Learn the Concept of Potential Energy with Meaning, Types, Formula, and More

To help you build a foundation, consider reviewing this table before approaching the concept of potential energy.

Topic	Key Points to Master
Scalar (Dot) Product	A.B = AB cos θ is the formula for Scalar Product Understand results for unit vectors (i.i = 1, i.j = 0).
Work (W)	W = F.d (Work = Force ⋅ Displacement) is the formula for Work Be aware of positive, negative, and zero work, as necessary conditions for work done.  Calculate variable force using integration, in the form of W = ∫F(x)dx
Kinetic Energy and Work-Energy Theorem	K = ½mv²  is the formula Apply ∆K = W_net (Change in Kinetic Energy = Net Work Done) as the main formula
Conservative & Non-Conservative Forces	In a conservative force, such as gravity or spring force, work is independent of the path.  Work is path-dependent in non-conservative forces, such as friction. So, there is no association of potential energy.
Basic Calculus	F(x) = -dV(x)/dx is the basic Force-Potential Energy Relation to remember.  Be comfortable using integration to find work done by a variable force.
Kinematics & Newton's Laws	Be proficient with F=ma, which is the 2nd Law of Motion. Remember the basic Kinematic Equations that define motion.

This guide on potential energy for Class 11 physics will cover all you need to learn for exams. It will also potentially prepare you to solve related PE problems quickly, which are necessary for any aspirant aiming to crack JEE Main. 

In physics, potential energy $\left(U\right)$ is the negative of the work done by a conservative force in moving a body from a reference position to a given position. 

The relationship between potential energy and conservative force is given on the basis of the universally understood potential energy formula

$U_f-U_i=-{\int }_i^f \vec{F}\cdot d\vec{r}$

Where,

•   $\vec{F}$ is the conservative force vector, and $\vec{r}$ is the displacement vector along the path from i to f. Together, they represent the dot product between the force and displacement, which helps us in calculating the small amount of work done over every small step of the path.
• Uf - Ui (or ΔU) in the potential energy equation tells how much change there is in potential energy. While Ui is the potential energy at the initial (or reference) position i, Uf is the potential energy at the final position f. One tip to remember here is that physics is most often concerned with this change rather than an absolute value.
• - (Negative Sign) directly implements the negative of part of the definition of potential energy. It's like the inverse relationship.
• ∫ᵢᶠ:  is a definite integral from the initial position i to the final position f, which is a mathematical tool for summing up quantities over a path. That's quite necessary as the force might not be constant.

Also, remember, potential energy is a property of the system.

Potential energy is not a feature of a single object. Because of that, potential energy always depends on the choice of reference point.

From the above equation of potential energy, you can say that  $U$  is often set to zero for convenience.

Potential Energy Formula: A Special Case of Gravity

As we now know that $U_f-U_i=-{\int }_i^f \vec{F}\cdot d\vec{r}$ , the potential energy formula that is commonly used is

PE = mgh

or,

U = mgh

• Potential energy can also be expressed as the product of mass, acceleration due to gravity, and height, when we are considering uniform gravitational force. 
• Note that the negative sign in the original formula above cancels with the gravitational force.
• Gravity is uniform with mg as constant, so it comes out of the integral.
• We can choose any reference point for zero potential energy.

We will explain this gravitational potential energy in more detail below. 

Unit of Potential Energy

The SI unit of Potential Energy is the Joule.

Dimensional Formula for Potential Energy 

The dimensional formula for potential energy is ${\mathrm{M}}^1{\mathrm{L}}^2{\mathrm{T}}^{-2}$ .   

Here, M is the mass of the dimension. 
m has dimensions of M (mass)
g has dimensions of LT⁻² (acceleration = length/time²)
h has dimensions of L (length/height)

This understanding of potential energy in the mathematical form is quite important for numerical problems that appear in both school and competitive exams. We will simplify the concept of potential energy below.  

 

In Section 5.7 of the Work, Energy, and Power chapter, you have an explanation of potential energy in this manner. 

Potential energy is the 'stored energy' of the position or configuration of an object or body. 
Another way of saying it is when work is done against a force, this work can get 'stored up' as potential energy. 

 

We can find two types of potential energy based on how energy is stored. These are also readily observable phenomena, just with different ways of representing them in physics using maths.   

The main forms of potential energy in Class 11 are

• Gravitational Potential Energy
• Elastic Potential Energy

Gravitational Potential Energy

Simply put, we observe gravitational potential energy in a situation where we lift an object higher; the more 'stored energy' it will have. This is the explanation of the potential energy that we were talking earlier. 

