questions

Group A — Multiple choice (1 mark each) (Textbook exercises.)

1. The condition of a body that does not change its position with respect to its surroundings is [1]amotionbvelocitycrestdspeed
2. The relative velocity of two bodies moving in opposite directions is [1]av_A/v_Bbv_A-v_Bcv_A\times v_Bdv_A+v_B
3. The shortest distance between two points in a particular direction is [1]aspeedbdisplacementcaverage velocitydrelative velocity
4. Average velocity is calculated for [1]anon-uniform motionbuniform motionczero accelerationda body at rest
5. The SI unit of acceleration is [1]am/sbm/s²cN m²/kg²dm²
6. Which statement is true? [1]auniform motion gives negative accelerationbnon-uniform motion has no accelerationcpositive acceleration is retardationdthe rate of change of velocity is acceleration
7. Which is a second-class lever? [1]ashovelbdhikicnutcrackerdhammer
8. The principle of a lever in balance is [1]aE × E.D. = L × L.D.bE = LcE.D. = L.D.dMA = efficiency
9. The work done by the effort is called [1]ainput workboutput workcfrictiondgravitational work
10. Which is not affected by friction? [1]amechanical advantagebvelocity ratiocefficiencydinput work
11. Which statement is true? [1]athe product of load and effort is VRbthe effort moves less than the loadcMA and VR have unitsdefficiency is output work ÷ input work in percent
12. The SI unit of pressure is [1]aN mbN/m²cm/s²dN m²/kg²
13. Which statement is true? [1]aless area, more pressurebarea does not affect pressurecpressure rises with areadpressure is the same for any force
14. Atmospheric pressure at sea level is [1]a760 mmHgb700 mmHgc1000 mmHgd750 mmHg
15. The pressure of water at a depth of 60 m (g=9.8 m/s², d=1000 kg/m³) is [1]a588 Pab5880 Pac58 800 Pad588 000 Pa
16. Atmospheric pressure is measured by a [1]athermometerbbarometercmanometerdlactometer
17. The pressure of compressed air is measured by a [1]apressure gaugebrain gaugeclactometerdthermometer
18. The air brake works on [1]aatmospheric pressurebcompressed aircliquid pressuredupthrust

Answers to Group A

1. (c).

2. (d).

3. (b).

4. (a).

5. (b).

6. (d).

7. (c) — the load (nut) is between hinge and hand.

8. (a).

9. (a).

10. (b).

11. (d).

12. (b).

13. (a).

14. (a).

15. (d) — 60\times1000\times9.8=588\,000 Pa.

16. (b).

17. (a).

18. (b).

Group B — Short answer (2 marks each)

1. Differentiate:(a) distance and displacement(b) average velocity and relative velocity(c) acceleration and retardation. [3]
2. Define reference point, relative velocity and retardation. [3]
3. Differentiate:(a) mechanical advantage and velocity ratio(b) input work and output work. [2]
4. Give reasons:(a) The efficiency of a lever is never 100 %.(b) A wheelbarrow is easier to push when the load is moved towards the wheel.(c) MA is always less than VR. [3]
5. Which class of lever is a pair of scissors? What can a lever with MA more than 1 do? Write other advantages of a lever besides making work easy. [2]
6. Give reasons:(a) Football boots have studs.(b) A ploughshare is made sharp.(c) The bottom of a dam is wider than its top.(d) A bucket fills faster on the ground floor than upstairs. [4]
7. How can we show that a liquid exerts pressure? Write two applications of liquid pressure. [2]
8. What is atmospheric pressure? Write two uses of it. [2]

Answers to Group B

1. (a) See the table in 6.1. (b) Average velocity is the total displacement ÷ total time of one body in non-uniform motion; relative velocity is the velocity of one body with respect to another. (c) See the table in 6.1.4.

2. A reference point is a fixed point, line or place with which the position of a body is compared. Relative velocity is the velocity of a body with respect to another body. Retardation is the rate of decrease of velocity (negative acceleration).

