Section 4.5: Elasticity and Hooke’s Law
Elastic materials deform under applied forces and return to their original shape when the forces are removed. Hooke’s Law describes the linear relationship between force and deformation for elastic objects.
Hooke's Law:
\[ F = k \Delta x \]
where \(F\) is the applied force, \(k\) is the spring constant, and \(\Delta x\) is the displacement from equilibrium.
Potential Energy Stored in a Spring:
\[ PE_s = \frac{1}{2} k (\Delta x)^2 \]
Example 1
A spring with constant \(k = 200 \, \text{N/m}\) is stretched 0.1 m. Find the restoring force.
\( F = k \Delta x = 200 \cdot 0.1 = 20 \, \text{N} \)
Example 2
How much potential energy is stored in the spring in Example 1?
\( PE_s = \frac{1}{2} k (\Delta x)^2 = 0.5 \cdot 200 \cdot 0.1^2 = 1 \, \text{J} \)
Practice Problems
- A spring has \(k = 150 \, \text{N/m}\). It is compressed 0.2 m. Find the force applied.
- Find the potential energy stored in the spring from Problem 1.
- A spring stretches 0.05 m under 10 N. Determine the spring constant \(k\).
- A block attached to a spring oscillates. Maximum displacement 0.1 m, \(k = 250 \, \text{N/m}\). Compute max potential energy.
- A spring is stretched 0.15 m, storing 4.5 J of energy. Find spring constant \(k\).