A-level Physics (Advancing Physics)/Gravitational Potential

Gravitational potential is the gravitational potential energy given to objects per unit mass:

$V_{grav}={\frac {E_{grav}}{m}}={\frac {-{\frac {GMm}{r}}}{m}}={\frac {-GM}{r}}$

For short changes in distance Δh (Δh << r), gravitational potential energy is given by:

$E_{grav}\approx mg\Delta h$,

where

$g={\frac {-GM}{r^{2}}}$

So, for short changes of distance, gravitational potential is equal to gΔh. Gravitational potential is measured in Jkg⁻¹.

Equipotentials

On a field diagram, lines can be drawn which, like contours on a map, show all the points which have the same gravitational potential. These lines are known as equipotentials. Equipotentials are always perpendicular to field lines, and get closer together as the field strength increases, and the density of field lines increases. Over short distances, equipotentials are evenly spaced.

Summary of Gravity

You should now know (if you did the gravity section in the right order) about four attributes of gravitational fields: force, field strength, potential energy and potential. These can be summarised by the following table:

Force $F_{grav}={\frac {-GMm}{r^{2}}}$	→ integrate → with respect to r and × by −1	Potential Energy $E_{grav}={\frac {-GMm}{r}}$
↓ per unit mass ↓
Field Strength $g={\frac {-GM}{r^{2}}}$	→ integrate → with respect to r and × by −1	Potential $V_{grav}={\frac {-GM}{r}}$

Questions

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)

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

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?

4. Express gravitational potential in terms of gravitational force.

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

Worked Solutions