Gravitational potential — practice questions

1. First, calculate the radial distance 'r' from Earth's centre: r = R + altitude = $6.37 \times 10^6 m$+ $600 \times 10^3 m$= $6.97 \times 10^6 m$.
2. Next, calculate the gravitational potential ($\phi$): $\phi = -\frac{GM}{r} = -\frac{(6.67 \times 10^{-11} \text{ N m}^2 \text{kg}^{-2}) \times (5.97 \times 10^{24} \text{ kg})}{6.97 \times 10^6 \text{ m}}$.
3. $\phi = -5.71 \times 10^7 \text{ J kg}^{-1} (to 3 s.f.)$.
4. Finally, calculate the gravitational potential energy (E_p) of the satellite: $E_p = m\phi = (2.5 \text{ kg}) \times (-5.71 \times 10^7 \text{ J kg}^{-1})$.
5. $E_p = -1.43 \times 10^8 \text{ J (to 3 s.f.).}$

The work done on the satellite is equal to the change in its gravitational potential energy (ΔE_p).

1. Calculate the initial radial distance (r_1): r_1 = Earth's radius + initial altitude r_1 = (6.37 × 10^6 m) + (500 × 10^3 m) = 6.87 × 10^6 m.
2. Calculate the final radial distance (r_2): r_2 = Earth's radius + final altitude r_2 = (6.37 × 10^6 m) + (35,786 × 10^3 m) = 4.2156 × 10^7 m.
3. Calculate the initial gravitational potential energy (E_p1): E_p1 = -GMm / r_1 E_p1 = - (6.67 × 10^-11 × 5.97 × 10^24 × 1200) / (6.87 × 10^6) E_p1 = -6.958 × 10^10 J.
4. Calculate the final gravitational potential energy (E_p2): E_p2 = -GMm / r_2 E_p2 = - (6.67 × 10^-11 × 5.97 × 10^24 × 1200) / (4.2156 × 10^7) E_p2 = -1.132 × 10^10 J.
5. Calculate the work done (change in GPE): Work Done = ΔE_p = E_p2 - E_p1 Work Done = (-1.132 × 10^10 J) - (-6.958 × 10^10 J) Work Done = 5.826 × 10^10 J.

The work done to move the satellite to the higher orbit is 5.83 × 10^10 J (to 3 s.f.). Note that this does not include the change in kinetic energy required for the orbit.