define gravitational potential at a point as the work done per unit mass in bringing a small test mass from infinity to the point

Gravitational Fields, Potential & Potential Energy

Learning Objectives

• Define the gravitational field g and the gravitational potential φ at a point.
• Derive the expressions g = GM/r² and φ = –GM/r from Newton’s law of gravitation.
• Relate potential energy U to potential ( U = mφ ) and to the field ( U = –∫ g·dr ).
• Apply these concepts to calculate escape velocity, orbital speed and the energy of a satellite.
• Choose an arbitrary reference point for potential and discuss the effect on sign conventions.
• Understand a simple experimental method for measuring g and estimating φ.

Key Formulae

1. Gravitational Field

• Definition: The gravitational field g at a point is the force per unit test mass, \(\mathbf g = \mathbf F/m\).
• From Newton’s law of gravitation, \(\displaystyle \mathbf F = -\frac{GMm}{r^{2}}\hat{\mathbf r}\), giving

  \(\displaystyle \mathbf g(r)= -\frac{GM}{r^{2}}\hat{\mathbf r}\).

• Field lines point toward the mass; the magnitude decreases as \(1/r^{2}\).
• Near the Earth’s surface the field is approximately constant:

  \(\displaystyle g_{\oplus}\approx 9.81\ \text{m s}^{-2}\).

2. Gravitational Potential

Definition: The gravitational potential φ at a point is the work done per unit mass in bringing a small test mass from infinity to that point (the zero of potential is chosen at ∞).

Derivation of \(\phi = -GM/r\)

1. Force on a test mass m at distance r: \(F = GMm/r^{2}\) (directed inward).
2. The external agent must apply an opposite force; the infinitesimal work required is

   \(dW = -F\,dr = -\dfrac{GMm}{r^{2}}\,dr\).

3. Integrate from \(r=\infty\) (where \(W=0\)) to the final distance r:

   \[

   W = -\int_{\infty}^{r}\frac{GMm}{r^{2}}\,dr

   = -GMm\Bigl[-\frac{1}{r}\Bigr]_{\infty}^{r}

   = -\frac{GMm}{r}.

   \]

4. Potential is work per unit mass:

   \[

   \phi(r)=\frac{W}{m}= -\frac{GM}{r}.

   \]

Key Points

• φ is a scalar (J kg⁻¹); it can be added algebraically.
• Because the zero is at infinity, φ is negative for isolated masses. Choosing a different reference (e.g., the Earth’s surface) can make φ positive.

3. Gravitational Potential Energy

• For a test mass m, \(U = m\phi = -\dfrac{GMm}{r}\) (J).
• It is the negative of the work done by the external agent in moving the mass from infinity to r.
• Relation to the field:

  \[

  U = -\int_{\infty}^{r}\mathbf g\cdot d\mathbf r.

  \]

4. Applications

4.1 Escape Velocity

To escape to infinity the kinetic energy must equal the magnitude of the gravitational potential energy at the surface:

\[

\frac12 mv_{\text{esc}}^{2}= \frac{GMm}{R}\;\Longrightarrow\;

v_{\text{esc}}=\sqrt{\frac{2GM}{R}}.

\]

4.2 Circular Orbital Speed

For a satellite in a circular orbit of radius r, the required centripetal acceleration is provided by the gravitational field:

\[

\frac{v_{\text{orb}}^{2}}{r}= \frac{GM}{r^{2}}

\;\Longrightarrow\;

v_{\text{orb}}=\sqrt{\frac{GM}{r}}.

\]

4.3 Energy of a Circular Orbit

Kinetic energy \(K = \tfrac12 mv_{\text{orb}}^{2}= \dfrac{GMm}{2r}\).

Total mechanical energy:

\[

E_{\text{tot}} = K + U = \frac{GMm}{2r} - \frac{GMm}{r}= -\frac{GMm}{2r}.

\]

The negative sign shows the orbit is a bound state.

5. Reference‑Point Flexibility

• The syllabus states that *any* point may be chosen as the zero of potential.
• If the zero is taken at the Earth’s surface (\(r=R_{\oplus}\)), the potential becomes

  \(\phi' = -\dfrac{GM}{r}+ \dfrac{GM}{R_{\oplus}}\), which is zero at the surface and positive above it.

• Only differences \(\Delta\phi\) matter for work; changing the reference adds a constant to every potential value but does not affect physical predictions.

6. Experimental Determination of g and φ

A simple classroom experiment:

1. Measure the period T of a simple pendulum of length L. The small‑angle approximation gives

   \(\displaystyle g = \frac{4\pi^{2}L}{T^{2}}\).

2. Integrate the measured field to obtain the potential difference between two heights h₁ and h₂:

   \[

   \Delta\phi = -\int{h{1}}^{h{2}} g\,dh \approx -g\,(h{2}-h_{1}).

   \]

3. Choose the lower height as the reference (φ = 0) and calculate φ at the higher point.

This satisfies the A‑Level requirement for experimental skills (AO3) and links the measured field to the concept of potential.

7. Symbol Table

8. Example – Potential at the Earth’s Surface

Given:

• Earth’s radius \(R_{\oplus}=6.37\times10^{6}\ \text{m}\)
• Earth’s mass \(M_{\oplus}=5.97\times10^{24}\ \text{kg}\)

Potential (zero at ∞):

\[

\phi{\oplus}= -\frac{GM{\oplus}}{R_{\oplus}}

\approx -\frac{6.674\times10^{-11}\times5.97\times10^{24}}{6.37\times10^{6}}

\approx -6.3\times10^{7}\ \text{J kg}^{-1}.

\]

Potential energy of a 1 kg mass at the surface:

\[

U = m\phi_{\oplus} \approx -6.3\times10^{7}\ \text{J}.

\]

Escape velocity from the Earth:

\[

v{\text{esc}} = \sqrt{\frac{2GM{\oplus}}{R_{\oplus}}}

\approx 11.2\ \text{km s}^{-1}.

\]

9. Common Misconceptions

1. “Potential is always negative.” Only true when the zero is at infinity. A different reference can give positive values.
2. “Potential = potential energy.” φ is energy per unit mass (J kg⁻¹); U = mφ (J).
3. “Field and potential are the same thing.” The field is the spatial gradient of the potential: \(\mathbf g = -\nabla\phi\). One is a vector, the other a scalar.
4. “Potentials of several bodies do not add.” Because φ is a scalar, the total potential is the algebraic sum of the individual contributions.
5. “Only the magnitude of g matters for work.” Work depends on the component of g along the displacement; the sign (direction) is crucial.

10. Summary

The gravitational field g describes the force per unit mass, while the gravitational potential φ describes the work per unit mass required to bring a test mass from infinity to a point. For a point mass M:

• \(\displaystyle \mathbf g = -\frac{GM}{r^{2}}\hat{\mathbf r}\)
• \(\displaystyle \phi = -\frac{GM}{r}\)
• \(\displaystyle U = m\phi = -\frac{GMm}{r}\)

These quantities are linked by \(\mathbf g = -\nabla\phi\). Choosing a convenient reference point for potential does not affect physical results, only the numerical value of φ. The concepts are applied to escape velocity, orbital motion, and energy calculations—core topics for Cambridge AS & A‑Level Physics.