Newton's Laws of Motion

Lessons

Senior 1 · Physics

Newton's Laws of Motion

5 lessons

Lesson 5 of 5

Universal Gravitation and Applications of Newton's Laws

25 min

The mango and the Moon

A story says that Newton, watching an apple fall, asked a bold question: if the Earth pulls an apple from the top of a tree, does its pull reach as far as the Moon? His answer was yes. The same force that brings a ripe mango down in Rubavu keeps the Moon moving round the Earth, keeps the Earth moving round the Sun, and holds RwaSat-1 and every other satellite in orbit. This lesson finishes the unit with that universal force, and then puts all three laws to work.

Definition

Newton's law of universal gravitation: every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

F = G\frac{m_1 m_2}{r^2}

where $F$ is the force in newtons, $m_{1}$ and $m_{2}$ are the masses in kilograms, $r$ is the distance between their centres in metres, and $G$ is the universal gravitational constant, $G = 6.67 \times 10^{-11}\,\mathrm{N\,m^2\,kg^{-2}}$ .

Figure 5.1: The two masses pull each other with forces of the same size in opposite directions, a third-law pair. The small mass feels exactly the same size of force as the large one.

What the formula tells you

More mass, more force. Double one mass and the force doubles; double both and it becomes four times as big.
More distance, much less force. The force depends on $\frac{1}{r^2}$ : an inverse-square law. Double the distance and the force falls to a quarter; triple it and the force falls to a ninth.
$G$ is tiny. The pull between everyday objects is far too small to notice. Only when one of the masses is enormous, like a planet or a star, is the force large.

Figure 5.2: The inverse-square law. Each time the distance grows, the force falls by the square of the factor.

Observing it: why is the pull between people too small to feel?

You cannot do this experiment with a school balance, but Henry Cavendish did it in London in 1798. He hung a light rod with a small lead ball on each end from a thin wire, and brought two large lead balls close to them. The rod twisted by a tiny angle: the large balls were pulling the small ones. From the twist he worked out the value of $G$ . The numbers in Worked example 1 show why ordinary objects never seem to attract each other.

Worked example 1: two pupils

Umutoni (50 kg) and Habimana (60 kg) sit 1.0 m apart. Find the gravitational force between them.

F = G\frac{m_1 m_2}{r^2} = 6.67 \times 10^{-11} \times \frac{50 \times 60}{(1.0)^2} = 2.0 \times 10^{-7}\,\mathrm{N}

That is about the weight of a grain of dust. If they move to 2.0 m apart, $r^{2}$ becomes 4 times bigger, so the force falls to $\frac{2.0 \times 10^{-7}}{4} = 5.0 \times 10^{-8}\,\mathrm{N}$ .

Worked example 2: where 9.8 comes from

The Earth has a mass of $5.97 \times 10^{24}\,\mathrm{kg}$ and a radius of $6.37 \times 10^{6}\,\mathrm{m}$ . Find the gravitational force on a 1.0 kg mass at its surface.

F = 6.67 \times 10^{-11} \times \frac{5.97 \times 10^{24} \times 1.0}{(6.37 \times 10^{6})^2} = 9.8\,\mathrm{N}

So 1 kg weighs 9.8 N: this is the value of $g$ from Lesson 3. The law of gravitation explains where $g$ comes from, and why it is smaller on the Moon, which has much less mass.

Orbits: falling round the Earth

Figure 5.3: Without gravity the Moon would fly off in a straight line (first law). Gravity pulls it towards the Earth all the time, so its path bends into a nearly circular orbit.

The Earth pulls the Moon with a force of about $2 \times 10^{20}\,\mathrm{N}$ . The same idea keeps satellites in orbit: they move sideways so fast that, as they fall towards the Earth, the curved surface falls away beneath them. Weather satellites, the GPS satellites that guide motos and buses in Kigali, and RwaSat-1 all stay up this way.

Tip

Try it: in PhET: Gravity Force Lab: Basics (version 1.1.22), double one mass and watch the force double; then double the distance and watch it fall to a quarter.

Applications of Newton's laws

Combine the three laws with the definition of acceleration from Unit 2, $a = \frac{v - u}{t}$ , and you can explain many everyday safety rules. When a moving body is stopped, the force needed is

F = ma = \frac{m(v - u)}{t}

For the same change of speed, a longer stopping time means a smaller force.

Situation	Law used	Physics
Seat belts, and not overloading a bus	1st	Inertia keeps a passenger moving when the vehicle stops; the belt supplies the stopping force.
Crumple zones and airbags	2nd	The front of the car crushes and the airbag gives way, so the passenger stops over a longer time and feels a smaller force.
Catching a ball, landing from a jump with bent knees	2nd	Drawing the hands back or bending the knees makes the stopping time longer, so the force on the hands or legs is smaller.
Braking distance of heavy lorries	2nd	A larger mass needs a larger force for the same deceleration.
Walking, swimming, rockets	3rd	Push backwards or downwards on something; it pushes you forwards or upwards.
Satellites, the Moon, tides on the oceans	gravity	Every mass attracts every other mass.

