Gravitational Potential Energy and Gravitational Potential
Gravitational Potential Energy ($E_p$)
Gravitational potential energy
Gravitational potential energy is the energy stored due to the position of an object in a gravitational field.
Gravitational potential energy is always negative because it is defined relative to a point at infinity, where the potential energy is zero.
The formula for gravitational potential energy between two masses $m_1$ and $m_2$ separated by a distance $r$ is: $$E_p = -G \frac{m_1 m_2}{r}$$ where:
- $G$ is the gravitational constant ($6.67 \times 10^{-11} \, \text{N m}^2 \text{kg}^{-2}$).
- $r$ is the distance between the centers of the masses.
Consider a satellite of mass $1500 \, \text{kg}$ orbiting Earth at a distance of $7000 \, \text{km}$ from its center. Calculate its gravitational potential energy.
Solution
Substituting the given values:
$$E_p = -G \frac{M_{\text{Earth}} \cdot m_{\text{satellite}}}{r}$$
$$E_p = -\frac{6.67 \times 10^{-11} \times 6.0 \times 10^{24} \times 1500}{7.0 \times 10^6}$$
$$ \approx -8.57 \times 10^{10} \, \text{J}$$
Gravitational Potential ($V_g$)
Gravitational potential
Gravitational potential ($V_g$) is the work done per unit mass to bring a small test mass from infinity to a point in a gravitational field.
It is given by:
$$V_g = -G \frac{M}{r}$$
where:
- $M$ is the mass creating the gravitational field.
- $r$ is the distance from the center of the mass.
Gravitational potential is a scalar quantity and is measured in joules per kilogram (J kg⁻¹).
Calculate the gravitational potential at a distance of $10,000 \, \text{km}$ from Earth’s center.
Solution
$$V_g = -G \frac{M_{\text{Earth}}}{r} $$
$$= -\frac{6.67 \times 10^{-11} \times 6.0 \times 10^{24}}{1.0 \times 10^7} $$
$$\approx -4.0 \times 10^7 \, \text{J kg}^{-1}$$
Relationship Between $E_p$ and $V_g$
The gravitational potential energy $E_p$ of a mass $m$ at a point in a gravitational field is related to the gravitational potential $V_g$ at that point by:
$$E_p = m V_g$$
- A $200 \, \text{kg}$ satellite is at a point where the gravitational potential is $-5.0 \times 10^6 \, \text{J kg}^{-1}$.
- The gravitational potential energy is:
$$E_p = m V_g = 200 \times (-5.0 \times 10^6) = -1.0 \times 10^9 \, \text{J}$$
Field Strength as a Potential Gradient
Gravitational Field Strength ($g$)
Gravitation field strength
The gravitational field strength ($g$) at a point is defined as the force per unit mass experienced by a small test mass placed at that point.
It is given by:
$$g = \frac{GM}{r^2}$$
Gravitational field strength is a vector quantity and is measured in newtons per kilogram (N kg⁻¹).
Potential Gradient
- The gravitational field strength can also be expressed as the negative gradient of the gravitational potential: $$g = -\frac{\Delta V_g}{\Delta r}$$
- This means that the field strength is the rate of change of potential with respect to distance.
The negative sign indicates that the gravitational field points in the direction of decreasing potential.
If the gravitational potential decreases by $2.0 \times 10^6 \, \text{J kg}^{-1}$ over a distance of $500 \, \text{m}$, the field strength is:
$$g = -\frac{\Delta V_g}{\Delta r} $$
$$= -\frac{-2.0 \times 10^6}{500} = 4000 \, \text{N kg}^{-1}$$
Work in Gravitational Fields
Work Done Moving a Mass
- When a mass $m$ is moved in a gravitational field, the work done by an external force is equal to the change in gravitational potential energy.
- This can be expressed as:
$$W = m \Delta V_g$$
where $\Delta V_g$ is the change in gravitational potential between the initial and final positions.
A $500 \, \text{kg}$ satellite is moved from a point where $V_g = -4.0 \times 10^6 \, \text{J kg}^{-1}$ to a point where $V_g = -2.0 \times 10^6 \, \text{J kg}^{-1}$.
The work done is:
$$W = m \Delta V_g$$
$$ = 500 \times \left((-2.0 \times 10^6) - (-4.0 \times 10^6)\right)$$
$$ = 1.0 \times 10^9 \, \text{J}$$
- Remember that work done in a gravitational field is path-independent.
- It depends only on the initial and final positions.
- How is gravitational potential energy different from gravitational potential?
- Why is gravitational potential energy negative?
- How does the concept of potential gradient relate to gravitational field strength?
