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Potential Energy

Potential energy is defined only for conservative forces. In the space occupied by conservative forces, every point is associated with certain energy which is called the energy of position or potential energy. Potential energy generally is of three types: elastic potential energy, electric potential energy and gravitational potential energy.

Change in potential energy

Change in potential energy between any two points is defined in the terms of the work done by the associated conservative force in displacing the particle between these two points without any change in kinetic energy.
We can define a unique value of potential energy only by assigning some arbitrary value to a fixed point called the reference point. Whenever and wherever possible, we take the reference point at infinite and assume potential energy to be zero there, i.e., if take r1 = ∞ and r2 = r, then from (1),
In case of conservative force (field), potential energy is equal to the negative of work done in shifting the body from reference position to given position.
This is why in shifting a particle in a conservative field (say gravitational or electric), if the particle moves opposite to the field, work done by the field will be negative and so change in potential energy will be positive, i.e., potential energy will increase. When the particle moves in the direction of field, work will be positive and change in potential energy will be negative, i.e., potential energy will decrease.

Three-dimensional formula for potential energy

For only conservative fields, 39007.png equals the negative gradient 39001.png of the potential energy.
⇒ 38995.png
where 38989.png Partial derivative of U wrt x (keeping y and z constant)
38983.png Partial derivative of U wrt y (keeping x and z constant)
38977.png Partial derivative of U wrt z (keeping x and y constant)

Potential energy curve

A graph plotted between the potential energy of a particle and its displacement from the center of force is called potential energy curve. Figure 8 shows a graph of potential energy function U(x) for one-dimensional motion.
Fig. 8
As we know that negative gradient of the potential energy gives force.
∴  38965.png

Nature of force

  • Attractive force: On increasing x, if U increases, dU/dx = positive
    Then F is negative in direction, i.e., force is attractive in nature. In Fig. 8, this is represented in region BC.
  • Repulsive force: On increasing x, if U decreases, dU/dx = negative.
    Then F is positive in direction, i.e., force is repulsive in nature. In Fig. 8, this is represented in region AB.
  • Zero force: On increasing x, if U does not changes, dU/dx = 0.
    Then F is zero, i.e., no force works on the particle. Point BC, and D represent the point of zero force or these points can be termed as position of equilibrium.

Types of equilibrium

If net force acting on a particle is zero, it is said to be in equilibrium.
For equilibrium dU/dx = 0, but the equilibrium of particle can be of three types:
  1. Stable: When a particle is displaced slightly from a position, then a force acting on it brings it back to the initial position. It is said to be in stable equilibrium position. Since potential energy is minimum,
    ∴ 38928.png
    i.e., rate of change of 38922.png is positive.

A marble placed at the bottom of a hemispherical bowl (Fig. 9).


Fig. 9


  1. Unstable: When a particle is displaced slightly from a position, then a force acting on it tries to displace the particle further away from the equilibrium position, it is said to be in unstable equilibrium. Since potential energy is maximum.
    ∴ 38903.png
    i.e., rate of change of 38897.png is negative.

A marble balanced on top of a hemispherical bowl (Fig. 10).


Fig. 10


  1. Neutral: When a particle is slightly displaced from a position, then it does not experience any force acting on it and continues to be in equilibrium in the displaced position, it is said to be in neutral equilibrium. Since potential energy is constant.
    i.e., rate of change of 38873.png is zero.

A marble placed on horizontal table (Fig. 11).


Fig. 11


Expression for elastic potential energy

When a spring is stretched or compressed from its normal position (x = 0), work has to be done by external force against restoring force.
Let the spring is further stretched through the distance dx. Then, work done
Therefore, total work done to stretch the spring through a distance x from its mean position is given by
This work done is stored as the potential energy of the stretched spring.
∴ Elastic potential energy, 38842.png
∴ Elastic potential energy, 38812.png


Note: If spring is stretched from initial position x1 to final position x2, then
Work done = Increment in elastic potential energy 39503.png

Energy graph for a spring

If the mass attached with spring performs simple harmonic motion about its mean position (Fig. 12), then its potential energy at any position (x) can be given by
Fig. 12
So, for the extreme position,
This is maximum potential energy or the total energy of mass.
[Because velocity of mass = 0 at extreme  38756.png]
Now kinetic energy at any position,
From the above formula, we can check that
Fig. 13
It mean kinetic energy changes parabolically wrt position, but total energy always remains constant irrespective to position of the mass.

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