Motion and Force

Why does an apple fall straight down?

The story goes that in 1666 a young Isaac Newton was sitting in his mother’s garden in England when an apple fell from a tree. (It didn’t hit him on the head; that part was added later.) Newton wondered: why does the apple always fall straight down, towards the center of the Earth? Why not sideways, or up?

His answer: the Earth pulls the apple. Then he asked a bigger question. If the Earth’s pull reaches the top of an apple tree, and the top of a mountain, does it reach all the way to the Moon? Could the same pull that drops the apple be what keeps the Moon going round the Earth, and the planets round the Sun?

After years of work, Newton concluded that every object in the universe attracts every other object. He called this attraction gravitation, and in 1687 he published the universal law of gravitation in his book Principia.

Activity 7.1: what makes the pull stronger?

Try it yourself here. Each time you change something, the lab shows how many times bigger or smaller the force became.

Newton’s universal law of gravitation

Activity 7.1 found exactly what Newton’s law says:

The gravitational force between any two objects in the universe is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Let’s turn the sentence into a formula. Say the masses are m₁ and m₂, the distance between their centers is d, and the force is F (Figure 7.2).

The force is proportional to the product of the masses:

$$F \propto m_1 m_2 \qquad \text{(i)}$$

and inversely proportional to the square of the distance:

$$F \propto \frac{1}{d^2} \qquad \text{(ii)}$$

Putting (i) and (ii) together:

$$F \propto \frac{m_1 m_2}{d^2}$$

To turn “is proportional to” into “equals”, we put in a constant, G:

$$F = \frac{G\,m_1 m_2}{d^2} \qquad \text{(iii)}$$

G is the universal gravitational constant. With equation (iii), you can work out the pull between any two objects, from two marbles to two galaxies.

Two things to notice about the force:

• It is mutual: A pulls B exactly as hard as B pulls A (Newton’s third law). The Earth pulls you with your weight, and you pull the Earth with the same force.
• It is always an attraction, never a push, and it acts along the line joining the two centers.

The gravitational constant, G

What does G mean? Put m₁ = 1 kg, m₂ = 1 kg and d = 1 m into equation (iii):

$$F = \frac{G \times 1 \times 1}{1^2} = G$$

So G is the gravitational force between two masses of 1 kg each, placed 1 m apart (center to center).

Its value is

$$G = 6.67 \times 10^{-11}\ \text{N m}^2/\text{kg}^2$$

That is 0.0000000000667 N: two 1 kg bags of rice a meter apart pull each other with less than a ten-billionth of a newton. No wonder you never notice it.

G has the same value everywhere: whatever the objects are made of, whatever lies between them (air, water, a wall, empty space), and anywhere in the universe. That is why it is called universal.

Worked examples

How the force changes with mass and distance

The results of Activity 7.1 come straight out of the formula. Start with two objects of masses m₁ and m₂, a distance d apart:

$$F_1 = \frac{G m_1 m_2}{d^2}$$

What changes	Put into the formula	New force
One mass doubled (m₁ → 2m₁)	$F_2 = \dfrac{G (2m_1) m_2}{d^2} = 2 \times \dfrac{G m_1 m_2}{d^2}$	F₂ = 2F₁ (twice)
Both masses doubled	$F_2 = \dfrac{G (2m_1)(2m_2)}{d^2} = 4 \times \dfrac{G m_1 m_2}{d^2}$	F₂ = 4F₁ (4 times)
Distance halved (d → d/2)	$F_2 = \dfrac{G m_1 m_2}{(d/2)^2} = \dfrac{G m_1 m_2}{d^2/4} = 4 \times \dfrac{G m_1 m_2}{d^2}$	F₂ = 4F₁ (4 times)
Distance doubled (d → 2d)	$F_2 = \dfrac{G m_1 m_2}{(2d)^2} = \dfrac{1}{4} \times \dfrac{G m_1 m_2}{d^2}$	F₂ = ¼F₁ (a quarter)

A quick way to do any of these: multiply by whatever happens to a mass, and divide by the square of whatever happens to the distance.

What gravitation does

Gravitation is the weakest force we know of, but it never cancels out (there is no “negative mass” to push back), and it reaches across any distance. So for big things, planets, stars and galaxies, it is the force in charge.

