Edexcel A Level (IAL) Physics-5.34 Gravitational Potential for a Radial Field- Study Notes- New Syllabus

Gravitational Potential in a Radial Gravitational Field

Gravitational potential describes the energy per unit mass at a point in a gravitational field. For a radial gravitational field (around a point mass), gravitational potential depends on distance from the mass.

Definition of Gravitational Potential

Gravitational potential \( V_{\text{grav}} \) at a point is defined as:

• The work done per unit mass in bringing a small test mass from infinity to that point.
• No change in kinetic energy occurs during this process.

Reference point: Gravitational potential is taken as zero at infinity.

Gravitational Potential Due to a Point Mass

For a radial gravitational field produced by a point mass \( m \), the gravitational potential is:

\( V_{\text{grav}} = \dfrac{-Gm}{r} \)

• \( V_{\text{grav}} \) = gravitational potential (J kg⁻¹)
• \( G \) = gravitational constant
• \( m \) = mass producing the field (kg)
• \( r \) = distance from the centre of the mass (m)

 Meaning of the Negative Sign

• The gravitational force is attractive.
• Work must be done to move a mass away from the field.
• Potential increases (becomes less negative) as distance increases.

Key idea: At infinity, \( V_{\text{grav}} = 0 \).

 Variation of Gravitational Potential with Distance

• Gravitational potential increases with distance from the mass.
• As \( r \rightarrow \infty \), \( V_{\text{grav}} \rightarrow 0 \).
• The graph of \( V_{\text{grav}} \) against \( r \) is a curve, not a straight line.

Important distinction:

• Gravitational potential is a scalar quantity.
• It has no direction.

 Relation to Gravitational Field Strength

Gravitational field strength is related to the gradient of gravitational potential:

\( g = -\dfrac{dV_{\text{grav}}}{dr} \)

Using \( V_{\text{grav}} = \dfrac{-Gm}{r} \):

\( g = \dfrac{Gm}{r^2} \)

This confirms consistency with Newton’s law of gravitation.

Using the Equation in Calculations

• Ensure distance is measured from the centre of the mass.
• Use SI units throughout.
• Remember potential values are negative.

Rearranged forms:

\( m = \dfrac{-V_{\text{grav}} r}{G} \)

\( r = \dfrac{-Gm}{V_{\text{grav}}} \)

Exam Tips

• Do not confuse gravitational potential with potential energy.
• Always mention “per unit mass”.
• Explain the negative sign physically.
• State reference point clearly (infinity).