A basic example is a bowling ball.

• When it's on the ground or the floor, the ball does not have any energy.
• But when you lift it, it has stored energy. When you let that go from the top, that stored energy turns or converts into kinetic energy (energy of motion) as it falls.  

Mathematically, we can look at Gravitational Potential Energy for a mass $m$ at height $h$ above a reference level, such as the ground.

So the gravitational potential energy formula is

$U=mgh$

Key Points to Remember about Gravitational Potential Energy 

• $g=9.8\mathrm{m}/{\mathrm{s}}^2$
• A heavier object, in this case, a bowling ball, will store more energy than a lighter one, like a tennis ball, at the same height.

Elastic Potential Energy

We can also observe potential energy in stretchable objects. One fine example is a rubber band. We can stretch it or compress it, and that energy stored is elastic potential energy.  

When letting that rubber band go from a compressed or stretched state, that stored energy is released.

The same will also apply to a spring. 

For a spring displaced by $x$ from its natural length, so the elastic potential energy formula is

$U=$ $\frac{1}{2}k{x}^2$

Further, we have to understand what spring constant and displacement mean. 

Key Points to Remember about Elastic Potential Energy 

• k (spring constant): This measures how stiff the spring is. A very stiff spring (high k) stores a lot more energy than a weak, flimsy one.
• x (displacement): This is how far you stretch or compress the spring. Because this term is squared (x²), stretching a spring twice as far actually stores four times the energy.

 

 

 

 

It's important to remember that potential energy is associated only with conservative forces, such as gravity or spring force. The work done is not dependent on the path, and it's zero around a closed path.

Mathematically, we can say that a conservative force is related to potential energy in this way. 

$\vec{F}=-\frac{dU}{dr}\hat{r}$

For example, in a gravitational field, we have force and potential energy represented as

$F=-mg$

and

$U=mgh$

Alternatively, non-conservative forces, like friction (as per the Law of Inertia), do not have associated potential energy because their work depends on the path taken.

Let's unpack more about the relationship between potential energy and conservative forces.  

Why Potential Energy Works on Conservative Forces?

The concept of potential energy is tied to its reversible nature. It can show that the work done against a force can be recovered. For this to happen, force itself must behave in a particular way, and that too, it has to be predictable. 

One of the simplest ways to consider is the lifting of a book against gravity.

You do work on the book, that may seem that energy is not there. But there is. The energy has been stored as gravitational potential energy.

So when you let go, gravity does work on the book. That converts the stored potential energy perfectly back into kinetic energy. Meaning, this process is completely reversible. 

Why Non-Conservative Forces Have No Association with Potential Energy? 

In physics, forces such as friction are non-conservative. It's because the energy you use against them isn't stored. Instead, it's lost as heat.
The energy cannot be recovered the way it happens with conservative forces. So the amount lost depends on the path taken. That's why we cannot define a potential energy for friction.
Even when you look at the work-energy theorem, it accounts for the effect of non-conservative forces $W_{\mathrm{n}\mathrm{c}}=\left(K_f+U_f\right)-\left(K_i+U_i\right)$

 

 

 

Potential energy and kinetic energy are the sum of total mechanical energy. We can represent that as 

$E=K+U$

K stands for kinetic energy, and it has the equation.

$K=\frac{1}{2}m{v}^2$  

For your exams, just remember that in systems that only have conservative forces, mechanical energy is conserved: $K_i+U_i=K_f+U_f$

 

Potential energy is applicable to various scenarios, relating mostly to the types we learned earlier. They can be proven mathematically. 

Gravitational Systems

Potential energy applies to all types of gravitational systems. For a body in free fall or on an incline, gravitational potential energy converts to kinetic energy:

$mgh=\frac{1}{2}m{v}^2⟹v=\sqrt[]{2gh}$

This formula is useful for calculating velocities without detailed force analysis.

Spring-Mass Systems

In a spring system, elastic potential energy drives oscillation. For a spring compressed by a distance (or displacement from the equilibrium position) x, the stored energy is:

$U=\frac{1}{2}k{x}^2$

This energy converts to kinetic energy as the spring returns to its natural length.

Vertical Circular Motion

In vertical circular motion, gravitational potential energy varies with height. So for a particle at height $h=r(1+\mathrm{c}\mathrm{o}\mathrm{s}\theta )$ , we get energy conservation as 

$\frac{1}{2}m{v}^2+mgr(1+\mathrm{c}\mathrm{o}\mathrm{s}\theta )=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$

This determines velocities or tensions at different points. Also read about circular motion, a quick brush up on how Newtonian Laws apply.