Worked example 3: catching a football

A goalkeeper catches a 0.45 kg football moving at $10\,\mathrm{m\,s^{-1}}$ . Find the average force on her hands if she stops it (a) in 0.020 s with stiff arms, (b) in 0.20 s by drawing her hands back.

\text{(a)}\quad F = \frac{m(v - u)}{t} = \frac{0.45 \times (0 - 10)}{0.020} = -225\,\mathrm{N}

\text{(b)}\quad F = \frac{0.45 \times (0 - 10)}{0.20} = -22.5\,\mathrm{N}

The minus sign shows that the force is opposite to the motion. Drawing the hands back makes the stopping time 10 times longer and the force 10 times smaller.

Worked example 4: why seat belts stretch

A 70 kg passenger is travelling at $15\,\mathrm{m\,s^{-1}}$ (54 km per hour) when the bus crashes. Find the average stopping force if he is stopped in 0.10 s by hitting the seat in front, and in 1.0 s by a seat belt that stretches a little.

F_{\text{seat}} = \frac{70 \times 15}{0.10} = 10\,500\,\mathrm{N}, \qquad F_{\text{belt}} = \frac{70 \times 15}{1.0} = 1050\,\mathrm{N}

The belt reduces the force ten times, from more than the weight of a small car to about one and a half times his own weight.

Warning

Three slips to avoid. (1) In $F = G\frac{m_1 m_2}{r^2}$ the distance is SQUARED: doubling $r$ quarters the force, it does not halve it. (2) $r$ is measured between the CENTRES; for an object on the Earth's surface it is the Earth's radius. (3) Astronauts in orbit are not “beyond gravity”: at the height of the Space Station the Earth's pull is still about 90% of its value on the ground. They float because they and their spacecraft are falling round the Earth together.

Important

Safety in practice. When you test catching or landing, use a soft ball and land on a mat or on sand. Never test a seat belt or helmet idea with a moving vehicle.

Newtonian gravity (CrashCourse Physics). Watch for the inverse-square law and for why the Moon keeps “falling” round the Earth without ever landing.

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Practice

State Newton's law of universal gravitation.
Two lorries of 20 000 kg each are parked 10 m apart. Find the gravitational force between them.
The force between two bodies is 36 N. What does it become if the distance between them is tripled?
The force between two bodies is 10 N. What does it become if both masses are doubled and the distance is unchanged?
The Moon's mass is $7.35 \times 10^{22}\,\mathrm{kg}$ and its radius is $1.74 \times 10^{6}\,\mathrm{m}$ . Find the force on a 1.0 kg mass at its surface.
An egg of mass 0.060 kg falls and hits the floor at $5.0\,\mathrm{m\,s^{-1}}$ , stopping in 0.0050 s. Find the average force. Why might it survive landing on a thick cushion?
Explain, naming the law, why a high jumper lands on a thick foam mat and not on the ground.
Explain why the Moon does not fly off into space, and why it does not fall onto the Earth.

Show answersHide answers

Every particle attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centres: $F = G\frac{m_1 m_2}{r^2}$ .
$F = 6.67 \times 10^{-11} \times \frac{20\,000 \times 20\,000}{10^2} = 2.7 \times 10^{-4}\,\mathrm{N}$
$\frac{36}{3^2} = \frac{36}{9} = 4.0\,\mathrm{N}$
$10 \times 2 \times 2 = 40\,\mathrm{N}$
$F = 6.67 \times 10^{-11} \times \frac{7.35 \times 10^{22}}{(1.74 \times 10^{6})^2} = 1.6\,\mathrm{N}$ , so $g_{\text{Moon}} = 1.6\,\mathrm{N\,kg^{-1}}$ .
$F = \frac{0.060 \times 5.0}{0.0050} = 60\,\mathrm{N}$ . A cushion makes the stopping time much longer (say 0.10 s), so the force is much smaller (3.0 N).
Second law: the mat makes the stopping time longer, so the deceleration $\frac{v - u}{t}$ and the force $m a$ on the jumper are smaller.
The Earth's gravity pulls it towards the Earth, so it cannot move off in a straight line. It does not fall in because it is moving sideways so fast that as it falls its path curves round the Earth: it stays in orbit.

Key points

Universal gravitation: $F = G\frac{m_1 m_2}{r^2}$ , with $G = 6.67 \times 10^{-11}\,\mathrm{N\,m^2\,kg^{-2}}$ .
The force is proportional to each mass and inversely proportional to the SQUARE of the distance between the centres.
Gravity gives objects their weight, gives $g = 9.8\,\mathrm{N\,kg^{-1}}$ at the Earth's surface, and keeps moons and satellites in orbit.
Stopping a body over a longer time needs a smaller force: seat belts, airbags, crumple zones, bent knees and soft hands all use this.

1The mango and the Moon

2What the formula tells you

3Observing it: why is the pull between people too small to feel?

4Orbits: falling round the Earth

5Applications of Newton's laws

The mango and the Moon

What the formula tells you

Observing it: why is the pull between people too small to feel?

Orbits: falling round the Earth

Applications of Newton's laws