Q2.

(a)

(i) State the relationship between the gravitational potential energy, Ep, and the gravitational potential, V, for a body of mass m placed in a gravitational field.

On your data sheet Ep is given the symbol W

ΔW = mΔV You have to explain the relationship in words and symbols they have given you.

The relationship between them is that Ep equals the product of the mass m and the gravitational potential V

or Graviational potential is gravitational potential energy per unit mass (or per kg)

(1 mark)

(ii) What is the effect, if any, on the values of Ep and V if the mass m is doubled?

V depends on the planetary field that m is positioned in not the mass itself - no 'm' in the equation - just 'M'.

Ep (W on your data sheet) has 'm' in it - therefore will change.

The value of Ep is doubled

but the value of V is unchanged

(2 marks)

(b)

The diagram above shows two of the orbits, A and B, that could be occupied by a satellite in circular orbit around the Earth, E.

The gravitational potential due to the Earth of each of these orbits is:

orbit A	– 12.0 MJ kg–1
orbit B	– 36.0 MJ kg–1

(i) Calculate the radius, from the centre of the Earth, of orbit A.

V = -GM/r so r = -GM/V

r_A = - (6.67 x 10^-11 x 5.98 x 10²⁴)/(-12.0 x 10⁶) = 3.32 x 10⁷

r_A = 3.3 x 10⁷ m

(2 marks)

(ii) Show that the radius of orbit B is approximately 1.1 × 10⁴ km.

Here they do not want you to simply rework as above (although if you did they would still give you the marks!), They want you to use the fact that for a given planet of mass M, M and G are constant so the product Vr will be constant!

r_AV_A = r_BV_B

r_B = r_A x V_A/V_B

V_A/V_B = ¹/₃

r_B = 3.32 x 10⁷/3 = 1.11

r_B = 1.1 x 10⁷m QED

(1 mark)

(iii) Calculate the centripetal acceleration of a satellite in orbit B.

The centripetal force on the satellite comes from the gravitational pull of the planet. It is that which holds it in the circular path.

mv²/r = GmM/r²

centripetal acceleration = v²/r = GM/r²

The accleration is given the symbol 'g' in this case.

g = GM/r_B²

g = (6.67 x 10^-11 x 5.98 x 10²⁴)/(1.11 x 10⁷)²

g = 3.2 ms^-2

(2 marks)

(iv) Show that the gravitational potential energy of a 330 kg satellite decreases by about 8 GJ when it moves from orbit A to orbit B.

ΔEp = mΔV

ΔEp = 330 × (–12.0 –(–36.0)) × 10⁶

ΔEp = 330 × 24.0 × 10⁶

ΔEp = 7.9 × 10⁹ J ≈ 8 GJ QED

(1 mark)

(c) Explain why it is not possible to use the equation Ep = mgh when determining the change in the gravitational potential energy of a satellite as it moves between these orbits.

It is not possible to use the equation because g is not constant over the distance involved.

Or because g decreases as height increases

Or because work done per metre decreases as height increases

Or field is radial and/or not uniform

(1 mark)

(10 marks Total)