Energy - Potential Energy (U)

Potential energy is the ‘stored energy’ by virtue of the position or configuration of a body. An object stores energy when it is in a specific state or position, such as at a height. When we release the object, the stored energy, known as potential energy, initiates motion.

For example,

• A chalk held at a height possesses potential energy due to its elevated position.
• The energy required to lift the chalk to that height was stored as potential energy.

Mathematically, the potential energy V(x) is defined if the force F(x) can be written as

`F(x)=-(dv)/(dx)`
`∫_(x_i)^(x_f) F(x)=V_i-V_f`

This relation is valid only for conservative forces.
"The energy possessed by any object by virtue of its position or configuration is called its potential energy."

There are three important types of potential energies:
(i) Gravitational Potential Energy: If a body of mass m is raised through a height h against gravity, then its gravitational potential energy = mgh.

(ii) Elastic Potential Energy: If a spring of spring constant k is stretched through a distance x, then elastic potential energy of the spring = `1/2 kx^2`

The variation of potential energy with distance is shown in the figure.

Potential energy is defined only for conservative forces. It does not exist for non-conservative forces. Potential energy depends upon the frame of reference.

(iii) Electric Potential Energy: The electric potential energy of two point charges `q_1` and `q_2`.  Elastic potential energy is never negative, whether due to extension or compression. Separated by a distance r in vacuum is given by

`U = 1 / (4πΣ0) * (q_1q_2) / r`

Here is a `1 / (4πΣ0) = 9.0 * 1010 (Nm^2)/ C^2 ` constant.

When an object is lifted to a height $h$ above the Earth's surface, work is done against the force of gravity to move the object. The object stores this work as potential energy. The relationship between work and potential energy can be derived as follows:

To lift an object of mass m, a force equal to mg (where $g$ is the acceleration due to gravity) must be applied. This force counteracts the gravitational pull on the object. 

Work is defined as the product of force and displacement in the direction of the force.

W = force × displacement

$$W=mg\times h$$

$$W=mgh$$

$$The\; potential\; energy\; stored\; in\; the\; object\; depends\; on\; three\; factors:$$

• Mass (m): Heavier objects store more potential energy.
• Height (h): The higher the object is lifted, the more potential energy it stores.
• Gravitational Acceleration (g): The strength of gravity affects the potential energy.

This relationship, P.E.=mgh, shows that when an object is displaced to a height h, energy equal to mgh is stored in it as potential energy. This stored energy can later be converted into other forms, such as kinetic energy, when the object falls.