Potential Energy: Definition, Types, Formula and Solved Examples

Potential energy definition states the energy of an object based on its position with respect to the various parts of a system. The objects gain energy stored in them when they are displaced from the positions of equilibrium, before being knocked out of equilibrium, by elastic rebound, gravity, or chemical reaction. 

Potential energy is broadly classified as gravitational potential energy, elastic potential energy and electrical potential energy. The Joule, denoted by the letter J, in the International System of Units (SI), is the SI unit of energy.

Read More: Principle of Conservation of Energy

Key Terms: Energy, Equilibrium, Potential Energy, Kinetic Energy, Gravity, Chemical Reaction,Gravitational, Elastic, Electric, Work Done

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Potential Energy

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Potential energy is a type of stored energy that is affected by the relative positions of various components in a system. Potential energy is a property of a system, not an individual entity or particle, for example, as the Earth and the elevated ball are separated, the system has greater potential energy.

Potential Energy Examples

• Newton's first law of motion, sometimes known as the law of inertia, is applied to a rubber band.
• It now expands and enlarges in length as you exert force on it.
• This is because the molecules inside the rubber band have previously saved energy, and when you apply force, all of the molecules start moving because they gain kinetic energy and the band extends.
• Potential energy gets its name from the fact that it can be changed into another kind of energy.
• This type of energy can be turned into kinetic, mechanical, thermal, and light . 

Read More: Types of Motion and Laws

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Potential Energy Formula

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The force acting on the two objects determines the potential energy formula. The formula for gravitational force is:

W = mgh

Where, m is the mass in kilograms 

g is the gravitational acceleration,

h is the height in meters

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Potential Energy Units

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The potential energy is symbolized by U, V, or PE. Gravitational potential energy has the units as: kg m² / s²

It is measured using the unit Joule (J). The dimensional formula for potential energy is given as: ML²T^-1

Read More: NCERT Solutions Work, Energy and Power

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Types of Potential Energy

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There are 3 major types of potential energy-

1. Gravitational Potential Energy
2. Elastic Potential Energy
3. Electric Potential Energy

Gravitational Potential Energy

• Gravitational potential energy is the energy stored in an object that is maintained in a vertical position by gravity pulling it down.
• An object's gravitational potential energy is proportional to its height and mass.
• The more gravitational potential energy an object has, the heavier it is and the higher it is above the ground.
• As one's weight and height increase, so does their gravitational potential energy.
• For example- This is how hydroelectric power is generated. The falling water spins a turbine, which pushes electrons around and creates an electric current.

The formula is given by 

Force x displacement = work done on the box.

W = m x g x h = mgh

Because the work done on the box = mgh, the energy gained by the box = mgh, which in this case is the potential energy E, according to the rule of conservation of energy.

Note:

• The gravitational energy of an object is determined by its displacement or the difference between its beginning and final height.
• As a result, the path taken by the object to achieve its current height is ignored.
• The gravitational potential energy for both blocks A and B in the example above will be the same.

Elastic Potential Energy

• Elastic potential energy is held in devices like trampolines, rubber bands, and bungee cords that can be stretched or squeezed.
• Springs act in the same way as springs, but they can be stretched or compressed.
• When you wind a wind-up watch, potential energy is stored in an internal spring, which is then gently used to power the watch.
• The more elastic potential energy an object has, the more it can stretch.
• An object that accumulates elastic potential energy has a high elastic limit, yet all elastic objects have a load limit.
• The object will not return to its original shape if it is deformed beyond the elastic limit.

The formula is given by:

Elastic Potential Energy (U)= 1/2kx²

Where k=spring force constant

x= string stretch length in meters

Electric Potential Energy

• The electric potential energy of an object is possessed by the means of two elements.
• Here, the charge is possessed by the object itself and the relative position of an object with respect to other electrically charged objects.
• The magnitude of electric potential is based on the amount of work done in moving the object from one point to another against the electric field.

A charge placed in an electric field will have potential energy and will be measured by the work done by moving the charge from infinity to that point against the electric field. If two charges q₁ and q₂ are separated by a distance d, the electric potential energy of the system is;

U = [1/(4πεo)] × [q₁q₂/d]

Read More: Work, Energy and Power Important Questions

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Potential Energy Examples

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Listed below are some of the classic examples of potential energy seen in our day-to-day lives. 

Gravitational Potential Energy

• Water that has been held back by a dam.
• An automobile parked at the summit of a hill.
• Water from a river cascades to the top of a waterfall.
• Before the fruit falls to the ground, it must be ripe.

