Potential Energy: Definition, Formula, Derivation of Gravitational Potential Energy

Have you ever wondered what energy does water stored in a tank possess when it is placed at a height h above the surface of the earth? Or a car parked on a hilltop several kilometers above the ground? Is there energy at all in the first place? Yes there is energy, and it is evident from the fact that when you let the hand brake of the car free or allow the water to come out, you can see them gaining kinetic energy. From where did this kinetic energy come? It came at the cost of potential energy.

The higher an object is located above the surface of the earth, the higher is its potential energy. In the following diagram for instance, the ball m, when located at the top of a tower of height h has a higher potential energy than when it was located at the bottom of the tower. Another way to imagine potential energy would be this: imagine you lifted a bottle of water from the ground up to your desk. The work done by your hand is stored as potential energy in the bottle.

Fig. A ball of mass m at two different energy levels

Definition of Potential Energy
Gravitational Potential Energy Derivation
Practice Problems of Potential Energy
FAQs of Potential Energy

Definition of Potential Energy

Potential energy (P.E) can be defined as the energy possessed by a body by virtue of its position in a conservative force field; i.e a field in which work done by the forces do not depend upon the path taken in moving the body. If a conservative force F with arrow acting on a body displaces it through a distance d bar r, then the change in P.E. can be written as

U_i- U_f= ∫ .r; here U_i and U_f indicate the initial and final potential energy.

Note: Change in potential energy is △U = U_f- U_i=-W; where W = ∫ .r is the work done by the conservative force.

Gravitational Potential Energy Derivation

Let us assume a ball of mass m is moved to a height h vertically. The work done in shifting the ball through a height is stored as gravitational potential energy inside the ball.

Change in P.E in this process for a small displacement dy can be written as mgdy

Taking the ground as reference level as the potential energy datum, where the P.E is zero; i.e U_i=0, the above equation becomes

U_f- U_i= U_f- 0 = mgh

Practice Problems

Q1) Calculate the potential energy stored in a bowl of mass 100 g when it is raised through a height of 0.2 m. (Take g=10 ms^-2)

Solution) Given, m=0.1 kg

h=0.2 m

Work done = potential energy stored in the bowl = mgh = 0.1 × 10 × 0.2 = 0.2 J

Q2) A chain of mass M and length l hangs on a frictionless table. ²⁄₃rd of the chain lies on the table. The work done in pulling the entire chain up the table will be?

Q3) A uniform rod of mass M and length l is held in position as shown in the figure. Find the potential energy of the rod.

Solution)

Given, mass of the rod = M

Length of the rod = l

Consider an element of length dx and mass dm at a distance x from the bottom of the rod. Then

Q4) Find the gravitational potential energy of the chain with reference level at the center of the hemisphere.

Solution) Let dm be the mass of an element of length dx and m be the mass of the chain.

FAQs

Question1. Can potential energy be defined in non-conservative field?
Answer. Potential energy can only be defined in conservative fields.

Question2. What is the expression for gravitational potential of a particle of mass m raised through a height h?(g -acceleration due to gravity)
Answer. mgh

Question3. Is potential energy a scalar or a vector quantity?
Answer. P.E is a scalar quantity.

Question4. What is the dimensional formula for potential energy?
Answer. [ML²T^-2].