Gravitation

Question 1

Gravitational field definition

Answer 1

A region where a MASS experiences a force.

Question 2

Gravitational field strength, g, definition

Answer 2

Gravitational Force per unit mass …. g=F/m.

(Since the force felt by any object due to the gravitational field of the Earth is called weight W, on Earth we have g=W/m → W=mg)

Question 3

(a) Kepler’s laws of planetary motion

Answer 3

1. ellipse
2. areas
3. T^3 ∝ r^2

(EAT! Kepler kipper …)

1. Planets move in elliptical orbits with the sun at one focus.
2. The sun-planet line sweeps out equal areas in equal times.
3. T^3 ∝ r^2 where T is orbital period and r = mean distance from the sun.

Question 4

(b) Gravitational force between two masses equation and important points x2

Answer 4

Equation: F = -Gm1m2/r^2

1. MINUS SIGN because the force is ALWAYS ATTRACTIVE and attractive forces are -ve by definition.
2. Force exerted on m1 by m2 is equal and opposite to …. blah. Consequence of N3L.

Question 5

(c) Derive r^3 ∝ T^2 for a circular orbit.

Answer 5

F=-Gm1m2/r^2.

Also, F =ma
=m1(ω^2r)

And ω = 2π /T

So -Gm1m2/r^2 = m1(2π /T)^2*r …. cancel m1 and rearrange … G, m2 and π are all constants so you get r^3 ω T^2.

NB: m1 is the mass doing the orbiting, so it’s the one feeling the force and therefore accelerating towards the centre of its orbit (mass 2), so it’s the one in the F=ma equation.

Question 6

(g), (d) GPE stuff … where’s zero, graphs …?

Answer 6

1. Zero GPE is taken at infinity … so all GPEs are negative because GPE lost as falling towards a mass.
2. Look at area under force-distance graph to find Gpe gained/ lost moving from one point to another.

Question 7

(e) Gravitational field strength due, g, due to a …

Answer 7

… spherical point mass, at a distance r from its centre:

F = Gm1m2/r^2

g=F/m

SO: g=Gm/r^2

Question 8

GPE equation

Answer 8

GPE = integral of F(grav) dr

= -Gm1m2/r

Question 9

Gravitational potential, V

Answer 9

GPE per unit mass, i.e. V = -Gm/r

Question 10

(h) Escape velocity definition and derivation

Answer 10

The velocity needed to be given to an object on the surface of the planet in order for it to escape the gravitational field of the planet and have ZERO KE at that time.

❗ Once the object has been given this velocity e.g. by a cannon, no more energy is put into the system and it is just a projectile … loses KE and gains GPE.
Just do it.

sqrt(2GM/r)

Question 11

(i) Distance of geostationary orbit from centre of Earth

Answer 11

Use r^3 prop to T^2 equation