Elastic potential energy is the energy stored when an elastic object is stretched or compressed. Springs, rubber bands, bows, and trampolines all store energy this way when they are deformed from their natural shape. This idea matters because it connects force, motion, and energy in many real systems, from car suspensions to launching devices.
For an ideal spring, the stored energy depends on how stiff the spring is and how far it is stretched or compressed.
Key Facts
- Hooke's law for an ideal spring: F = kx
- Elastic potential energy in a spring: E_e = 1/2 kx^2
- Spring constant units: k is measured in N/m
- Extension or compression x is measured from the natural length of the spring.
- The area under a force-extension graph gives the work done on the spring.
- For an ideal spring, the force-extension graph is a straight line through the origin, so area = 1/2 base times height = 1/2 xF.
Vocabulary
- Elastic potential energy
- Energy stored in an object when it is stretched, compressed, or bent and can return to its original shape.
- Spring constant
- A measure of a spring's stiffness, represented by k, equal to the force needed per meter of extension.
- Extension
- The change in length of a spring or elastic object from its natural length.
- Restoring force
- The force exerted by an elastic object that acts toward its original shape or natural length.
- Hooke's law
- The rule that the force in an ideal spring is proportional to its extension, written as F = kx.
Common Mistakes to Avoid
- Using the total spring length as x instead of the change in length is wrong because x must be measured from the natural length.
- Forgetting the 1/2 in E_e = 1/2 kx^2 is wrong because the spring force grows from zero to kx, so the energy is the triangular area under the graph.
- Using centimeters directly in the formula is wrong because k is usually in N/m, so x must be converted to meters.
- Thinking elastic potential energy doubles when x doubles is wrong because the energy depends on x^2, so doubling x makes the energy four times larger.
Practice Questions
- 1 A spring has k = 200 N/m and is stretched by 0.15 m. Calculate the elastic potential energy stored in the spring.
- 2 A spring stores 4.0 J of elastic potential energy when stretched by 0.20 m. Find the spring constant k.
- 3 Two identical springs are stretched, one by 5 cm and one by 10 cm. Explain which stores more elastic potential energy and by what factor.