Elastic Potential Energy

• A spring that is coiled.
• The string on an archer's bow is pushed back.
• A rubber band that has been stretched.
• Soccer balls before they are kicked by players.

Read More: Torque and Couple

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Potential Energy and Work

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Potential energy can also be defined as the amount of work done to move an object to a specified position. Thus if we are to move a car from position A to position B, the potential energy will be the work done to move the car from point A to B.  The force applied to carry out this work must be equal in magnitude but opposite in direction to the movement of the object. 

Consider a ball falling from a height h as shown in the figure below. The gravitational force acts on the ball in the same direction as that of displacement of the ball.

Potential Energy and Work

The angle between these will hence be zero. Then we can say

W= ΔK= K₂- K₁

W= FΔr cosΘ

= mgh

Therefore, mgh= K₂-K₁ = ½ mv₂²

Here W- work done on the ball in moving it to the bottom. Now, considering both the ball and the earth, the equation can be re-written as:

0= ΔK- W_g

_- W_g = – mgh= ΔU_g

From the above, we see that no work is done on the system but there is a change in the gravitational potential energy, U. This potential change is in the opposite direction to the force applied.

Read More: Linear Momentum Formula

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Kinetic Energy and Potential Energy

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Suppose a truck going up the tip of the mountain and after that, went down the mountain to reach the ground. Primarily, it will store energy that is the potential energy, and then it will start moving up the mountain, which is the kinetic energy. 

Therefore, the total energy utilized by it is given as:

KE_i + PE_i 

Now, since it is at the tip of the mountain, and it again has final stored energy or potential energy as it starts moving down the mountain. This stored energy transforms into kinetic energy. Now, the total energy used in the final case which is as follows:

KE_f + PE_f

Hence, whatever total energy was utilised at the beginning is equivalent to the energy used finally. So, the desired establishment of a relation between kinetic and potential energy is:

KE_i + PE_i = KE_f + PE_f

Read More: Heat Transfer and Mechanism

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Things to Remember

• Potential energy is the stored energy that depends on the relative position of an object. 
• A steel ball has more potential energy when raised above the ground than it has after falling on the ground.
• There are three types of potential energy: gravitational, potential, and electric potential energy.
• Elastic potential energy is held in devices like trampolines, rubber bands, and bungee cords that can be stretched or squeezed. Elastic Potential Energy (U)= 1/2kx²; where k=spring force constant, x= string stretch length in meters. 
• The electric potential energy of an object is possessed by the means of two elements.
• If two charges q₁ and q₂ are separated by a distance d, the electric potential energy of the system is; U = [1/(4πεo)] × [q₁q₂/d]

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Previous Years Questions 

1. A ball dropped from a height of 2m rebounds to a height of 1.5m after hitting the ground.
2. A ball is dropped from a height h. What height will it rise after jumping twice from the ground….
3. A ball of mass m is dropped from a height h on a platform fixed at the top of a vertical spring, Then the spring constant is..
4. An engine pumps water continuously through a hose. What is the rate at which kinetic energy is imparted to water?​
5. A car of mass 1500kg is lifted up a distance of 30m by crane A in 0.5 minutes. The ratio of their power is.…
6. The work done in the first one second by the external force is….​.
7. Under the action of a constant force, a particle is experiencing a constant acceleration. The power is….
8. The total work done by the forces in SI unit is..…
9. A particle is released from a height S .The height and speed of the particle at that instant are respectively​…​
10. A rod of mass mm and length l Potential energy of the rod in this position is​
11. A body of mass m=1kg is moving in a medium and experiences a fractions force $F = - kv,Then, the value of k is​
12. The position vector of the particle is →r(t)=acosωt^i+asinωt^jr, where aa and ω are real constants of suitable dimensions. The acceleration is
13. A spring stores 1J of energy for a compression of 1mm. The additional work to be done to compress it further by 1mm is​
14. According to the work-energy theorem, the work done by the net force on a particle is equal to the change in its​
15. The efficiency of the pump should be (Take g=10 ms−2g=10 ms−2 )​
16. Which of the following is not an inelastic collision?​
17. what will be the ratio of velocity of two balls after collision ?​
18. The work done in pulling up a block of wood weighing 2kN for a length of 10m on a smooth plane inclined at an angle of 15∘ with the horizontal is
19. Two springs of spring constant 1500Nm−1and3000Nm−1 respectively are stretched with the same force. They will have potential energy in ratio
20. A particle is released from a height h. Height and speed of the particle at that instant are​​​