Elastic Potential Energy
Understanding stored energy in stretched or compressed objects - springs, rubber bands, and more for GCSE, Grade 9-10, Year 9-10
What is Elastic Potential Energy?
Elastic potential energy (EPE) is the energy stored in elastic materials when they are stretched or compressed. This energy is 'potential' because it has the potential to do work when the material returns to its original shape.
Spring Constant (k)
The stiffness of the spring. Higher k means a stiffer spring that's harder to stretch.
Extension (x)
How far the spring is stretched or compressed from its natural length.
Energy Stored (EPE)
EPE increases with the square of extension. Double extension = 4× energy!
Key Concept
Elastic potential energy is stored when work is done to deform an elastic object. When the force is removed, this stored energy is released as the object returns to its original shape. This is described by Hooke's Law for materials within their elastic limit.
The EPE Formula
Elastic Potential Energy Formula
Where elastic potential energy (EPE) equals one-half times spring constant (k) times extension squared (x²)
Hooke's Law Connection
Elastic potential energy is related to Hooke's Law, which states: F = k × x, where:
- F = Force applied to stretch/compress the spring (N)
- k = Spring constant (N/m)
- x = Extension/compression (m)
The EPE formula (½kx²) comes from calculating the work done to stretch the spring, which is the area under the force-extension graph.
Derivation of the Formula
The EPE formula comes from calculating the work done to stretch a spring. Since the force needed increases as the spring stretches, we need to use integration or calculate the area under the force-extension graph.
Derivation Steps
Hooke's Law
The force needed to stretch a spring is proportional to the extension: F = k × x
Work Done
Work done (W) = average force × distance. For a changing force, work is the area under the force-extension graph.
Force-Extension Graph
The graph of F = kx is a straight line through the origin. The area under this line to extension x is a triangle.
Calculate Area
Base = extension (x), Height = force at full extension (kx)
Simplify
This area equals the work done to stretch the spring, which is stored as elastic potential energy.
Important Note
The EPE formula only applies when the material obeys Hooke's Law (within the elastic limit). Beyond this limit, the material may be permanently deformed or break, and the formula doesn't apply.
Interactive Spring System Simulation
Explore how mass, spring constant, and gravity affect elastic potential energy in a hanging spring-mass system.
Spring System Calculations
The mass stretches the spring until the spring force (kx) equals the weight (mg). At equilibrium: kx = mg, so x = mg/k.
Graphical Representation
EPE increases with the square of extension, creating a parabolic relationship. The force-extension graph is linear (Hooke's Law).
EPE vs. Extension
EPE increases with extension squared. Double extension = 4× EPE!
Force vs. Extension (Hooke's Law)
Force increases linearly with extension. Slope = spring constant (k).
Real-World Examples of EPE
Elastic potential energy is all around us. Here are some common examples:
Archery Bows
When an archer pulls back the bowstring, they store EPE in the bent bow. Releasing transfers this energy to the arrow as kinetic energy.
Calculation: For k = 200 N/m bow pulled back 0.3 m: EPE = ½ × 200 × (0.3)² = 9 J
Car Suspension
Springs in car suspension store EPE when compressed by bumps. This energy is then gradually released, giving a smoother ride.
Calculation: For k = 20,000 N/m spring compressed 0.05 m: EPE = ½ × 20000 × (0.05)² = 25 J
Trampolines
When you jump on a trampoline, the springs stretch and store EPE. This energy then pushes you back up on the next bounce.
Calculation: For k = 1000 N/m trampoline stretched 0.2 m: EPE = ½ × 1000 × (0.2)² = 20 J
Solved Example Problems (GCSE Level)
Example 1: Basic EPE Calculation
GCSE FoundationA spring has a spring constant of 80 N/m. It is stretched by 0.15 m from its natural length. Calculate the elastic potential energy stored in the spring.
Step 1: Write the formula
EPE = ½ × k × x²
Step 2: Identify known values
k = 80 N/m, x = 0.15 m
Step 3: Substitute values into formula
EPE = ½ × 80 × (0.15)²
Step 4: Calculate extension squared
(0.15)² = 0.15 × 0.15 = 0.0225
EPE = ½ × 80 × 0.0225
Step 5: Perform multiplication
½ × 80 = 40
40 × 0.0225 = 0.9
Step 6: State the answer with units
EPE = 0.9 J
Example 2: Finding Spring Constant from EPE
GCSE HigherA spring stores 2.0 J of elastic potential energy when stretched by 0.1 m. Calculate the spring constant of the spring.
Step 1: Write the formula
EPE = ½ × k × x²
Step 2: Rearrange to solve for k
k = (2 × EPE) ÷ x²
Step 3: Identify known values
EPE = 2.0 J, x = 0.1 m
Step 4: Substitute values into formula
k = (2 × 2.0) ÷ (0.1)²
Step 5: Calculate numerator and denominator
2 × 2.0 = 4.0
(0.1)² = 0.01
k = 4.0 ÷ 0.01
Step 6: Perform division
4.0 ÷ 0.01 = 400
Step 7: State the answer with units
Spring constant = 400 N/m
Example 3: Energy Conversion - Spring Launcher
Grade 10A spring with constant 500 N/m is compressed 0.08 m and used to launch a 0.02 kg ball horizontally. Assuming all EPE converts to kinetic energy, calculate the speed of the ball as it leaves the spring.
Step 1: Calculate EPE stored in spring
EPE = ½ × k × x² = ½ × 500 × (0.08)²
EPE = ½ × 500 × 0.0064 = 1.6 J
Step 2: Set EPE equal to kinetic energy
Assuming all EPE converts to KE: EPE = KE = ½ × m × v²
1.6 = ½ × 0.02 × v²
Step 3: Rearrange to solve for v²
v² = (2 × EPE) ÷ m = (2 × 1.6) ÷ 0.02
v² = 3.2 ÷ 0.02 = 160
Step 4: Take square root to find v
v = √160 ≈ 12.65
Step 5: State the answer with units
Speed = 12.7 m/s (to 3 significant figures)
Step 6: Interpretation
The compressed spring stores 1.6 J of EPE, which converts to kinetic energy, launching the ball at 12.7 m/s.
Practice Problems (Unsolved)
Test your understanding with these GCSE-level problems. Try to solve them yourself before checking the answers!
Problem 1: Rubber Band Energy
GCSE FoundationA rubber band has a spring constant of 40 N/m. It is stretched by 0.25 m. Calculate the elastic potential energy stored in the rubber band.
Problem 2: Finding Extension from EPE
GCSE HigherA spring with constant 200 N/m stores 6.25 J of elastic potential energy. Calculate the extension of the spring.
Problem 3: Spring-Mass System
Grade 10 ChallengeA 0.8 kg mass hangs from a spring with constant 160 N/m. Calculate:
- The extension of the spring at equilibrium (use g = 10 N/kg)
- The elastic potential energy stored in the spring at this extension
EPE Calculator
Use this calculator to solve for any variable in the EPE equation (EPE = ½ × k × x²).