Gravitational potential
1. Gravitational potential:
- Gravitational potential is the potential energy per unit mass at a given point in a gravitational field. It’s a scalar quantity, and is measured in units of energy per unit mass, typically joules per kilogram (J/kg) or meters squared per second squared (m²/s²).
- [math] \Delta E_{p} = mg \Delta h [/math]
- Gravitational potential has several key features:
– It’s a scalar field, meaning it has no direction, only magnitude.
– It’s a conservative field, meaning the work done is independent of the path taken.
– It’s a long-range force, acting over vast distances. - Gravitational potential is crucial in understanding various phenomena, such as:
– Orbital motion
– Gravitational waves
– Tides
– Gravitational redshift
– Geopotential - However, we choose to define ∞ as the point of zero potential for all planets and stars. If we chose any other point as zero, such as the surface of the Earth, we would get a more complicated set of equations when we deal with potentials near to other planets.
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Example
- (1)
- Gravitational potential energy change
- Refer to Figure 1.
- What is the gravitational potential energy change in moving a 2kg mass from A to B?
- Solution:
- The mass moves along an equipotential, so the change is 0.
- (2)
- What is the gravitational potential energy change in moving a 2kg mass from A to C?
- Solution:It does not matter which path the mass takes, the potential change from A to C is 100 Jkg-1. So, the potential energy change is
- [math]\Delta E_p = m \Delta V \\
\Delta E_p = 2 \, \text{kg} * 100 \, \text{J kg}^{-1} \\
\Delta E_p = 200 \, \text{J}
[/math]
2. Gravitational potential difference:
- Gravitational potential difference refers to the difference in gravitational potential energy between two points in a gravitational field. It’s a measure of the energy required to move an object from one point to another against the force of gravity.
- Gravitational potential is given the symbol V, and gravitational potential difference is given the symbol ΔV. Since [math] \Delta E_{p} = mg \Delta h [/math], it follows that
- [math] \Delta V = \frac{\Delta E_{p}}{m} = g \Delta h [/math]
- So
- [math] \Delta V = g \Delta h \\
\text{Units of gravitational potential: } J kg^{-1} [/math]
⇒ Example
(1)
The equation in the main text can be used to calculate the magnitude of potential changes. In Figure 1, what is the gravitational potential difference between being on the ground and being at a height of 80 m?
Solution:
[math]
\Delta V = g \Delta h \\
\Delta V = 5 \, \text{N kg}^{-1} \times 80 \, \text{m} \\
\Delta V = 400 \, \text{J kg}^{-1} [/math]
3. Equipotential surfaces:
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⇒ Gravitational field:
- A gravitational field is a mathematical representation of the gravitational force around a mass or a distribution of mass. It’s a vector field that describes the strength and direction of the gravitational force at any point in space.
- Key aspects:
– Gravitational field is a vector field, denoted by g (or G)
– It’s defined as the force per unit mass at a given point
– Units: N/kg (Newtons per kilogram) or m/s² (meters per second squared)
– Gravitational field lines can be used to visualize the field - A gravitational field g is linked to the gravitational potential gradient by the equation
- [math] g = -\frac{\Delta V }{\Delta h} [/math]
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⇒Work done in a moving mass:
- Work done in moving mass m:
- [math] \Delta W = m \Delta V [/math]
- This equation states that the work done (∆W) in moving a mass (m) through a gravitational potential difference (∆V) is equal to the product of the mass and the potential difference.
- In other words:
– ∆W is the work done in moving the mass
– m is the mass being moved
– ∆V is the change in gravitational potential - Unit: Joules (J)
- This equation is a fundamental concept in understanding how gravity affects the motion of objects.
- It shows that the work done in moving an object depends on both the mass of the object and the change in gravitational potential.
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⇒ Equipotential surface:
- A surface where the gravitational potential (V) is constant, meaning that every point on the surface has the same potential value.
- Properties:
- Level surface: Equipotential surfaces are always level, meaning they are perpendicular to the direction of the gravitational field (g).
- No work done: No work is done when moving an object along an equipotential surface, since the potential difference (∆V) is zero.
- When moving an object along an equipotential surface, no work is done because the potential difference (∆V) is zero. This means that the force of gravity is perpendicular to the direction of motion, so no energy is transferred between the object and the gravitational field.
- Think of it like this:
– You’re standing on a level surface (an equipotential surface) and you move a ball horizontally. No work is done because the force of gravity (acting downward) doesn’t contribute to the motion (which is horizontal).
