Change in Potential Energy When Some Work is Done

Potential energy is the energy possessed by an object due to its position or configuration. When work is done on or by an object, there can be a change in its potential energy. This concept is crucial in understanding various physical phenomena, from the simple motion of a ball thrown into the air to the complex interactions of celestial bodies.

Understanding Work and Energy

Before diving into the change in potential energy, let's define work and energy:

• Work (W): In physics, work is done when a force (F) causes a displacement (d) in the direction of the force. The formula for work is:

[ W = F \cdot d \cdot \cos(\theta) ]

where ( \theta ) is the angle between the force and the displacement vectors.

• Energy: Energy is the capacity to do work. There are many forms of energy, but we will focus on mechanical energy, which includes kinetic energy (energy of motion) and potential energy (energy of position).

Potential Energy

Potential energy comes in various forms, such as gravitational potential energy, elastic potential energy, and electric potential energy. We will focus on gravitational potential energy, which is relevant to an object's position in a gravitational field.

• Gravitational Potential Energy (U): The energy an object possesses due to its height above a reference level. The formula is:

[ U = m \cdot g \cdot h ]

where ( m ) is the mass of the object, ( g ) is the acceleration due to gravity, and ( h ) is the height above the reference level.

Change in Potential Energy

When work is done on an object, it can lead to a change in the object's potential energy. The work done by external forces (excluding conservative forces like gravity) can change the potential energy of a system.

• Change in Potential Energy ((\Delta U)): The change in potential energy is equal to the work done by the external forces:

[ \Delta U = W_{\text{external}} ]

If the work is done against conservative forces, like lifting an object against gravity, the potential energy increases. If the work is done by conservative forces, like an object falling, the potential energy decreases.

Table of Differences and Important Points

Aspect	Work Done (W)	Potential Energy Change ((\Delta U))
Definition	Force applied over a distance	Energy due to position or configuration
Formula	( W = F \cdot d \cdot \cos(\theta) )	( \Delta U = W_{\text{external}} )
Dependency	Depends on force and displacement	Depends on initial and final positions
Conservation	Work is not a conserved quantity	Potential energy is conserved in a closed system with only conservative forces
Sign Convention	Positive when force and displacement are in the same direction	Positive when potential energy increases, negative when it decreases

Examples

Example 1: Lifting an Object

When you lift an object from the ground to a height ( h ), you do work against the gravitational force. The work done is equal to the increase in gravitational potential energy.

• Work done: ( W = m \cdot g \cdot h )
• Change in potential energy: ( \Delta U = m \cdot g \cdot h )

The work done is positive, and the potential energy of the object increases.

Example 2: Object Falling

When an object falls from a height ( h ), gravity does work on the object. The potential energy decreases as the object loses height.

• Work done by gravity: ( W = m \cdot g \cdot h )
• Change in potential energy: ( \Delta U = -m \cdot g \cdot h )

The work done by gravity is positive, but the change in potential energy is negative since the object is losing height.

Conclusion

The change in potential energy when work is done is a fundamental concept in physics that describes how energy is transferred or transformed within a system. Understanding this concept is essential for solving problems in mechanics, astrophysics, and various engineering fields. It is also a key principle in the conservation of energy, which states that energy cannot be created or destroyed, only transformed from one form to another.