Work Done on a Mass (AQA A Level Physics): Revision Note

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)

Change in gravitational potential energy

• Two points at different distances from a mass will have different gravitational potentials

  • This is because the gravitational potential increases with distance from a mass

• Therefore, there will be a gravitational potential difference $∆V$ between the two points

$∆V=V_f-V_i$

• Where:

  • $V_i$ = initial gravitational potential (J kg^–1)

  • $V_f$ = final gravitational potential (J kg^–1)

• The change in work done against a gravitational field is equal to the change in gravitational potential energy (GPE)

  • When $V=0$, then the GPE = 0

• It is usually more useful to find the change in the GPE of a system

  • For example, a satellite lifted into space from the Earth’s surface

• The change in GPE when a mass moves towards, or away from, another mass is given by:

$∆GPE=-\frac{GMm}{r_2}-\left(-\frac{GMm}{r_1}\right)$

$∆GPE=GMm\left(\frac{1}{r_1}-\frac{1}{r_2}\right)$

• Where:

  • $M$ = mass that is producing the gravitational field (e.g. a planet) (kg)

  • $m$ = mass that is moving in the gravitational field (e.g. a satellite) (kg)

  • $r_1$ = first distance of $m$ from the centre of $M$ (m)

  • $r_2$ = second distance of $m$ from the centre of $M$ (m)

• The change in potential $∆V$ is the same, without the mass of the object $m$:

$∆V=-\frac{GM}{r_2}-\left(-\frac{GM}{r_1}\right)$

$∆V=GM\left(\frac{1}{r_1}-\frac{1}{r_2}\right)$

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

Gravitational equipotential surfaces

• 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 (e.g. a planet), the equipotential lines:

  • are concentric circles around the planet

  • can become further apart as they move further away from the planet 

• In a radial field, equipotential lines can become further apart as they move further away from the planet because:

  • potential increases with distance away 

  • the gravitational field gets weaker with distance away

  • a greater distance needs to be moved to obtain the same change in potential, $∆V$

• 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 $∆V=0$