Section 2.2: Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy a mass has due to its position in a gravitational field. For a mass \( m \) at a distance \( r \) from the center of a planet of mass \( M \), the gravitational potential energy is:

Gravitational Potential Energy:

\( U = - G \frac{M m}{r} \)

  • \( U \) is the gravitational potential energy
  • \( G \) is the universal gravitational constant \( 6.674 \times 10^{-11} \text{ N·m²/kg²} \)
  • \( M \) is the mass of the planet
  • \( m \) is the mass of the object
  • \( r \) is the distance from the center of the planet
  • Negative sign indicates the energy is bound within the gravitational field

Example 1

Find the gravitational potential energy of a 500 kg satellite orbiting Earth at 400 km above the surface.

Distance from Earth's center: \( r = 6371 + 400 = 6771 \text{ km} = 6.771 \times 10^6 \text{ m} \)

\( U = - G \frac{M m}{r} = - 6.674 \times 10^{-11} \frac{5.97\times10^{24} * 500}{6.771\times10^6} \approx -2.94 \times 10^{10} \text{ J} \)

Example 2

Calculate the change in gravitational potential energy for a 1000 kg object moved from Earth's surface to 1000 km altitude.

Distance from center at surface: \( r_1 = 6371 \text{ km} = 6.371 \times 10^6 \text{ m} \)

Distance at altitude: \( r_2 = 6371 + 1000 = 7371 \text{ km} = 7.371 \times 10^6 \text{ m} \)

\( \Delta U = U_2 - U_1 = -G M m \left( \frac{1}{r_2} - \frac{1}{r_1} \right) \)

\( \Delta U = -6.674\times10^{-11} * 5.97\times10^{24} * 1000 * (1/7.371\times10^6 - 1/6.371\times10^6) \approx 8.26 \times 10^9 \text{ J} \)

Practice Problems

  1. Calculate the gravitational potential energy of a 200 kg satellite 300 km above Earth’s surface.
  2. A 1000 kg spacecraft is at 2000 km from Earth's center. Find its GPE.
  3. Change in GPE of a 500 kg object from surface to 1000 km altitude.
  4. Two masses, 1000 kg and 5000 kg, separated by 2×10^6 m. Compute GPE.
  5. Gravitational potential energy of 50 kg mass on top of a 100 m high tower.
  6. Satellite of mass 800 kg at 400 km altitude. Compute GPE.
  7. Change in GPE of 1500 kg object from 400 km to 800 km above Earth.
  8. Calculate GPE for a 1000 kg object on the Moon's surface (mass 7.35×10^22 kg, radius 1.737×10^6 m).
  9. Object of mass 500 kg moves from Earth's surface to 500 km altitude. Determine ΔU.
  10. A 100 kg mass is 2×10^7 m from a planet of mass 6×10^24 kg. Compute GPE.
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