– You’re on a sphere (an equipotential surface) and you move a ball along the surface. Again, no work is done because the force of gravity (acting toward the center) doesn’t contribute to the motion (which is along the surface).
- Same potential value: Every point on an equipotential surface has the same potential value, by definition.
- Perpendicular to field lines: Equipotential surfaces are perpendicular to the gravitational field lines, which helps visualize the field.
- Examples:
- Sphere: A sphere centered on a massive object (like a planet) is an equipotential surface.
- Plane: A horizontal plane in a uniform gravitational field is an equipotential surface.
- Curved surfaces: More complex curved surfaces can also be equipotential surfaces in various gravitational fields.
- Figure 1 also shows equipotential close to the surface of the planet.
- In the diagram these look like lines, but in three dimensions they are surfaces.
- On the diagram, equipotential surfaces have been drawn at intervals of 100Jkg. When an object moves along an equipotential, it means that the potential (and therefore the potential energy) stays the same.
Figure 1 Gravitational field lines and equipotential close to the surface of a planet- Equipotential surfaces are always at right angles to the gravitational field.
- When something moves at right angles to the field (and hence along an equipotential), no work is done by or against the gravitational field, so there is no potential energy change.
- When something moves along a field line, there is a change of gravitational potential energy.
- To be exact, in the link between gravitational field and potential, we should link them with this equation:
- [math] g = – \frac{\Delta V}{ \Delta h} [/math]
- The significance of the minus sign is that the potential gradient [math] \frac{\Delta V}{ \Delta h} [/math] is in a positive direction upwards, because the potential increases as the height above the planet increases. The gravitational field direction is downwards.
4. Gravitational potential in radial fields:
- The graph shows that a force, F, acts on the object at a distance r from the centre of the planet.
Figure 2 Graph of gravitational force acting on an object in the vicinity of a planet.- If the object is moved a small distance, Δr , further away from the planet, we can calculate the increase in the object’s gravitational potential energy as
- [math] \Delta E_p = \text{work done} = F \Delta r [/math]
- Δr is more usually written as mgΔh because the force acting on the mass is equal to its weight, mg.
- Using the Newton’s law of gravitation
- [math] \text{work done} = F \Delta r = \frac{GMm}{r^2} \Delta r [/math]
- So, the increase in gravitational potential energy in moving m from [math]r_1 [/math] to [math] r_2 [/math] is
- [math]\Delta E_p = \int_{r_1}^{r_2} \frac{GMm}{r^2} \, dr \\
\Delta E_p = \left[ -\frac{GMm}{r} \right]_{r_1}^{r_2} \\
\Delta E_p = GMm \left[ \frac{1}{r_1} – \frac{1}{r_2} \right] [/math] - We can also derive a formula for the increase in potential
- [math] \Delta V = \frac{\Delta E_p}{m} = GM \left[ \frac{1}{r_1} – \frac{1}{r_2} \right] [/math]
- We can calculate the potential change in moving from a distance to a point infinitely far away from the planet. When [math] r_2 = \infty, \quad \frac{1}{r_2} = 0 [/math] . So, the potential change in moving from [math]r_1 \, \text{to} \, \infty[/math] is
- [math] \Delta V = \frac{GM}{r_1} [/math]
- However, we choose to define as the point of zero potential for all planets and stars. If we chose any other point as zero, such as the surface of the Earth, we would get a more complicated set of equations when we deal with potentials near to other planets.
- So, since we define the potential as zero at infinity, it means the potential near to any planet is a negative quantity, because potential energy decreases as something falls towards a planet.
- Gravitational potential represents the amount of work done per unit mass to move an object from a reference point to a given point in the gravitational field. The reference point is often chosen as infinity or a point where the gravitational field is zero.
- Mathematically, gravitational potential is defined as:
- [math] \Delta V = \frac{GMm}{r} \qquad (2) [/math]
where:
– V is the gravitational potential
– G is the gravitational constant (6.67*10-11 Nm²kg-2)
– M is the mass of the object or source
– r is the distance from the center of the object or source 
Figure 3 This diagram shows gravitational field lines and equipotential near to a planet (radius of planet =10000 km)- This diagram shows two important linked points:
• The potential gradient is steeper close to the planet’s surface, because the equipotentials are closer together.
• The field lines are closer together near the surface, because the gravitational field strength g is stronger. - These two statements are linked through the equation you met earlier.
- [math] g = -\frac{\Delta V}{\Delta h} [/math]
- Or
- [math] g = -\frac{\Delta V}{\Delta r} [/math]
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These equations are exactly the same, except that h has been used for a change in height, and Δr has been used for a change in distance from the centre of a planet.