Force and Motion

Picture a scene. A house and a tree stay where they are. A boy kicks a football, a bird flies, a girl runs, a ball rolls, a bee buzzes from flower to flower, the hands of a clock turn, the pendulum of a tall clock swings, and the Earth goes round the Sun. A man leans on a table: it stays put until he pushes or pulls it.

Everything in the universe is either at rest or in motion. But to say which, you must compare it with something: a point, a place, or the things around it. That something is called the reference point (a whole set of surroundings used this way is a frame of reference, or reference frame). A flying bird, a moving bus and a walking person are all in motion compared with the ground, the trees and the houses around them.

A table at rest only starts moving when it is pushed or pulled. A force is needed to change the state of an object from rest to motion, or from motion to rest.

An object is in motion if it changes its position compared with a reference point or its surroundings. A flying airplane, a moving bus, a walking person and a rolling ball are all in motion compared with the ground.

An object is at rest if it does not change its position compared with a reference point or its surroundings. Houses, trees and buses in a parking area are at rest compared with the ground.

Now sit in a moving bus. Your friend in the next seat stays next to you the whole time: compared with the bus, she is at rest. But compared with the shops and trees outside, she is changing position all the time, so she is in motion. The same person is at rest and in motion at the same time, depending on the reference point. That is why we say rest and motion are relative.

Uniform and non-uniform motion

Watch a bus on the Arniko Highway. On a long, straight, empty stretch it might cover 15 meters in every second, second after second. In town, it speeds up, slows down for a cow, stops for passengers, and starts again.

• An object in uniform motion covers equal distances in equal intervals of time, however short those intervals are.
• An object in non-uniform motion covers unequal distances in equal intervals of time.

Almost all real motion is non-uniform: a bus, a motorcycle, a running dog, a falling mango. Uniform motion is what a vehicle does for a while on cruise control on a straight road.

Speed

Which runs faster, a deer or a cheetah? The cheetah: for a few seconds it can reach about 100 km/h. Some things move fast and some slowly. To compare them we measure the distance covered in one unit of time. That is the speed.

$$\text{speed} = \frac{\text{distance travelled } (d)}{\text{time taken } (t)}$$

The SI unit of speed is the meter per second (m/s). On the road we usually use kilometers per hour (km/h).

• A car moving at 20 m/s covers 20 meters every second.
• Sound in cold air at 0 °C travels at about 332 m/s: 332 meters every second. (On a warm day it is a little faster, about 343 m/s.)

Speed has a size (a magnitude) but no direction: “20 m/s” says how fast, not where to. A quantity like this, with magnitude only, is called a scalar quantity.

Distance and displacement

You walk from your house to the shop by the lanes, turning left and right: that’s 500 m of walking. But the shop is only 300 m from your house in a straight line, towards the north.

• Distance is the length of the whole path traveled, whatever its shape (500 m). It has magnitude only: a scalar.
• Displacement is the shortest, straight-line distance from the start to the end, in a particular direction (300 m north). It has magnitude and direction: a vector quantity.

Velocity

Velocity is the distance traveled in one unit of time in a particular direction. Put another way, it is the rate of change of displacement:

$$\text{velocity } (v) = \frac{\text{displacement, the distance travelled in a particular direction } (s)}{\text{time taken } (t)}$$

Its SI unit is also the meter per second (m/s). But a velocity is only complete with its direction: “20 m/s towards the east”. A quantity that needs both magnitude and direction is a vector quantity. So speed is a scalar, and velocity is a vector.

In daily life people use “speed” and “velocity” to mean the same thing. In science they are different quantities. A bus going round a bend at a steady 30 km/h has a constant speed, but its velocity is changing, because its direction is changing.

Think about these before you read on.

Any fixed point, line or place from which we compare the motion of an object is the reference point (or reference line). In Figure 6.2, the line MN is the reference line and P is the reference point. Car A moves away from it at 10 m/s towards the west, and car B at 15 m/s towards the east.

So far, every velocity was measured from the ground. But you can also measure it from another moving object. The velocity of a body compared with a reference point or with another body is called its relative velocity.

We write $v_{AB}$ for “the velocity of A relative to B”: how A seems to move to someone sitting in B. It is found by subtracting B’s velocity from A’s:

$$v_{AB} = v_A - v_B$$

A direction must be chosen as positive. If A and B go the same way, both velocities are positive. If they go opposite ways, one of them is negative, and the subtraction becomes an addition. Let’s see the three cases.

