Gravitational Potential Energy Equation (DP IB Physics: HL): Revision Note

Work Done on a Mass

• When a mass is moved against the force of gravity, work is required

  • This is because gravity is attractive, therefore, energy is needed to work against this attractive force

• The work done in moving a mass m is given by:

$∆W=m∆V_g$

• Where:

  • ΔW = change in work done (J)

  • m = mass (kg)

  • ΔV_g = change in gravitational potential (J kg⁻¹)

Gravitational Potential Energy Equation

• In a radial field, gravitational potential energy (GPE) describes the energy an object possesses due to its position in a gravitational field

• The gravitational potential energy of a system is defined as:

The work done to assemble the system from infinite separation of the components of the system

• Similarly, the gravitational potential energy of a point mass is defined as:

The work done in bringing a mass from infinity to a point

• The equation for GPE of two point masses m and M at a distance r is:

$E_p=-\frac{G{m}_1{m}_2}{r}$

• Where:

  • G = universal gravitational constant (N m² kg^–2)

  • m₁ = larger mass producing the field (kg)

  • m₂ = mass moving within the field of M (kg)

  • r = distance between the centre of m and M (m)

Gravitational potential energy increases as a satellite leaves the surface of the Moon (of mass M)

• Recall that Newton's Law of Gravitation relates the magnitude of the force F between two masses M and m:

$F=\frac{G{m}_1{m}_2}{r^2}$

• Therefore, a force-distance graph would be a curve, because F is inversely proportional to r², or:

$F\propto \frac{1}{r^2}$ 

• The product of force and distance is equal to work done (or energy transferred) 

• Therefore, the area under the force-distance graph for gravitational fields is equal to the work done

  • In the case of a mass m moving further away from a mass M, the potential increases

  • Since gravity is attractive, this requires work to be done on the mass m

  • The area between two points under the force-distance curve, therefore, gives the change in gravitational potential energy of mass m

Work is done on the satellite of mass m to move it from A to B, because gravity is attractive. The area under the curve represents the magnitude of energy transferred

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 ΔV 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:

$∆E_p=-\frac{G{m}_1{m}_2}{r_2}-\left(-\frac{G{m}_1{m}_2}{r_1}\right)$

$∆E_p=G{m}_1{m}_2\left(\frac{1}{r_1}-\frac{1}{r_2}\right)$

• Where:

  • m₁ = mass that is producing the gravitational field (e.g. a planet) (kg)

  • m₂ = mass that is moving in the gravitational field (e.g. a satellite) (kg)

  • r₁ = first distance of m₂ from the centre of m₁ (m)

  • r₂ = second distance of m₂ from the centre of m₁ (m)

• The change in potential ΔV_g is the same, without the mass of the object m₂:

$∆V_g=-\frac{G{m}_1}{r_2}-\left(-\frac{G{m}_1}{r_1}\right)$

$∆V_g=G{m}_1\left(\frac{1}{r_1}-\frac{1}{r_2}\right)$

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