The Potential Energy Formula gives the stored energy of an object due to its position or configuration, expressed as PE = mgh for gravitational potential energy and PE = ½kx² for elastic potential energy. This concept is covered in NCERT Class 9 Chapter 11 and Class 11 Chapter 6, and it forms a high-weightage topic in both CBSE board exams and competitive exams like JEE Main and NEET. This article covers every variant of the potential energy formula, a complete formula sheet, three progressive solved examples, CBSE exam tips for 2025-26, common mistakes, and JEE/NEET application patterns.
Key Potential Energy Formulas at a Glance
Quick reference for the most important potential energy formulas used in CBSE and competitive exams.
- Gravitational PE: \( PE = mgh \)
- Elastic PE (Spring): \( PE = \frac{1}{2}kx^2 \)
- Gravitational PE (Universal): \( U = -\frac{GMm}{r} \)
- Electric PE: \( U = \frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r} \)
- Work-Energy Theorem: \( W = \Delta KE = -\Delta PE \)
- Conservation of Energy: \( KE + PE = \text{constant} \)
- Relation between PE and Force: \( F = -\frac{dU}{dx} \)
What is Potential Energy Formula?
The Potential Energy Formula quantifies the energy stored in an object because of its position, shape, or state relative to a reference point. Potential energy is not energy in motion. It is energy waiting to be released. When a ball is held at a height, it stores gravitational potential energy. When a spring is compressed, it stores elastic potential energy. The moment these objects are released, the stored energy converts into kinetic energy.
In NCERT Class 9 Science, Chapter 11 (Work and Energy), students first encounter gravitational potential energy. The concept is revisited and extended in NCERT Class 11 Physics, Chapter 6 (Work, Energy and Power), where elastic potential energy and the universal gravitational potential energy formula are introduced. The SI unit of potential energy is the Joule (J), the same as all other forms of energy. Potential energy is a scalar quantity, meaning it has magnitude but no direction. Understanding the potential energy formula is fundamental to mastering the law of conservation of energy.
Potential Energy Formula — Expression and Variables
1. Gravitational Potential Energy Formula
The gravitational potential energy formula calculates the energy stored in an object raised to a height above the ground (reference level):
\[ PE = mgh \]
| Symbol | Quantity | SI Unit |
|---|---|---|
| PE | Gravitational Potential Energy | Joule (J) |
| m | Mass of the object | Kilogram (kg) |
| g | Acceleration due to gravity | m/s² (standard value: 9.8 m/s²) |
| h | Height above the reference level | Metre (m) |
2. Elastic Potential Energy Formula (Spring)
The elastic potential energy formula applies to springs and elastic materials stretched or compressed by a displacement x from their equilibrium position:
\[ PE = \frac{1}{2}kx^2 \]
| Symbol | Quantity | SI Unit |
|---|---|---|
| PE | Elastic Potential Energy | Joule (J) |
| k | Spring constant (stiffness) | N/m |
| x | Displacement from equilibrium | Metre (m) |
3. Universal Gravitational Potential Energy Formula
At large distances from Earth (used in Class 11 and JEE), the gravitational potential energy between two masses is:
\[ U = -\frac{GMm}{r} \]
| Symbol | Quantity | SI Unit |
|---|---|---|
| U | Gravitational Potential Energy | Joule (J) |
| G | Universal Gravitational Constant | N·m²/kg² |
| M | Mass of the larger body (e.g., Earth) | Kilogram (kg) |
| m | Mass of the smaller body | Kilogram (kg) |
| r | Distance between the centres of the two masses | Metre (m) |
Derivation of Gravitational Potential Energy Formula
The derivation of the gravitational potential energy formula follows directly from the definition of work done against gravity.
Step 1: To lift an object of mass m to height h, we apply an upward force equal to its weight: \( F = mg \).
Step 2: The work done against gravity over height h is: \( W = F \times h = mg \times h \).
Step 3: By the work-energy theorem, this work is stored as potential energy in the object.
Step 4: Therefore, \( PE = mgh \). This energy is recoverable when the object falls back to the reference level.
The derivation of the elastic potential energy formula uses integration of Hooke's Law: \( F = kx \). The work done to compress or stretch the spring by x is \( W = \int_0^x kx\, dx = \frac{1}{2}kx^2 \), which is stored as elastic PE.