(a) Same direction, same speed

Buses A and B leave the reference line PQ side by side on a two-lane road, both at 22 m/s. After 1 second each has gone 22 m, so they are still side by side (Figure 6.3). Bus A has not moved at all compared with bus B, and B has not moved compared with A. The passengers in both buses feel that they are at rest, as long as they look only at the other bus.

$$v_{AB} = v_A - v_B = 22\ \text{m/s} - 22\ \text{m/s} = 0$$

(b) Same direction, different speeds

Now bus A goes at 10 m/s and bus B at 15 m/s, the same way. After 1 second, A has gone 10 m and B 15 m, so B is 5 m ahead (Figure 6.4). Every second, B pulls 5 m further ahead. To the people in B, bus A seems to slide backwards at 5 m/s; to the people in A, bus B seems to creep forward at 5 m/s.

$$v_{AB} = v_A - v_B = 10\ \text{m/s} - 15\ \text{m/s} = -5\ \text{m/s}$$

The size of the relative velocity is 5 m/s, the difference of the two speeds. (The minus sign only says that A falls behind B; seen from A, B moves ahead at +5 m/s.)

(c) Opposite directions

Buses A and B move away from the reference line PQ in opposite directions, both at 20 m/s (Figure 6.5). Take A’s direction as positive. B goes the other way, so its velocity is negative: −$v_B$, where $v_B$ is its speed. Then

$$v_{AB} = v_A - (-v_B) = v_A + v_B$$

So when two bodies move in opposite directions, the size of their relative velocity is the sum of their speeds: 20 + 20 = 40 m/s. After 1 second they are 40 m apart. The people in each bus feel that they are going much faster than their own bus really is.

An object in uniform motion keeps the same velocity. In non-uniform motion the velocity keeps going up and down, so no single number tells you “the” velocity. Instead, we use the average velocity: the total displacement (total distance traveled in a particular direction) divided by the total time taken.

$$\text{average velocity } (\bar v) = \frac{\text{total displacement } (s)}{\text{total time taken } (t)} \qquad \bar v = \frac{s}{t}$$

Here $\bar v$ (read “v bar”) is the average velocity, $s$ the total displacement and $t$ the total time taken.

There is a second formula. If an object speeds up (or slows down) steadily, from an initial velocity $u$ at the start to a final velocity $v$ at the end, its average velocity is halfway between the two:

$$\bar v = \frac{\text{initial velocity } (u) + \text{final velocity } (v)}{2} = \frac{u + v}{2}$$

A car that speeds up steadily from 0 to 20 m/s has an average velocity of (0 + 20) ÷ 2 = 10 m/s. (This formula works only when the velocity changes steadily, at a constant rate. For any other journey, use total displacement ÷ total time.)

The velocity of a car, motorcycle or bus is usually non-uniform: it goes up and down. A bicycle speeds up going down a slope; a dropped stone keeps speeding up. The change in velocity in one unit of time is called acceleration. It is the rate of change of velocity. It is written $a$, and its SI unit is the meter per second squared (m/s²).

Suppose a bicycle moving with velocity $u$ speeds up to velocity $v$ in time $t$. Its velocity has changed by $v - u$, so

$$\text{acceleration} = \frac{\text{change in velocity}}{\text{time taken}} = \frac{\text{final velocity} - \text{initial velocity}}{\text{time taken}} \qquad a = \frac{v - u}{t}$$

An acceleration of 2 m/s² means the velocity goes up by 2 m/s every second: 0, 2, 4, 6 m/s…

Retardation

When the driver brakes, the bus slows down: its final velocity is less than its initial velocity ($v < u$). Then $v - u$ is negative, so the acceleration is negative. The rate of decrease of velocity is called retardation (or deceleration). In short: negative acceleration is retardation.

1. Choose the best alternative

2. Differentiate between

3. Answer the following questions

4. Solve these numerical problems

A boy pushes a long iron bar under a big rock, rests it on a small stone, and presses down on the far end. The rock lifts. A carpenter pulls a nail out with the claw of a hammer. Both use the same idea.

You have seen a beam balance, a dhiki, scissors, tongs, a wheelbarrow and a crowbar. Each of them turns about a fixed point to lift or move a load. They are all levers.

A lever is a rigid bar that can turn about a fixed point. That fixed point is called the fulcrum.

Every lever has five things to name (Figure 6.6):

• The fulcrum (F): the fixed point the lever turns about.
• The effort (E): the force we apply to the lever.
• The load (L): the object to be lifted or moved, or the force the lever has to overcome.
• The load distance (L.D.): the distance from the fulcrum to the load. That part of the bar is the load arm.
• The effort distance (E.D.): the distance from the fulcrum to the effort. That part of the bar is the effort arm.

