How to Calculate Gravitational Potential Energy with Examples
Gravitational potential energy is a form of mechanical energy associated with an object's position within a gravitational field. It quantifies the energy stored due to the height of an object above a reference level, resulting from the gravitational attraction between the object and the Earth.
Definition of Gravitational Potential Energy
Gravitational potential energy is defined as the energy possessed by an object due to its position relative to the Earth or another massive body in a gravitational field. This energy is due to the work done against gravity to change the object's vertical position.
Formula for Gravitational Potential Energy
The gravitational potential energy (GPE) of an object of mass $m$ at a height $h$ above the ground, near the Earth's surface where gravity is constant, is given by the formula:
$U = mgh$
Here, $U$ is the gravitational potential energy, $m$ is the mass of the object in kilograms, $g$ is the acceleration due to gravity (typically $9.81~\mathrm{m/s^2}$ on Earth), and $h$ is the height in meters.
For detailed concepts related to gravitation, refer to the Gravitation Overview.
Derivation of Gravitational Potential Energy
The gravitational potential energy can be derived from the work done in lifting an object against gravitational force. When a constant force $mg$ is applied to move an object vertically by a height $h$, the work done (and thus the energy stored) is $mgh$.
Gravitational Potential Energy in a Uniform Field
In a uniform gravitational field, which is a good approximation near the Earth's surface, gravitational potential energy depends linearly on height. The reference level, where $h=0$, is usually chosen as the Earth's surface or any convenient point.
Gravitational Potential Energy in the Universal Law of Gravitation
At larger distances from the Earth where gravitational field strength varies, the gravitational potential energy for two masses $M$ (Earth) and $m$ (object) separated by a distance $r$ is calculated using:
$U = -G\dfrac{Mm}{r}$
Here, $G$ is the universal gravitational constant. This negative value indicates that gravitational potential energy is considered zero at infinite separation.
Practice questions involving the universal law can be found in the Gravitation Practice Paper.
Dependence of Gravitational Potential Energy
Gravitational potential energy depends on the object's mass, height from the reference point, and the local gravitational acceleration. An increase in any of these factors leads to a corresponding increase in potential energy.
| Quantity | Unit (SI) |
|---|---|
| Mass ($m$) | Kilogram (kg) |
| Height ($h$) | Meter (m) |
| Gravitational field strength ($g$) | m/s$^2$ |
| Potential Energy ($U$) | Joule (J) |
Unit of Gravitational Potential Energy
The unit of gravitational potential energy in the International System of Units (SI) is the joule (J). One joule is equal to one kilogram meter squared per second squared ($1~\mathrm{J} = 1~\mathrm{kg \, m^2\, s^{-2}}$).
Solved Example on Gravitational Potential Energy
A mass of $10~\mathrm{kg}$ is suspended at a height of $5~\mathrm{m}$ above the ground. The gravitational potential energy possessed by the mass relative to the ground is:
$U = mgh = 10~\mathrm{kg} \times 9.81~\mathrm{m/s^2} \times 5~\mathrm{m} = 490.5~\mathrm{J}$
For more problem-solving practice, students can attempt the Gravitation Important Questions.
Key Points on Gravitational Potential Energy
- Energy depends on mass, height, and gravity
- SI unit is joule (J)
- Defined for any reference level
- Negative in universal law due to reference at infinity
To assess your understanding, use the available Gravitation Mock Test 1 and Gravitation Mock Test 2.
FAQs on Understanding Gravitational Potential Energy
1. What is gravitational potential energy?
Gravitational potential energy is the energy an object possesses due to its position in a gravitational field, usually relative to the Earth's surface.
- It depends on the object's mass (m), height above the ground (h), and gravitational acceleration (g).
- It is calculated with the formula: Potential Energy (PE) = m × g × h
- Measured in joules (J).
2. How do you calculate gravitational potential energy?
To calculate gravitational potential energy, multiply the object's mass, gravitational acceleration, and height above ground:
- Use the formula: PE = m × g × h
- Where m is mass in kilograms (kg)
- g is the acceleration due to gravity (9.8 m/s² on Earth)
- h is height in meters (m)
3. What factors affect gravitational potential energy?
Three main factors affect gravitational potential energy:
- Mass of the object: More mass means more energy.
- Height above ground: Higher position increases energy.
- Value of gravity (g): Often constant (9.8 m/s² on Earth), but changes with location.
4. Why does an object at a higher position have more gravitational potential energy?
An object at a higher position has more gravitational potential energy because its height increases the work done against gravity, storing more energy.
5. What is the SI unit of gravitational potential energy?
The SI unit of gravitational potential energy is the joule (J).
- 1 joule = 1 kg·m²/s²
- It is the same as the unit for work and other forms of energy.
6. Give a real-life example of gravitational potential energy.
A real-life example of gravitational potential energy is a book kept on a shelf.
- It has energy due to its elevated position above the ground.
- If the book falls, this energy changes to kinetic energy.
7. How is gravitational potential energy related to work done?
The work done to lift an object against gravity equals the gravitational potential energy gained.
- Lifting a mass m to height h uses work: Work = m × g × h
- This work is stored as potential energy.
8. What happens to gravitational potential energy when an object falls?
When an object falls, its gravitational potential energy changes into kinetic energy as it moves towards Earth's surface.
- Total energy is conserved, transforming from potential to kinetic form.
9. Can gravitational potential energy be negative?
Yes, gravitational potential energy can be negative, depending on the reference point.
- If the zero level is set above the object, energy values can be negative.
- The negative sign shows the object is below the reference height.
10. Is gravitational potential energy a scalar or vector quantity?
Gravitational potential energy is a scalar quantity because it has magnitude but no direction.
11. State the formula for gravitational potential energy with its terms explained.
The formula for gravitational potential energy is PE = m × g × h:
- PE: Potential energy in joules (J)
- m: Mass (kg)
- g: Acceleration due to gravity (9.8 m/s² on Earth)
- h: Height above reference level (m)