Complete Physics Formula Sheet: Work, Energy and Power
| Formula Name | Expression | Variables | SI Units | NCERT Chapter |
|---|---|---|---|---|
| Gravitational Potential Energy | \( PE = mgh \) | m = mass, g = 9.8 m/s², h = height | Joule (J) | Class 9, Ch 11; Class 11, Ch 6 |
| Elastic Potential Energy | \( PE = \frac{1}{2}kx^2 \) | k = spring constant, x = displacement | Joule (J) | Class 11, Ch 6 |
| Universal Gravitational PE | \( U = -\frac{GMm}{r} \) | G = 6.67×10²²¹¹ N·m²/kg², M, m = masses, r = distance | Joule (J) | Class 11, Ch 8 |
| Kinetic Energy | \( KE = \frac{1}{2}mv^2 \) | m = mass, v = velocity | Joule (J) | Class 9, Ch 11; Class 11, Ch 6 |
| Work Done | \( W = Fs\cos\theta \) | F = force, s = displacement, θ = angle | Joule (J) | Class 9, Ch 11; Class 11, Ch 6 |
| Conservation of Mechanical Energy | \( KE + PE = \text{constant} \) | KE = kinetic energy, PE = potential energy | Joule (J) | Class 11, Ch 6 |
| Power | \( P = \frac{W}{t} \) | W = work done, t = time | Watt (W) | Class 9, Ch 11; Class 11, Ch 6 |
| Force from Potential Energy | \( F = -\frac{dU}{dx} \) | U = potential energy, x = position | Newton (N) | Class 11, Ch 6 |
| Electric Potential Energy | \( U = \frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r} \) | q&sub1;, q&sub2; = charges, r = distance, ε&sub0; = permittivity | Joule (J) | Class 12, Ch 2 |
| Gravitational Potential | \( V = -\frac{GM}{r} \) | G = gravitational constant, M = mass, r = distance | J/kg | Class 11, Ch 8 |
Potential Energy Formula — Solved Examples
Example 1 (Class 9-10 Level): Gravitational Potential Energy of a Book
Problem: A book of mass 2 kg is placed on a shelf 1.5 m above the floor. Calculate the gravitational potential energy stored in the book. (Take g = 10 m/s²)
Given:
- Mass, m = 2 kg
- Height, h = 1.5 m
- g = 10 m/s²
Step 1: Write the gravitational potential energy formula: \( PE = mgh \)
Step 2: Substitute the given values: \( PE = 2 \times 10 \times 1.5 \)
Step 3: Calculate: \( PE = 30 \text{ J} \)
Answer
The gravitational potential energy stored in the book is 30 Joules. This energy will convert into kinetic energy if the book falls off the shelf.
Example 2 (Class 11-12 Level): Elastic Potential Energy of a Compressed Spring
Problem: A spring with a spring constant of 500 N/m is compressed by 0.08 m from its natural length. Calculate the elastic potential energy stored in the spring. Also find the force exerted by the spring at this compression.
Given:
- Spring constant, k = 500 N/m
- Compression, x = 0.08 m
Step 1: Write the elastic potential energy formula: \( PE = \frac{1}{2}kx^2 \)
Step 2: Substitute values: \( PE = \frac{1}{2} \times 500 \times (0.08)^2 \)
Step 3: Calculate \( x^2 \): \( (0.08)^2 = 0.0064 \text{ m}^2 \)
Step 4: Calculate PE: \( PE = \frac{1}{2} \times 500 \times 0.0064 = 250 \times 0.0064 = 1.6 \text{ J} \)
Step 5: Find the spring force using Hooke's Law: \( F = kx = 500 \times 0.08 = 40 \text{ N} \)
Answer
Elastic Potential Energy stored = 1.6 Joules. The spring exerts a restoring force of 40 N at this compression.
Example 3 (JEE/NEET Level): Conservation of Energy on a Curved Track
Problem: A ball of mass 0.5 kg is released from rest at the top of a frictionless curved track at a height of 4 m. Find (a) the potential energy at the top, (b) the kinetic energy at the bottom, and (c) the speed of the ball at the bottom. (g = 10 m/s²)
Given:
- Mass, m = 0.5 kg
- Initial height, h = 4 m
- Initial velocity = 0 (released from rest)
- g = 10 m/s²
- Track is frictionless (no energy loss)
Step 1: Calculate the potential energy at the top using \( PE = mgh \):
\( PE = 0.5 \times 10 \times 4 = 20 \text{ J} \)
Step 2: At the top, KE = 0 (ball starts from rest). Total mechanical energy = PE + KE = 20 + 0 = 20 J.
Step 3: Apply conservation of mechanical energy. At the bottom, h = 0, so PE = 0. Therefore, all energy converts to KE:
\( KE_{\text{bottom}} = 20 \text{ J} \)
Step 4: Use the kinetic energy formula to find speed: \( KE = \frac{1}{2}mv^2 \)
\( 20 = \frac{1}{2} \times 0.5 \times v^2 \)
\( 20 = 0.25 v^2 \)
\( v^2 = 80 \)
\( v = \sqrt{80} = 4\sqrt{5} \approx 8.94 \text{ m/s} \)
Answer
(a) PE at the top = 20 J. (b) KE at the bottom = 20 J. (c) Speed at the bottom = 4√5 ≈ 8.94 m/s. This problem demonstrates the complete conversion of potential energy into kinetic energy on a frictionless surface.
CBSE Exam Tips 2025-26
- Always define your reference level. Potential energy is always measured relative to a reference point. In most CBSE problems, the ground is taken as the reference (h = 0). State this assumption clearly in your answer to earn full marks.
- Use g = 9.8 m/s² unless told otherwise. CBSE board papers sometimes specify g = 10 m/s² for easier calculation. Always check the question before substituting. We recommend writing the value you use at the top of your solution.