Types of lever

Levers are sorted into three classes by what is in the middle: the fulcrum, the load or the effort.

(a) First-class lever

In a first-class lever, the fulcrum is between the load and the effort. The effort is at one end and the load at the other. Examples: a beam balance, a see-saw, scissors, a pair of pliers, a dhiki, a crowbar, and the claw of a hammer pulling a nail.

(b) Second-class lever

In a second-class lever, the load is between the fulcrum and the effort. The fulcrum is at one end and the effort at the other. Examples: a nutcracker, a lemon squeezer, a bottle opener and a wheelbarrow (the wheel’s axle is the fulcrum, the load sits in the tray, and you lift the handles).

(c) Third-class lever

In a third-class lever, the effort is between the fulcrum and the load. The fulcrum is at one end and the load at the other. Examples: fire tongs (chimta), tweezers, a shovel, a broom, a fishing rod, and a hammer when you swing it to hit a nail (your wrist is the fulcrum).

A third-class lever needs more effort than the load, because the effort is closer to the fulcrum. What it gives you instead is speed and reach: a small movement of your hand makes the far end move a long way, fast. Your own forearm is a third-class lever: the elbow is the fulcrum, the biceps muscle pulls just below it, and the load is in your hand.

Working principle of a lever

This is the principle of the lever. When a lever is balanced,

$$\text{effort} \times \text{effort distance} = \text{load} \times \text{load distance}$$
$$E \times \text{E.D.} = L \times \text{L.D.}$$

It tells us how to lift a heavy load with a small effort: make the effort distance long and the load distance short. If the effort distance is short, a big effort is needed. Every lever, of every class, balances by this rule.

Mechanical advantage

A lever lets a small effort lift a big load. How many times bigger? The ratio of the load to the effort is called the mechanical advantage (MA). It tells you how many times the lever multiplies your effort.

$$\text{mechanical advantage (MA)} = \frac{\text{load } (L)}{\text{effort } (E)}$$

MA is a ratio of two forces (newtons ÷ newtons), so it has no unit.

For example, if a person needs an effort of 150 N to lift a load of 600 N with a lever:

$$\text{MA} = \frac{L}{E} = \frac{600\ \text{N}}{150\ \text{N}} = 4$$

An MA of 4 means the lever lifts a load 4 times as heavy as the effort.

• If MA > 1, the lever multiplies the effort: the load is bigger than the effort (a crowbar, a wheelbarrow).
• If MA < 1, the effort has to be bigger than the load (tongs, a broom); such a lever gives speed or convenience instead.

MA is found from the real forces, so friction affects it. Friction at the fulcrum means you must push a little harder, which lowers the mechanical advantage.

Velocity ratio

When a lever lifts a load, the load and the effort both move, but not by the same distance. Push the long end of a crowbar down by 30 cm, and the rock at the short end rises only 10 cm. The effort moves farther, in the same time, so it moves faster than the load. The ratio of the velocity of the effort to the velocity of the load is called the velocity ratio (VR).

The effort and the load move for the same time $t$, so the times cancel and VR is just a ratio of distances:

$$\text{VR} = \frac{\text{velocity of effort}}{\text{velocity of load}} = \frac{\text{distance moved by effort} \div t}{\text{distance moved by load} \div t} = \frac{\text{distance moved by effort}}{\text{distance moved by load}}$$

For a lever, the two ends turn through the same angle, so the distance each one moves is in proportion to its distance from the fulcrum. That gives a handy formula:

$$\text{VR} = \frac{\text{effort distance (E.D.)}}{\text{load distance (L.D.)}}$$

VR depends only on the shape and size of the lever (where the fulcrum is), not on the forces, so friction does not affect it. But friction does lower the MA. So in every real machine, the mechanical advantage is less than the velocity ratio.

Efficiency

Lifting a load means doing work (work = force × distance moved in the direction of the force).

• The work done by the effort on the machine is the input work: input work ($W_i$) = effort × distance moved by the effort.
• The work done by the machine on the load is the output work: output work ($W_o$) = load × distance moved by the load.

