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Out Into Space

Gravitational potential energy in a radial field

  • 3 things to know
  • Components 01, 02 and 03

Overview

Recall questions on E = −GMm/r and why it is negative, the zero at infinite separation, how potential energy changes with distance, satellite and spacecraft calculations, constant energy in a circular orbit, when mgh is enough, escape velocity v = √(2GM/r) and why it does not depend on mass, negative total energy in a bound orbit, and the energy–distance graph.

What you need to know

Answer each one before you reveal it.

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Key term

Why is gravitational potential energy defined as negative?

Key term

Why can mgh be used near the surface while −GMm/r is needed for a rocket leaving the Earth?

Key term

What is the gravitational potential energy of a mass m at a distance r from a mass M?

Questions from this subtopic

Each one with the model answer underneath.

  1. 1

    What is the gravitational potential energy of a mass m at a distance r from a mass M?

    Model answer

    The energy is E = −GMm/r.

  2. 2

    Why is gravitational potential energy defined as negative?

    Model answer

    The zero is taken at infinite separation, and work must be done against the attraction to reach infinity, so the energy at any finite separation is less than zero.

  3. 3

    What is the gravitational potential energy of two masses infinitely far apart?

    Model answer

    It is zero, which is the chosen reference level.

  4. 4

    How does the gravitational potential energy of a satellite change as it moves further from a planet?

    Model answer

    It increases towards zero, in other words it becomes less negative.

  5. 5

    A 1200 kg satellite is 4.22 × 10⁷ m from the Earth's centre, where GM is 3.98 × 10¹⁴ N m² kg⁻¹. What is its gravitational potential energy?

    Model answer

    E = −GMm/r = −3.98 × 10¹⁴ × 1200 / 4.22 × 10⁷ = −1.13 × 10¹⁰ J.

  6. 6

    A 2000 kg spacecraft moves from 2.0 × 10⁷ m to 4.0 × 10⁷ m from the Earth's centre, where GM is 3.98 × 10¹⁴ N m² kg⁻¹. What is the change in its gravitational potential energy?

    Model answer

    The energy rises from −3.98 × 10¹⁰ J to −1.99 × 10¹⁰ J, an increase of 1.99 × 10¹⁰ J.

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