7.2.4 Work Done on a Mass

• When a mass is moved against the force of gravity, work is done
• The work done in moving a mass m is given by:

∆W = m∆V

• Where:
  • ∆W = change in work done (J)
  • m = mass (kg)
  • ∆V = change in gravitational potential (J kg^-1)

• This change in work done is equal to the change in gravitational potential energy (G.P.E)
  • When V = 0, then the G.P.E = 0

• The change in G.P.E, or work done, for an object of mass m at a distance r₁ from the centre of a larger mass M, to a distance of r₂ further away can be written as:

• Where:
  • M = mass that is producing the gravitational field (eg. a planet) (kg)
  • m = mass that is moving in the gravitational field (eg. a satellite) (kg)
  • r₁ = first distance of m from the centre of M (m)
  • r₂ = second distance of m from the centre of M (m)

• Work is done when an object in a planet's gravitational field moves against the gravitational field lines ie. away from the planet

Gravitational potential energy increases as a satellite leaves the surface of the Moon

Maths Tip

• Multiplying two negative numbers equals a positive number, for example:

Step 1:            Write down the work done (or change in G.P.E) equation

Step 2:            Determine values for r₁ and r₂

r₁ is the radius of Mars = 3400 km = 3400 × 10³ m

r₂ is the radius + altitude = 3400 + 700 = 4100 km = 4100 × 10³ m

Step 3:            Substitute in values

ΔG.P.E =  643.076 × 10⁶  = 640 MJ (2 s.f.)

• Equipotential lines (2D) and surfaces (3D) join together points that have the same gravitational potential
• These are always:
  • Perpendicular to the gravitational field lines in both radial and uniform fields
  • Represented by dotted lines (unlike field lines, which are solid lines with arrows)

• In a radial field (eg. a planet), the equipotential lines:
  • Are concentric circles around the planet
  • Become further apart further away from the planet

• In a uniform field (eg. near the Earth's surface), the equipotential lines are:
  • Horizontal straight lines
  • Parallel
  • Equally spaced

• No work is done when moving along an equipotential line or surface, only between equipotential lines or surfaces
  • This means that an object travelling along an equipotential doesn't lose or gain energy and ΔV =  0

Gravitational equipotential lines in a non-uniform and uniform gravitational field