The output work divided by the input work, written as a percentage, is the efficiency of the machine. It is written η (the Greek letter eta).

$$\text{efficiency } (\eta) = \frac{\text{output work}}{\text{input work}} \times 100\% = \frac{L \times \text{distance moved by load}}{E \times \text{distance moved by effort}} \times 100\%$$

Rearrange the fraction, and it becomes MA divided by VR:

$$\eta = \frac{L / E}{\text{distance moved by effort} / \text{distance moved by load}} \times 100\% = \frac{\text{MA}}{\text{VR}} \times 100\%$$

For a lever, the distances moved are in the same proportion as the arms (E.D. ÷ L.D.), so its efficiency can also be found from the arm lengths:

$$\eta = \frac{L \times \text{L.D.}}{E \times \text{E.D.}} \times 100\%$$

Friction always turns part of the input work into heat, so the output work is always a little less than the input work. That is why MA is always less than VR, and why the efficiency of a real machine is always less than 100%. An imaginary machine with no friction at all, with 100% efficiency and MA equal to VR, is called an ideal or perfect machine.

1. Choose the best alternative

2. Differentiate between

3. Give reasons

4. Answer the following questions

5. Solve these numerical problems

Both activities say the same thing. When a force acts on a large area, its effect is small; when the same force acts on a small area, its effect is big. This effect of a force on an area is called pressure. Pressure is the force acting perpendicularly (at right angles) on a unit area of a surface.

Measurement of pressure

Pressure grows when the force grows. And for the same force, the smaller the area, the greater the pressure. So pressure depends on the size of the force and on the area it acts on. If a force $F$ acts on a surface of area $A$, the pressure $P$ on it is

$$\text{pressure} = \frac{\text{force}}{\text{area}} \qquad P = \frac{F}{A}$$

In SI, force is measured in newtons (N) and area in square meters (m²), so the SI unit of pressure is the newton per square meter (N/m²), also called the pascal (Pa), after the French scientist Blaise Pascal.

One pascal is the pressure of a force of 1 N acting perpendicularly on an area of 1 m².

$$1\ \text{Pa} = 1\ \text{N/m}^2$$

A pascal is tiny: a sheet of paper lying flat on a table presses on it with about 1 Pa. So larger pressures are often given in kilopascals (1 kPa = 1000 Pa).

Applications of pressure in daily life

Sometimes we want more pressure, so we make the area small:

• A plowshare is made of iron and has a sharp edge, so even a modest pull drives it into the soil.
• Sickles, knives and axes are sharpened so that their thin edge cuts and chops easily.
• A nail and a screw have pointed tips, so they go into wood or a wall easily.

Sometimes too much pressure is a problem, so we make the area large:

• The back wheels of a tractor are big and wide, so it doesn’t sink into a soft, plowed field.
• Buses have pairs of tyres on their back wheels, and heavy trucks have many wheels, so their weight is spread over a bigger area of road, and they don’t damage it.
• A porter’s namlo (head strap) and a school bag’s straps are made wide, so they don’t cut into the skin.

Pressure of a liquid

A liquid has weight, so, like a solid, it presses on whatever it rests on. But a liquid can flow, so unlike a solid, it presses in all directions: down on the bottom of the container, sideways on its walls, and even upwards on anything put in it. The pressure exerted by a liquid on a unit area of its container (or of anything in it) is called liquid pressure.

Measuring the pressure of a liquid

Take a container whose base has an area $A$, filled to a height $h$ with a liquid of density $d$ (Figure 6.14). The acceleration due to gravity there is $g$. Let’s find the pressure on the base.

The force on the base is the weight $W$ of all the liquid above it. So the pressure on the base is

$$P = \frac{\text{weight of the liquid column } (W)}{\text{area of the base } (A)} = \frac{W}{A}$$

Weight is mass × acceleration due to gravity, $W = mg$:

$$P = \frac{m \times g}{A}$$

Mass is density × volume, $m = d \times V$:

$$P = \frac{d \times V \times g}{A}$$

The liquid is a block of base $A$ and height $h$, so its volume is $V = A \times h$:

$$P = \frac{d \times A \times h \times g}{A}$$

The $A$ cancels, and

$$P = h\,d\,g$$

So the pressure of a liquid depends on three things only: the depth $h$ of the liquid, the density $d$ of the liquid, and the acceleration due to gravity $g$. It grows with depth, and it is bigger in a denser liquid. The area has canceled out: the pressure at a depth does not depend on how wide the container is.

