7.2.3

# Gravitational Potential

Test yourself

## Gravitational Potential

Understanding gravitational potential lets us consider the energy changes that objects moving in gravitational fields experience.

### Zero value

• The zero value of gravitational potentials is defined as being at an infinite distance away from the mass generating the field.
• This is done so that when a small test mass is placed in the gravitational field, the test mass has work done on it by the original mass and gains kinetic energy (KE).
• To gain that KE, the object must have lost gravitational potential energy (GPE). In other words, the gravitational potential has become more negative.

### Example

• Imagine a rubber sheet with a heavy object in the middle.
• The sheet is deformed downwards. The undeflected sheet is defined to be at zero potential.
• You can find the gravitational potential difference by subtracting the initial gravitational potential from the final gravitational potential.
• E.g. In the diagram, the object has dropped from a potential of -30 J/kg to -50 J/kg.
• The change is -50 - (-30) = -20 J/kg.
• There has a been a drop in potential of 20 J/kg.

### Work done on a moving mass

• The work done on a moving mass in that gravitational potential can be found by multiplying the change in potential by the mass.
• On the previous slide, if the mass of the object was 2.0 kg, then the work done on the mass would be 2.0 × 20 = 40 J.
• We could say that the 2.0 kg mass has lost 40 J of GPE, or that the large mass has done 40 J of work on the mass.

### Work done 2

• If an object moves along an equipotential surface, then the potential has not changed and no work has been done on the object.
• This is like satellites in circular orbit around a planet.
• The planet does no work on the satellites. So there is no loss in potential, and no loss in GPE (the radius of the orbit does not change). This means the satellite does not change KE (and so has constant speed).

## Gravitational Potential Graphs

Gravitational potential graphs, gravitational field strengths and changes in gravitational potentials are all inter-linked.

### Gravitational potential

• The gravitational potential surrounding a planet or point mass is given by the formula $V = -\frac{GM}{r}$.
• The minus sign is important because it signifies an attractive force.

### Gravitational potential

• The graph shows the variation of gravitational potential with distance from a point mass.

### Gravitational field strength

• The graph shows the variation of gravitational field strength with distance from a point mass.

• The -1 × (gradient of the gravitational potential) graph gives the variation of the gravitational field strength with distance.
• In a similar way, the gradient of a displacement-time graph gives the velocity-time graph.

### Area under graph

• The reverse operation (i.e. finding the area between the curve and the distance axis) for a gravitational field strength against distance graph will give the change in potential.