A-level Physics (Advancing Physics)/Gravitational Potential/Worked Solutions

G = 6.67 x 10⁻¹¹ m³kg⁻¹s⁻²

g = 9.81 ms⁻²

1. What is the gravitational potential at the Earth's surface? (mass of Earth = 5.97 x 10²⁴ kg,radius of Earth = 6371 km)

$V_{grav}={\frac {-GM}{r}}={\frac {-6.67\times 10^{-11}\times 5.97\times 10^{24}{MJkg}^{-1}}{6371000}}=62.5{\mbox{ MJkg}}^{-1}$

2. Taking the Earth's surface as V_grav = 0, what is the gravitational potential 2m above the ground?

$V_{grav}\approx g\Delta h=9.81\times 2=19.62{\mbox{ Jkg}}^{-1}$

3. A 0.2 kg firework reaches a gravitational potential relative to the ground of 500Jkg⁻¹. If the firework is 30% efficient, how much energy was expended to get there?

$V_{grav}={\frac {E_{grav}}{m}}$

$E_{grav}=mV_{grav}=0.2\times 500=100{\mbox{ J}}$

However, this is only 30% of the energy expended, so:

$E_{expended}={\frac {100}{0.3}}\approx 333{\mbox{ J}}$

4. Express gravitational potential in terms of gravitational force.

$V_{grav}={\frac {\int {F_{grav}}\;dr}{m}}$

5. Draw the equipotentials and field lines surrounding the Earth.