Hooke's Law – Springs and Elasticity
Understand the relationship between force and extension in elastic materials
Springs and Elasticity
Force, extension, and stored energy
Hooke's Law states that the force needed to extend a spring is directly proportional to the extension, provided the elastic limit is not exceeded:
Force = spring constant × extension
F = k × x
- F = Force applied (Newtons, N)
- k = Spring constant (N/m) - measure of stiffness
- x = Extension (metres, m) - increase beyond natural length
A higher spring constant means a stiffer spring that requires more force to extend.
Elastic deformation: The material returns to its original shape when the force is removed. Hooke's Law applies in this region.
Plastic deformation: The material is permanently stretched and doesn't return to its original shape. This happens when the force exceeds the elastic limit.
Force-Extension Graph
- • Straight line through origin = Hooke's Law region
- • Gradient = spring constant (k)
- • Graph curves beyond elastic limit
When a spring is stretched, work is done and energy is stored as elastic potential energy:
Elastic PE = ½ × k × x²
Eₚ = ½kx²
This energy can be released when the spring is let go, converting to kinetic energy (e.g., in a spring-loaded toy).
Natural length: 100 units
Current length: 120 units
Extension: 20.0 units
Controls
Higher k = stiffer spring
Hooke's Law Calculation
F = k × x
40 = 20 × x
x = 40 ÷ 20 = 2.00 m
Elastic limit: 100 N
Force vs Extension Graph
Hooke's Law
Question:
A spring has a spring constant of 40 N/m. (a) What force is needed to extend it by 0.25 m? (b) How much elastic potential energy is stored?
Answer:
(a) Force required:
F = k × x
F = 40 × 0.25 = 10 N
(b) Elastic potential energy:
Eₚ = ½ × k × x²
Eₚ = ½ × 40 × 0.25²
Eₚ = ½ × 40 × 0.0625
Eₚ = 1.25 J
A spring has a spring constant of 25 N/m. What force is needed to extend it by 0.4 m?