Applications of liquid pressure in daily life

The pressure of a liquid increases with depth, and that matters in many places:

• a.A dam is built much thicker at the bottom than at the top, and the base of a water tank is made thicker, because the water pushes hardest at the greatest depth.
• b.A deep-sea diver wears a special suit (or goes down in a strong submarine), because deep in the sea the water pushes with a very great pressure.
• c.A water tank is kept as high as possible on a house. The taps are then far below the water level, so the water comes out with a greater pressure.
• d.A bottle for mercury has a thicker bottom than bottles for other liquids, because mercury is 13.6 times as dense as water, so its pressure at the same depth is 13.6 times as great.

Air pressure

Have you pumped up a football with a pump, or seen the tubes of bicycle and motorcycle tyres filled with air? Have you seen an oxygen cylinder in a hospital? In each, air (or oxygen) has been squeezed into a small space. A gas can be compressed: its volume made smaller. A machine that does this is called a compressor.

When air is compressed, its pressure rises, and it also gets warm (feel the barrel of a bicycle pump after pumping hard). The pressure of compressed air is greater than the pressure of the air around us.

Compressed air stores energy, so it is used to carry energy in industry and on the road:

• It drives air hammers, drills and wrenches (the machine a mechanic uses to undo the nuts of a bus wheel in seconds), and works automatic air cylinders that move parts in factories.
• Air brakes on large trains, buses and trucks use compressed air to push the brakes on, and stop these heavy vehicles safely.
• It powers cleaning sprays, spray-painting guns and air rifles, and is used more and more.

The pressure of compressed air is measured with a pressure gauge (Figure 6.16), the dial you see on a tyre pump at a petrol pump. Gauges read in bar or psi (pounds per square inch); 1 bar = 100,000 Pa, and 1 millibar (mbar) = 100 Pa. Weather reports give the air pressure in millibars, and atmospheric pressure is also given in millimeters of mercury (mmHg), as you will see next.

Atmospheric pressure

Why does the card hold up the water? And what crushed the can? The air. In Activity 6.7, the air pressure acting from below holds the card up. In Activity 6.8, before heating, the air pressure inside and outside the can was the same, so the can stayed as it was. When the water boiled, the steam pushed most of the air out of the can. After the lid was closed and the can cooled, the steam condensed back into a few drops of water, leaving almost nothing inside to push outwards. The pressure inside fell far below the pressure outside, and the air outside crushed the can.

The Earth is wrapped in a layer of air hundreds of kilometers thick, called the atmosphere. Air is light, but it does have weight, and all that air above us presses down on the ground and on everything on it. The pressure of the air of the atmosphere is called atmospheric pressure: the force of the air on each unit area of the Earth’s surface.

• Atmospheric pressure is greatest at sea level, and gets less as you go higher, because there is less air above you.
• At sea level it is about 760 mmHg (it holds up a column of mercury 760 mm tall), which is about 100,000 Pa (1.01 × 10⁵ Pa). This is called the standard atmospheric pressure.
• Atmospheric pressure is measured with a barometer.
• Air flows from places of higher pressure to places of lower pressure: differences in atmospheric pressure over the Earth’s surface make the winds.

100,000 Pa means that the air pushes on every square centimeter of you with a force of about 10 N, like a 1 kg bag of rice resting on it. We aren’t crushed because the air and the fluids inside our bodies push outwards just as hard.

Airplanes fly at heights of about 10 km, where the air pressure is less than a third of that at sea level. Too little to breathe. So the cabin of a passenger plane is pressurized: air is pumped in to keep the pressure inside close to that on a high hill town, so the passengers can breathe normally during the flight. Atmospheric pressure affects every object and every living thing on the Earth.

Manometer

The pressure of a fluid (a liquid or a gas) can be measured with a manometer (also called a pressure gauge). The simplest kind, the U-tube manometer, is a glass or plastic tube bent into a U, partly filled with a liquid such as colored water.

When both ends are open to the air, the liquid stands at the same level in both arms. Now connect one arm to a gas or liquid whose pressure you want to measure. If its pressure is greater than the air’s, it pushes the liquid down in that arm and up in the other. The difference in height $h$ between the two levels measures the extra pressure: the bigger the pressure, the bigger $h$.

Applications of atmospheric pressure

We use atmospheric pressure all the time:

• Filling a syringe with medicine, and a fountain pen with ink.
• Drinking through a straw.
• Pumping air into a bicycle tube.
• Lifting water with a hand pump (tube well).

In each, pulling out a piston (or sucking) makes the pressure inside less than the air pressure outside, and the atmosphere then pushes the liquid or gas in. Without atmospheric pressure, none of these would work.

1. Choose the best alternative

2. Give reasons

3. Answer the following questions

4. Solve these numerical problems