Gravitational potential energy Knowledge-Based Study

Study notes

1. What is Gravitational Potential Energy?

   Gravitational potential energy ($E_p$) is the energy stored in an object due to its vertical position or height within a gravitational field.

   Whenever work is done against the force of gravity to raise an object to a higher point, energy is transferred to that object and stored as gravitational potential energy.

   • Energy Store: The higher an object is lifted above a chosen reference level (such as the ground), the more gravitational potential energy it gains.
   • Energy Release: When the object falls, this stored potential energy is converted into kinetic energy as it accelerates downward.

   Calculating Gravitational Potential Energy

   Near the surface of the Earth, where the gravitational field strength is approximately uniform, gravitational potential energy is given by the equation:

   $$\Delta E_p = m \cdot g \cdot \Delta h$$

   Where:

   • $E_p$ (or $\text{GPE}$) is the gravitational potential energy, measured in joules (J).
   • $m$ is the mass of the object, measured in kilograms (kg).
   • $g$ is the acceleration due to gravity (or gravitational field strength), approximately $9.8\text{ N/kg}$ or $9.8\text{ m/s}^2$ on Earth.
   • $\Delta h$ is the vertical height change relative to a reference level, measured in metres (m).

   Factors Influencing $E_p$

   From the formula $\Delta E_p = mgh$, we observe that gravitational potential energy is directly proportional to:

   • Mass ($m$): Doubling the mass of an object doubles its stored gravitational potential energy at the same height.
   • Height ($h$): Doubling the height doubles the stored energy.
   • Gravitational field strength ($g$): An object on Jupiter has greater $E_p$ than the same object at the same height on Earth, because Jupiter's gravity is stronger.

   Path Independence

   Gravitational potential energy depends solely on the vertical displacement ($h$), not on the path taken. Whether an object is lifted vertically straight up or pushed up a long frictionless ramp, the change in gravitational potential energy is identical as long as the initial and final vertical heights are the same.

   Choosing a Reference Level

   Potential energy is always measured relative to an arbitrary reference point where height is defined as zero ($h = 0$).

   • Common choices for the reference level include the floor, the ground, or a tabletop.
   • Only the change in potential energy ($\Delta E_p$) has physical significance in calculations involving energy conservation or work.

   Relationship to Work Done

   The work done by an external force to lift an object at constant speed equals the gain in gravitational potential energy:

   $$W = F \cdot d = (mg) \cdot h = \Delta E_p$$

   Conversely, when an object falls under gravity alone, the gravitational force does positive work on the object, decreasing its gravitational potential energy and increasing its kinetic energy.