3. (a) MA = load ÷ effort, affected by friction, always less than VR; VR = effort distance ÷ load distance, not affected by friction. (b) Input work = effort × effort distance (work put into the machine); output work = load × load distance (useful work done by it); output is less than input.

4. (a) Some input work is always lost against friction, so output work is less than input work. (b) The load distance becomes shorter, so by E\times E.D.=L\times L.D. less effort is needed. (c) Friction reduces MA but does not change VR.

5. First class (the pivot is between blade and handle). A lever with MA > 1 lifts a load larger than the effort (multiplies force). Other advantages: changes the direction of the force, increases speed or distance of movement (third class), and allows work at a convenient place (tongs keep hands away from fire).

6. (a) The small area of the studs gives large pressure, so they grip the ground. (b) A sharp edge has small area, so a small force gives large pressure to cut the soil. (c) Liquid pressure increases with depth, so the bottom must be strongest. (d) The depth of water below the tank (h) is greater at the ground floor, so pressure hdg and the flow are greater.

7. Make holes at different heights in a bottle full of water; water comes out from all holes (pressure acts sideways) and spurts farthest from the lowest hole (pressure increases with depth). Applications: dams thick at the bottom; tanks kept high; divers' suits.

8. The pressure exerted by the weight of air on unit area of the earth's surface (760 mmHg at sea level). Uses: drinking with a straw, filling a syringe or fountain pen, pumping tyres.

Group C — Numerical and long answer (4 marks each)

1. A bus covers 4.2 km in 7 minutes. Find its average velocity. [2]
2. A car starts from rest and reaches 30 m/s in 12 s. Find its acceleration. [2]
3. A car starts from rest with an acceleration of 5 m/s² for 6 s. Find its final velocity and average velocity. [3]
4. Brakes give a car a retardation of 2 m/s² and it stops in 4 s. Find its initial velocity. [2]
5. Two motorcycles move at 50 km/h and 60 km/h. Find their relative velocity in the same and opposite directions. [2]
6. An effort of 75 N lifts a load of 300 N placed 25 cm from the fulcrum. Find the effort distance. [2]
7. Sarad (550 N) sits 1.5 m from the centre of a see-saw. How far from the centre must Nirmal (300 N) sit to balance him? [2]
8. An effort of 300 N lifts a load of 900 N with a 2 m first-class lever; the load is 50 cm from the fulcrum and the effort is at the other end. Find MA, VR and efficiency. [3]
9. Prove that P=hdg. [4]

Answers to Group C

1. s=4.2 km =4200 m, t=7 min =420 s. \bar{v}=\frac{4200}{420}=10 m/s.

2. a=\frac{v-u}{t}=\frac{30-0}{12}=2.5 m/s².

3. v=u+at=0+5\times6=30 m/s. Average velocity =\frac{u+v}{2}=\frac{0+30}{2}=15 m/s. (The answer printed in the textbook, 15 m/s and 7.5 m/s, would be correct only for an acceleration of 2.5 m/s²; with 5 m/s² the correct values are 30 m/s and 15 m/s.)

4. Retardation =\frac{u-v}{t}, so 2=\frac{u-0}{4} and u=8 m/s.

5. Same direction: 60-50=10 km/h. Opposite directions: 60+50=110 km/h.

6. E\times E.D.=L\times L.D.: 75\times E.D.=300\times25, so E.D.=\frac{7500}{75}=100 cm =1 m.

7. 300\times d=550\times1.5, so d=\frac{825}{300}=2.75 m.

8. L.D.=0.5 m, E.D.=2-0.5=1.5 m. MA=\frac{900}{300}=3; VR=\frac{1.5}{0.5}=3; \eta=\frac{3}{3}\times100\%=100\% (an ideal lever, friction neglected).

9. See the derivation box in 6.6.1: V=Ah, m=dAh, F=mg=Ahdg, P=\frac{F}{A}=hdg.