1. It holds the universe together. It made the Sun, the planets and the galaxies, and it keeps the planets moving round the Sun.
2. It causes the tides. The Moon is tiny compared with the Sun, but it is so much closer that its pull on the Earth’s oceans is the bigger one. The Moon pulls the side of the sea nearest it more strongly than the Earth’s center, and the Earth’s center more strongly than the far side, so the oceans bulge on both sides. As the Earth turns, most coasts get two high tides a day.
3. It holds us and everything else on the ground, and brings back whatever we throw up.

Gravity

The Earth, the other planets and the moons pull nearby objects towards their centers. The pull of a planet or moon on an object near it is called the force of gravity. It is also called the object’s weight.

Newton’s law says that this pull gets weaker the farther away you go, and becomes negligible very far away. So each planet and moon has a gravitational field, a region around it where its gravity can be felt.

We see the Earth’s gravity at work all day long:

1. Everything has weight.
2. The Earth keeps its atmosphere: gravity stops the air from escaping into space. (The Moon’s gravity is too weak to hold an atmosphere.)
3. Things fall when dropped.
4. Rivers flow downhill, from the Himalaya to the Terai and on to the Bay of Bengal.
5. A falling object speeds up: it accelerates.

Activity 7.2: measuring the acceleration of a falling stone

Acceleration due to gravity

In Activity 7.2 the stone, once released, is pulled down by the Earth’s gravity all the way. A force that keeps acting makes an object go faster and faster: it accelerates. When air resistance is small enough to ignore, this acceleration is about 9.8 m/s².

The acceleration of a freely falling object, produced by the force of gravity, is called the acceleration due to gravity. It is written g, and its SI unit is m/s² (also written m s⁻²).

What does g = 9.8 m/s² mean? A freely falling object gains 9.8 m/s of speed every second:

Time after letting go	0 s	1 s	2 s	3 s	4 s
Speed	0 m/s	9.8 m/s	19.6 m/s	29.4 m/s	39.2 m/s
Distance fallen	0 m	4.9 m	19.6 m	44.1 m	78.4 m

An object thrown straight up does the opposite: it loses 9.8 m/s every second until its speed is zero at the top, and then it falls back down.

Activity 7.3: paper, paper ball and stone

Try the same race on the screen, with the air on and off.

The feather and coin experiment

The same thing can be shown on Earth, by taking the air away.

A long glass tube is connected to a vacuum pump. A feather and a coin lie at the bottom (Figure 7.8).

• With air in the tube, turn it quickly upside down. The coin falls fast; the feather floats down slowly.
• Pump the air out, then turn the tube upside down again. The feather and the coin fall together, side by side.

Why? With air in the tube, the feather’s large, fluffy surface meets a lot of air resistance for its tiny weight, which slows it down. The coin is heavy for its size, so air hardly slows it. With no air, there is no air resistance, and both fall with the same acceleration, g.

Conclusion of the feather and coin experiment: in the absence of air resistance, all objects fall with the same acceleration. The value of g does not depend on the mass of the falling object.

Calculating g

Why should a heavy stone and a light one fall with the same acceleration? Newton’s two laws together give the answer.

Suppose an object of mass m is on the surface of a planet of mass M and radius R. By the law of gravitation, the planet pulls it with a force

$$F = \frac{G M m}{R^2} \qquad \text{(i)}$$

This force gives the object an acceleration g. By Newton’s second law (F = ma, with a = g):

$$F = m g \qquad \text{(ii)}$$

Both are the same force, so (i) = (ii):

$$m g = \frac{G M m}{R^2}$$

The object’s mass m is on both sides, so it cancels:

$$g = \frac{G M}{R^2} \qquad \text{(iii)}$$

This is the formula for the acceleration due to gravity, and look what is not in it: the mass of the falling object. g depends only on the planet: its mass M and its radius R. That is exactly what Galileo, the feather and coin, and David Scott found.

A heavy stone is pulled harder than a light one, but it is also harder to speed up, by exactly the same factor. The two effects cancel.

Mass and radius both matter

Since G and M are fixed for a planet, equation (iii) says

$$g \propto \frac{1}{R^2}$$

g is inversely proportional to the square of the radius. A bigger mass makes g bigger; a bigger radius makes it smaller, and the radius counts twice (squared).

