Universal gravitation and Kepler's laws

In the human sciences high school (the Italian liceo delle scienze umane), Year 4 physics, with only a few hours a week, reaches gravitation between November and December, after work, energy and momentum. The test falls six to eight weeks from now, at the end of November or the start of December, so the evening when the teacher writes “Newton” and “Kepler” on the board is already close. The trouble is that by then the formulas are few but they all arrive together, with powers of ten and odd units. Here you will find the minimum you need, and the exercises that I see come back every year in my lessons. No derivatives or integrals: the four operations, squares and a calculator are enough.

The law of universal gravitation

Any two bodies with mass attract each other. The force is the same on both (third law), and it is:

where $r$ is the distance between the centres of the two bodies and $G=6.67\cdot 10^{-11}\mathrm{N}\cdot {\mathrm{m}}^2/k{\mathrm{g}}^2$. The two things to remember are simple. If you double one of the masses, the force doubles. If you double the distance, the force becomes a quarter, not a half: the distance is squared.

Why don’t we get pulled towards the classmate at the next desk? Let’s work it out.

Exercise 1. Two people, of 60 kg and 70 kg, stand 1 m apart. What is the force of attraction?

Less than a millionth of a newton. Gravity is the weakest of the forces: we feel it only when one of the masses is enormous, like a planet.

g is not a universal constant

The weight of a body of mass $m$ near the Earth is the gravitational force with which the Earth attracts it. Writing $M$ for the mass of the Earth and $R$ for its radius:

The mass $m$ cancels: that is why all bodies fall with the same acceleration. With the Earth’s data ($M=5.97\cdot 10^{24}$ kg, $R=6.37\cdot 10^6$ m):

The number we all know comes out of the planet’s data. At altitude the formula stays the same, but in place of $R$ you must put the distance from the centre, $r=R+h$.

Exercise 2. What is $g$ at a height of 400 km, where the International Space Station orbits? And how much does a 70 kg person “weigh” there?

The distance from the centre is $r=6.37\cdot 10^6+4.00\cdot 10^5=6.77\cdot 10^6$ m.

Weight on the ground is $70\cdot 9.81\approx 687$ N, at 400 km it is $70\cdot 8.69\approx 608$ N: about 89% of before. Astronauts float not because gravity is missing up there, but because the station and the astronauts are in free fall together. The typical mistake is to say “there is no gravity in space”: in a test that is a lost mark.

Exercise 3. At what height does $g$ become a quarter of its value at the surface?

If $g$ is a quarter, $r^2$ must be four times larger, so $r=2R$. The height above the ground is $h=2R-R=R\approx 6370$ km, and $g=9.81/4\approx 2.45m/{\mathrm{s}}^2$. Note the trick: you do not need to redo the sums with $G$ and $M$, you just reason about the proportion.

Kepler’s three laws

Kepler worked them out from data on the planets, before Newton explained why they hold. In a test you are almost always asked for the statement and an example.

1. First law. Planets travel in elliptical orbits, and the Sun occupies one of the two foci. The point closest to the Sun is called perihelion, the farthest aphelion.
2. Second law. The segment joining the planet to the Sun sweeps out equal areas in equal times. Practical consequence: near the Sun the planet moves faster, far away it slows down. The Earth is faster in January than in July.
3. Third law. For planets orbiting the same star:

where $T$ is the orbital period and $r$ the semi-major axis of the orbit, which for nearly circular orbits is the mean radius.

A warning I always repeat: the constant holds for bodies that orbit the same star. Planets around the Sun share one constant, the Earth’s satellites share another.

Exercise 4. A hypothetical planet has an orbit with a mean radius 4 times that of the Earth. How long does its year last?

The radius is in the ratio 4, so the cube is $4^3=64$. From the third law $T^2\propto r^3$, so $T^2$ is 64 times the Earth’s and $T=\sqrt{64}=8$ Earth years. You can do it in your head, with no constants: when the problem talks about “ratios” and does not give $G$, this is the way.

Exercise 5. Mars is on average 1.52 times as far from the Sun as the Earth is. How long does its year last?

The real value is about 1.88 years: the third law really works. With the same method Jupiter, at 5.2 times the Earth’s distance, has $T=5.2\cdot \sqrt{5.2}\approx 11.9$ years.

Orbital speed

A satellite in a circular orbit is falling towards the Earth, but it has such a high sideways speed that the ground curves away beneath it. The gravitational force acts as the centripetal force:

The satellite’s mass cancels again: the speed depends only on the star and the distance. The farther away it is, the slower it goes.

Exercise 6. Speed and period of the International Space Station at an altitude of 400 km ($r=6.77\cdot 10^6$ m).

The period is the length of the orbit divided by the speed:

About 16 orbits a day. Converting seconds to minutes is the step where I see the most students fall: divide by 60 and check that the result makes sense.

From the third law to the distance of the Moon

Putting the two formulas together and substituting $v$ into the period gives the third law with its constant:

Exercise 7. The Moon goes once round the Earth in about 27.3 days. How far away is it?

First the seconds: $T=27.3\cdot 86,400\approx 2.36\cdot 10^6$ s. Then isolate $r^3$:

The real mean distance is about 384,000 km: from the period alone and the Earth’s mass we have arrived at the distance of the Moon.

The mistakes I see most

• Distance from the surface instead of distance from the centre. If an exercise says “at 400 km altitude”, $r$ must contain $R+400$ km.
• Forgetting to square. Double the distance, the force is divided by four.
• Units. Kilometres must be converted to metres before substituting into $G$: $6.77\cdot 10^6$ m, not 6770.
• Confusing mass and weight. Mass does not change with altitude, weight does.

If you need to review the underlying quantities, the post on work and energy and the one on momentum and collisions cover the syllabus that comes before gravitation. If graphs give you trouble, start from motion graphs in Year 4, and for an overall picture of frequent mistakes see typical physics mistakes in high school. For those who want to go further, with transfer orbits at university level, there is orbital mechanics: Kepler, Hohmann and delta-v. The overall picture of the subject is on the physics page.

Before the test

Redo the exercises above without looking at the solutions, then try the free physics test practice paper: the correct answer is shown straight away, and you can see where you are losing marks. If you want to work on it with me, the first introductory call is free and lasts 30 minutes: write to me through contact.