# Chapter 10

Chapter 10 – Gravitation for people to look at. Drop me a comment if you notice anything wrong. 😀

1. State Kepler’s Laws. Where necessary, illustrate your answer by means of diagrams.
One: The planets move in elliptical orbits, with two foci. The Sun is one focus, whilst the other focus remains empty.
Two: The radius vector sweeps to same distance in the same amount of time.
Three: The squares of the sidereal periods (years) of the planets are proportional to the cubes of their mean distances from the Sun. ( T₁/T₂ )² = ( r₁/r₂ )³

2. Explain, using a diagram, why the Moon may be said to be ‘falling’ continuously towards the Earth. Why does it not fly off in a straight line?
The Moon continues to circle the Earth because there is constantly gravity. The gravity is pulling it in as it orbits, which means it will not just fly off at a tangent. Should the Earth suddenly disappear, then the Moon would fly off, until there was a new force acting upon it. 3. Describe the manner in which the Earth and the Moon move round the barycentre. What difference would it make if the Earth and the Moon were equal mass?
Because of Newton’s third law, every action has an opposite reaction. This means that if there were two exactly identical planets, they would orbit each other, or they would orbit a certain point exactly half way between them, called a barycentre. But the Earth and Moon are not the same mass. This means the barycentre moves. It will move towards the more massive planet, until they orbit each other equally. For the Earth and Moon, the barycentre is somewhere within the Earth’s globe.

4. Give an explanation of the phenomenon of the tides, explaining the terms ‘spring tide’ and ‘neap tide’.
The tides are controlled by the Moon. The Moon pulls on the water towards the Moon, but also, on the other side of the Earth, the water bulges because there is less gravity from the Moon to pull it back. That is a normal tide, which happens twice a day because of the Earth’s rotation. A Spring Tide is when the Moon and Sun both pull at the same place, when they are en-lined. This makes an even bigger bulge, so the high tide is highest at this point. A Neap Tide is when the Moon and Sun are at right angles, making the tide more even around the Earth. This is when the high tide is lowest. Each of these happen twice a month.

5. Venus moves round the Sun at a mean distance of 108,000,000 km, the Earth at a mean distance of 150,000,000 km. What is the ratio of the force of gravity on the Earth compared with that on Venus?
We don’t have to work out the top part of the equation (Gm₁m₂) because they will be the same for both planets. I only have to work out the bottom part (d²).

F=Gm₁m₂/d²        d=108,000,000km
=108,000,000 ²
=1.1664E16  (next inverse it –> x-1)
=8.573E-17
F=Gm₁m₂/d²        d=150,000,000km
=150,000,000 ²
=2.25E16  (next inverse it –> x-1)
=4.44E-17

We then need to divide Venus by Earth.

=8.573E-17 / 4.44E-17
=1.929
=2 6. Given that the distance between the Earth and the Sun is 1 a.u, and the orbital period is 1 year, calculate the orbital period of Mars, whose distance from the Sun is 1.5 a.u.

( T₁/T₂ )² = ( r₁/r₂ )³                            T₁= ?        (Mars)

T₁ = √ ( r₁/r₂ )³ T₂                                T₂= 1 year    (Earth)

T₁ = √ (1.5/1) ³ x1                                    r₁ = 1.5 a.u.    (Mars)

T₁ = √ 3.375                                             r₂ = 1 a.u.    (Earth)

T₁ = 1.8371173070874

T₁ = 1.837 years