Jupiter shows both at work. Its mass is about 319 times the Earth’s, but its radius is about 11 times the Earth’s. So its g is

$$\frac{g_\text{Jupiter}}{g_\text{Earth}} = \frac{319}{11^2} = \frac{319}{121} = 2.6$$

only 2.6 times the Earth’s, not 319 times. Its huge mass is spread out, and its surface is far from its center.

Why g is different in different places on the Earth

The Earth isn’t a perfect ball. It spins once a day, and the spin has made it bulge at the equator and flatten slightly at the poles. The radius at the equator is about 6378 km; at the poles it is about 6357 km, 21 km less.

Since g ∝ 1/R², g is larger at the poles, where R is smaller:

Place	Distance from the center	g
Equator	about 6378 km	9.78 m/s²
Poles	about 6357 km	9.83 m/s²
Average, used in calculations	6371 km	9.81 m/s² (often rounded to 9.8)

So a stone dropped at the North Pole falls a little faster than one dropped in Kenya or Indonesia, on the equator.

g decreases with height

The farther you are from the Earth’s center, the weaker its pull. On the surface, at distance R from the center, $g = \dfrac{GM}{R^2}$. At distance d from the center:

$$g_1 = \frac{G M}{d^2}$$

A satellite at a height h above the surface is at a distance d = R + h from the center, so

$$g_1 = \frac{G M}{(R + h)^2}$$

Since (R + h)² is bigger than R², g₁ is smaller than g on the surface.

On a mountain, too, g is a little less than at its foot, because the top is farther from the Earth’s center. But the difference is small, because even Everest is tiny next to the Earth.

Mass

The mass of an object is the quantity of matter in it. Its SI unit is the kilogram (kg), and it is a scalar quantity (it has size but no direction).

A 1 kg lump of iron has 1 kg of iron in it whether it is on the Earth, in the Space Station, on the Moon or on Mars. Mass does not depend on the place. Every particle of matter, even an electron, has some mass (never zero), and so even the smallest mass feels the pull of gravity.

Weight

The Earth pulls everything on its surface towards its center. The weight of an object is the force of gravity acting on it. Since weight is a force:

• its SI unit is the newton (N);
• it is a vector quantity (it has a direction): it always points towards the center of the planet or moon.

Weight is measured with a spring balance: the object pulls a spring down, and the stretch of the spring shows the force.

By Newton’s second law, the force of gravity on an object of mass m, which gives it an acceleration g, is

$$W = m g$$

So the weight depends on two things: the object’s mass and g at that place.

1. Weight is proportional to mass (with g the same): $W \propto m$. A 10 kg sack of rice weighs 98 N; a 20 kg sack weighs 196 N. To lift something, you have to pull up with at least its weight, which is why a big stone is harder to lift than a small one.
2. Weight is proportional to g (with the mass the same): $W \propto g$. So the same object weighs a little less where g is smaller: at the equator, on a mountain top, on the Moon.

Weight on other worlds

On the Moon, g is about one sixth of the Earth’s, $g_m \approx \tfrac{1}{6} g_e$, so everything weighs about 6 times less.

That has some fun results. With the same leg muscles, you could jump about 6 times higher on the Moon. A weightlifter could lift about 6 times the mass she lifts on Earth, because the same muscles produce the same upward force, and on the Moon that force holds up six times the mass.

Here is the weight of a 50 kg student around the Solar System, using W = mg:

Place	g (m/s²)	Weight of 50 kg (N)
Moon	1.62	81
Mercury	3.70	185
Mars	3.71	185.5
Venus	8.87	443.5
Earth	9.81	490.5
Jupiter	24.8	1240

(The values are from NASA’s planetary fact sheets.)

Mass	Weight
The quantity of matter in an object.	The force of gravity on an object.
SI unit: kilogram (kg).	SI unit: newton (N).
Scalar: no direction.	Vector: points towards the center of the planet.
The same everywhere.	Changes with g: less at the equator, on mountains, on the Moon.
Never zero.	Can be zero: a balance falling freely reads zero (weightlessness), and far out in space, far from any planet or star, weight is almost zero.
Measured with a beam balance.	Measured with a spring balance.

