Physics (9630) | Gravitational potential

🌍 Gravitational Potential: Your Guide to Energy in the Cosmos

Hello! Welcome to one of the most abstract but essential topics in A-level Physics: Gravitational Potential. Don't worry if this feels tricky—it deals with energy and fields on a massive scale (literally!).

We are moving beyond the simple \(E_p = mgh\) you learned previously. That formula only works near Earth's surface where the field is uniform. In this chapter, we explore how energy works in radial fields, like those around planets, where the field strength constantly changes.

Why is this important? Understanding gravitational potential is key to calculating how much energy is needed to launch a satellite, or how fast objects move when falling from great distances.

1. Defining Gravitational Potential (\(V\))

When dealing with vast gravitational fields, we need a standard way to measure the energy stored at any point in space. We use Gravitational Potential.

What is Gravitational Potential?

The Gravitational Potential (\(V\)) at a point in a gravitational field is defined as the work done per unit mass required to move a small test mass from infinity to that point.

• Symbol: \(V\)
• Unit: Joules per kilogram (\(\text{J kg}^{-1}\))
• Analogy: Think of it like a price tag for that location. If \(V = -50 \text{ J kg}^{-1}\), it means it took \(50 \text{ J}\) of work to move every \(1 \text{ kg}\) of mass into that location from space.

Key Takeaway: Potential is essentially the GPE calculated per kilogram. It tells you how much energy a unit of mass *would* have at that location.

2. The Crucial Zero Point: Infinity

This is often the most confusing part! Unlike \(E_p = mgh\) where \(h=0\) (zero potential) is usually the ground, in A-level Physics we define the zero point for gravitational potential differently.

Why is Zero Potential at Infinity?

We define the zero of gravitational potential to be at infinity ($r = \infty$).

• When a mass is infinitely far away from a planet, the gravitational force acting on it is zero (remember \(F \propto 1/r^2\)).
• If the force is zero, no work is done on the mass, so the potential energy is logically zero.
• By defining \(V = 0\) at infinity, we establish a fixed, universal reference point.

The Negative Sign Convention

When a mass moves from infinity (\(V=0\)) towards a planet, the gravitational field does work on the mass (it pulls it in). Because energy is being given up by the field, the mass is now in an energy deficit.

Therefore, gravitational potential (\(V\)) is always negative inside a gravitational field.

Memory Aid: "If you are stuck inside a gravitational well, you have a negative potential because you need to spend positive energy (do positive work) to get back out to freedom (zero potential)."

3. Work Done and Potential Difference

The relationship between potential difference and work done is fundamental. Since potential is work done per unit mass, if you want to find the total work done (\(\Delta W\)) for a mass (\(m\)), you just multiply:

Equation for Work Done

Work done in moving a mass \(m\) through a potential difference \(\Delta V\):

\[\Delta W = m \Delta V\]

Where:

• \(\Delta W\) is the work done (or energy transferred) in Joules (J).
• \(m\) is the mass being moved (kg).
• \(\Delta V\) is the potential difference between the start and end points (\(\text{J kg}^{-1}\)).

Example: If a satellite of mass \(100 \text{ kg}\) moves from a potential of \(-50 \text{ MJ kg}^{-1}\) to \(-40 \text{ MJ kg}^{-1}\) (moving further from the Earth), the change in potential \(\Delta V\) is \(-40 - (-50) = +10 \text{ MJ kg}^{-1}\). The work done on the satellite is \(\Delta W = 100 \text{ kg} \times 10 \text{ MJ kg}^{-1} = 1000 \text{ MJ}\). This is the energy required to lift it.

Common Mistake to Avoid: \(\Delta V\) is always calculated as \(V_{final} - V_{initial}\). If the result is positive, positive work was done (energy supplied). If negative, the field did the work (energy released).

4. The Gravitational Potential in a Radial Field

For a point mass \(M\) (or a large spherical mass like a planet), the gravitational potential \(V\) at a distance \(r\) from the centre is given by:

The Radial Field Formula

\[V = -\frac{GM}{r}\]

Where:

• \(V\) is the gravitational potential (\(\text{J kg}^{-1}\)).
• \(G\) is the Gravitational Constant (\(6.67 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}\)).
• \(M\) is the mass creating the field (kg).
• \(r\) is the distance from the centre of the mass (m).

Notice that the potential is inversely proportional to $r$ (\(V \propto 1/r\)), and crucially, it retains the negative sign.

Graphical Representations of \(V\) and \(g\)

It is vital to be able to sketch and interpret the graphs showing how gravitational field strength (\(g\)) and potential (\(V\)) vary with distance (\(r\)).

A. Gravitational Field Strength (\(g\)) vs. \(r\)

Recall that \(g = \frac{GM}{r^2}\).

• \(g\) is always positive (representing the magnitude of the force inwards).
• It follows an inverse square law (\(g \propto 1/r^2\)).
• It decreases rapidly as \(r\) increases.

B. Gravitational Potential (\(V\)) vs. \(r\)

Recall that \(V = -\frac{GM}{r}\).

• \(V\) is always negative.
• It follows an inverse relationship (\(V \propto 1/r\)).
• The curve approaches the \(V=0\) axis (infinity) asymptotically, but does so less steeply than the \(g\) curve.

Key Takeaway: \(V\) is always negative and gets closer to zero (less negative) as \(r\) increases.

5. Equipotential Surfaces

Since potential is energy per unit mass, points that have the same potential must have the same energy state.

Definition and Significance

An Equipotential Surface is a surface connecting all points in a gravitational field that have the same gravitational potential (\(V\)).

• For a spherical mass (like a planet), the equipotential surfaces are concentric spheres surrounding the mass.
• Analogy: Equipotential lines are like contour lines on a topographic map. They show regions of constant height (constant GPE). The closer the lines, the steeper the slope (the stronger the field).

Work Done on Equipotentials

Moving a mass along an equipotential surface requires zero work.

Why? Because if you move from point A to point B on the same surface, \(V_A = V_B\). Therefore, the potential difference \(\Delta V = 0\). Since \(\Delta W = m \Delta V\), the work done \(\Delta W\) must also be zero.

Important Relationship: Equipotential surfaces are always perpendicular to the gravitational field lines. Field lines show the direction of force (the steepest downhill path), and equipotentials cross this path at 90 degrees.

6. Linking Potential and Field Strength

There is a direct mathematical link between the gravitational potential (\(V\)) and the gravitational field strength (\(g\)).

Potential Gradient

The gravitational field strength \(g\) is equal to the negative of the potential gradient.

The potential gradient describes how steeply the potential changes with distance.

\[g = -\frac{\Delta V}{\Delta r}\]

(For small changes, \(\Delta V / \Delta r\) is mathematically equivalent to the gradient of the V-r graph, \(dV/dr\)).

Understanding the Negative Sign

The negative sign is crucial:

• Gravitational field strength (\(g\)) is a vector pointing inwards (towards the mass).
• The potential (\(V\)) gets less negative as \(r\) increases, meaning the gradient \((\Delta V / \Delta r)\) is always positive.
• To make the inwards pointing vector \(g\) correct, we must multiply the positive gradient by negative one.

In simple terms: The field points in the direction where the potential is decreasing most rapidly (becoming more negative).

Don't forget! This means that if you are given a graph of \(V\) against \(r\), the field strength \(g\) at any point is simply the negative gradient of that curve.

FINAL KEY TAKEAWAY: Gravitational Potential is a scalar quantity (just a number, defined relative to infinity) that governs the energy in the field. Gravitational Field Strength is a vector quantity (magnitude and direction) that describes the force caused by the field. They are linked by the potential gradient.