- Show the formula before substituting. CBSE marking schemes award one mark for writing the correct formula. Never skip this step, even if the calculation seems simple.
- Distinguish between types of PE. In 3-mark or 5-mark questions, CBSE may ask you to compare gravitational and elastic potential energy. Prepare a short comparison table as part of your revision.
- Practice conservation of energy problems. The combination of PE and KE using the conservation of mechanical energy is a favourite 3-mark question in Class 9 and Class 11 CBSE exams. Practise at least 10 such problems before your 2025-26 board exam.
- Include units in every step. Missing units in the final answer leads to a half-mark deduction in most CBSE marking schemes. Always write “Joules (J)” as the unit for potential energy.
Common Mistakes to Avoid with the Potential Energy Formula
- Mistake 1: Using the wrong reference level. Many students calculate height from a random point. Always measure h from the clearly defined reference level (usually the ground or the lowest point of motion). The correct approach is to define h = 0 at the start of every problem.
- Mistake 2: Forgetting the negative sign in universal gravitational PE. The formula \( U = -GMm/r \) has a negative sign. Students often drop this sign, which completely changes the physical meaning. The negative sign indicates that the system is bound — energy must be added to separate the two masses.
- Mistake 3: Confusing spring constant units. The spring constant k is measured in N/m, not N/m² or kg/s². Using wrong units for k leads to an incorrect answer for elastic PE. Always check that k is in N/m before substituting into \( PE = \frac{1}{2}kx^2 \).
- Mistake 4: Squaring only the number, not the unit in elastic PE. When calculating \( x^2 \), students sometimes write the number squared but forget that the unit also becomes m². The correct substitution is \( x^2 = (0.08 \text{ m})^2 = 0.0064 \text{ m}^2 \).
- Mistake 5: Treating potential energy as a vector. Potential energy is a scalar quantity. It has no direction. Some students mistakenly assign a direction to PE when solving inclined plane problems. The correct approach is to use only the vertical height h, not the length along the incline.
JEE/NEET Application of Potential Energy Formula
In our experience, JEE aspirants find potential energy problems among the most conceptually rich topics in mechanics. The potential energy formula appears in JEE Main and JEE Advanced almost every year, and NEET regularly tests energy conservation in biological and physical contexts. Here are the key application patterns:
Pattern 1: Energy Conservation on Inclined Planes and Projectiles
JEE frequently combines the gravitational potential energy formula with projectile motion. A typical problem asks for the speed of a projectile at a certain height, given its launch speed. The approach is straightforward. Apply \( KE_1 + PE_1 = KE_2 + PE_2 \). This gives \( \frac{1}{2}mv_1^2 + mgh_1 = \frac{1}{2}mv_2^2 + mgh_2 \). The mass cancels, and you solve for the unknown velocity. Our experts suggest practising at least 15 such problems from previous JEE papers.
Pattern 2: Spring-Mass Systems and Simple Harmonic Motion
NEET and JEE both test elastic potential energy in the context of simple harmonic motion (SHM). In SHM, the total energy is \( E = \frac{1}{2}kA^2 \), where A is the amplitude. At any displacement x, the PE is \( \frac{1}{2}kx^2 \) and the KE is \( \frac{1}{2}k(A^2 – x^2) \). JEE Advanced problems often ask for the position where PE equals KE, which occurs at \( x = A/\sqrt{2} \). This is a high-frequency question type.
Pattern 3: Gravitational PE in Orbital Mechanics
JEE Advanced tests the universal gravitational potential energy formula \( U = -GMm/r \) in the context of satellite orbits and escape velocity. A key result is that the total mechanical energy of a satellite in a circular orbit of radius r is \( E = -\frac{GMm}{2r} \), which is half the potential energy. NEET tests a simpler version: the escape velocity formula \( v_e = \sqrt{2gR} \), derived by setting the total mechanical energy to zero. In our experience, students who understand the sign convention of gravitational PE score significantly better on these questions.
Pattern 4: Force from Potential Energy (JEE Advanced)
JEE Advanced frequently gives a potential energy function \( U(x) \) and asks for the force or the equilibrium position. The formula \( F = -\frac{dU}{dx} \) is essential here. For equilibrium, \( F = 0 \), so \( \frac{dU}{dx} = 0 \). Students must be comfortable differentiating polynomial and trigonometric PE functions. This topic appears in Class 11 NCERT Chapter 6 but is tested at a much deeper level in JEE Advanced.
FAQs on Potential Energy Formula
Explore More Physics Formulas
Understanding the Potential Energy Formula becomes even more powerful when studied alongside related concepts. We recommend exploring the Critical Velocity Formula, which uses energy conservation to find the minimum speed needed at the top of a circular path. You should also study the Induced Voltage Formula to understand energy transformations in electromagnetic systems. For a broader foundation, visit our Complete Physics Formulas Hub, where our experts have compiled every important formula from Class 9 to Class 12 NCERT, organised chapter-wise for efficient revision. For official NCERT textbook content, refer to the NCERT official website.