In Activity 7.3, the paper ball and the stone fell together because air hardly slowed them. When the only force acting on a falling object is gravity, it falls with acceleration g.

An object falling under the influence of gravity alone, with nothing else acting on it, is said to be in free fall. The acceleration of an object in free fall is g.

In real life, air gets in the way. Air rubs against a falling object (air resistance, a kind of friction that depends on the object’s shape and size), and air gives every object a tiny upthrust, the buoyant force you will meet in the unit on pressure. Both act upwards against gravity. So true free fall happens only in a vacuum. On the Moon, which has no air, every falling object is in free fall. On Earth, a stone or a paper ball falling a few meters is so little affected by air that we treat it as free fall.

Activity 7.6: making a parachute

Why parachutes, hailstones and seeds come down gently

Think of the model parachute of Activity 7.6, let go from rest. Its weight W pulls it down, and the air pushes up on the canopy with a force R, the air resistance. The faster it falls, the harder the air pushes. So:

1. At first it is slow, R is small, and the weight wins: it speeds up.
2. As it speeds up, R grows.
3. Soon R equals the weight. The two forces balance, the net force is zero, and so the acceleration is zero. From then on it falls at a steady speed, called the terminal velocity.

A real skydiver falls for a while before she opens her parachute, so she is already going fast, about 55 m/s (200 km/h). When the big canopy opens, R is suddenly much bigger than her weight: the net force points up, and she slows down, until R = W again. Either way, she ends at the same steady speed, the parachute’s terminal velocity of about 5 m/s: a safe landing speed, like jumping off a table. So falling with a parachute is not free fall: air resistance is doing the important work.

Hailstones work the same way. A hailstone from a cloud 2 km up would hit the ground at about 200 m/s if it fell freely, over 700 km/h, as fast as a jet airliner. Air resistance holds it to a steady speed instead: about 10 to 30 m/s for most hailstones. That is still enough to dent cars and ruin a field of wheat, but far less harmful than without air.

Seeds use the air too. The seeds of simal (the silk-cotton tree) hang in a tuft of white fluff, and sal seeds have wings like a little fan. Their large surface makes air resistance big compared with their weight, so they drift down slowly, and the wind carries them far from the parent tree before they land. Dandelion and kapok seeds work the same way.

Activity 7.7: weightlessness

When an object is in free fall, it presses on nothing and pulls on nothing: a spring balance falling with it reads zero. It seems to have no weight. This is called weightlessness.

Orbiting is falling

But if the Space Station is falling, why doesn’t it hit the ground? Newton answered that too, with a thought experiment. Imagine a cannon on top of a very high mountain, firing horizontally:

• A slow ball curves down and lands near the mountain.
• A faster ball lands farther away.
• Fire fast enough, and the ball curves down exactly as fast as the round Earth curves away beneath it. It keeps falling, but never comes any closer to the ground. It is in orbit.

The Space Station moves at about 7.7 km/s, going round the Earth once every 92 minutes, so the astronauts see about 16 sunrises a day. It is falling towards the Earth all the time, and missing it all the time. Everything inside falls with it, so everything floats. You can see where it is right now on our Space Station tracker; on a clear evening it often passes over Nepal as a bright, fast, steady dot.

In Grade 9 you met the equations of motion for an object moving with uniform acceleration a. A freely falling object moves with uniform acceleration g, so the same equations work: just put g in place of a, and h (height) in place of s.

For objects in linear motion	For objects in free fall
$v = u + at$	$v = u + gt$
$v^2 = u^2 + 2as$	$v^2 = u^2 + 2gh$
$s = ut + \tfrac{1}{2}at^2$	$h = ut + \tfrac{1}{2}gt^2$

Here u is the starting velocity, v the velocity after time t, and h the height fallen (or risen).

Two rules for the signs:

• An object dropped (let go, not thrown) starts with u = 0.
• For an object thrown up, gravity slows it down, so its acceleration is opposite to its motion: use g = −9.8 m/s². At the highest point, its velocity is momentarily v = 0.

1. Choose the correct option

2. Differentiate between

3. Give reasons

4. Answer these questions

5. Solve these problems

Use G = 6.67 × 10⁻¹¹ N m²/kg².