Gravitational Potential (φ)
What is Gravitational Potential?
Imagine you are at the bottom of a hill and you want to roll a ball to the top. You have to do work, or use energy, to push the ball upwards against gravity. The higher you push the ball, the more work you do. Gravitational potential is a very similar idea, but it's a property of the location itself, not the object you place there.
Formally, the gravitational potential (φ) at a point in a gravitational field is defined as the work done by an external agent in bringing a unit mass from infinity to that point, without any acceleration. Let's break this down:
- Work Done: This is the energy transferred. In this case, it's the energy needed to move the mass against the gravitational pull.
- Per Unit Mass: We divide the total work done by the mass of the object. This gives us a value that only depends on the location, not on the object we bring there. It's like finding the "cost" of moving one kilogram to that spot.
- From Infinity: We use infinity as our starting point because it's a place where the gravitational force is effectively zero. This gives us a consistent reference point of zero potential.
- Without Acceleration: This means we move the mass infinitely slowly, so all the work goes into overcoming gravity, and none into giving the object kinetic energy (speed).
Formula for a Point Mass: For a simple object like a planet or star (treated as a point mass M), the gravitational potential at a distance r from its center is given by:
$ \phi = -\frac{GM}{r} $
Where:
- φ is the gravitational potential (in J/kg).
- G is the Universal Gravitational Constant[1] (6.67430 × 10-11 N⋅m²/kg²).
- M is the mass creating the field (in kg).
- r is the distance from the center of the mass (in m).
The negative sign is crucial! It indicates that work is done by the field as an object falls in, and work must be done against the field to move an object out.
Why is Gravitational Potential Always Negative?
This is one of the most important and often confusing aspects of gravitational potential. The key is to remember our reference point: zero potential is defined at infinity.
Think of it like this: To bring a mass from infinity (where gravity is zero) to a point near a planet, gravity is pulling the mass inward. You, as the external agent, must resist this pull. However, if you let go, the gravitational field itself does the work and pulls the mass in. Because the field is doing the work to bring the mass in, the potential energy of the mass-mass system decreases. Since we set the potential at infinity to be zero, any point where the mass is bound by gravity must have a potential less than zero, hence it is negative.
An analogy is a well. The ground level is like infinity (zero potential). As you dig a well, the bottom of the well is at a negative depth relative to the ground. An object at the bottom of the well has less potential energy than an object on the ground. Similarly, any object in a gravitational field is "in the well" and has a negative potential.
Gravitational Potential vs. Gravitational Potential Energy
It's easy to mix up these two related ideas. The table below clarifies the difference.
| Aspect | Gravitational Potential (φ) | Gravitational Potential Energy (U or GPE) |
|---|---|---|
| Definition | Work done per unit mass. | Total work done on a specific mass. |
| Symbol & Unit | φ (Joules per kilogram, J/kg) | U or GPE (Joules, J) |
| Dependence | Depends only on the source mass and position. | Depends on the source mass, the test mass, and position. |
| Formula (Point Mass) | $ \phi = -\frac{GM}{r} $ | $ U = m\phi = -\frac{GMm}{r} $ |
| Analogy | The height of a location on a hill. | The potential energy of a specific object at that height (mgh). |
Applying Gravitational Potential: Orbits and Escape Velocity
Gravitational potential isn't just a theoretical idea; it has powerful practical applications, especially in understanding how objects move in space.
Example 1: Calculating the Energy of a Satellite
Let's calculate the gravitational potential and potential energy for a 500 kg satellite orbiting Earth (M = 5.97 × 1024 kg) at an altitude of 400 km above the surface. Earth's radius is about 6.37 × 106 m.
- Distance from Earth's center, r = Earth's radius + altitude = 6.37×106 m + 4.00×105 m = 6.77×106 m.
- Gravitational Potential, φ = -GM / r = - (6.67×10-11 × 5.97×1024) / (6.77×106) ≈ -5.88×107 J/kg.
- Gravitational Potential Energy, U = m × φ = 500 kg × (-5.88×107 J/kg) = -2.94×1010 J.
The large negative value confirms the satellite is bound to Earth.
Example 2: Understanding Escape Velocity
The escape velocity[2] is the minimum speed needed for an object to break free from a gravitational field without further propulsion. It can be derived using the concept of energy. For an object to just reach infinity, its total energy (Kinetic + Potential) must be zero at infinity.
Solving for velocity v gives the escape velocity formula:
$ v_{escape} = \sqrt{\frac{2GM}{r}} $
For Earth, using the values above, the escape velocity from the surface is about 11.2 km/s. Notice how this formula depends on the gravitational potential at the starting point.
Common Mistakes and Important Questions
Q: Why can't gravitational potential be positive?
A: It's a matter of definition and convention. We define gravitational potential to be zero at an infinite distance. Since gravity is an attractive force, you would need to do positive work (add energy) to move a mass away from a planet to infinity. Conversely, the gravitational field does positive work (the system loses potential energy) when a mass moves closer. Because the potential energy decreases as the mass comes in from infinity, the potential at any finite distance must be less than zero, i.e., negative.
Q: Does a more negative potential mean a stronger gravitational field?
A: Not necessarily. The value of the potential itself doesn't directly tell you the strength of the field. The strength of the gravitational field is given by the gravitational field strength (g), which is related to the rate of change of the potential with distance. In mathematical terms, the field strength is the negative gradient of the potential (g = -dφ/dr). A point can have a very negative potential (like near the Sun) and a very strong field, but the potential value alone doesn't convey the "slope" or "steepness" of the potential well.
Q: How is gravitational potential different from the "mgh" we learn in early physics?
A: U = mgh is a special, simplified version of gravitational potential energy for objects near the surface of the Earth, where the gravitational field strength g is approximately constant. The more general form is U = -GMm/r. The "mgh" formula calculates the change in potential energy over small height changes, and it sets the zero point at the ground level, not at infinity. The general potential φ = -GM/r works for any distance and is fundamental to all gravitational interactions.
Footnote
[1] Universal Gravitational Constant (G): A fundamental physical constant that appears in Newton's law of universal gravitation. It determines the strength of the gravitational force between two bodies. Its value is approximately 6.67430 × 10-11 N⋅m²/kg².
[2] Escape Velocity: The minimum speed needed for an object to escape from the gravitational influence of a primary body without any further propulsion, moving onward to an infinite distance without falling back.