A2-4. Hooke's Law
1. Elastic Restoring Force
When an elastic object (like a spring or a rubber band) is stretched or compressed, it exerts a force that attempts to return the object to its original, undeformed length. This is known as a restoring force.
Hooke's Law:
For many materials, the restoring force is directly proportional to the displacement (extension or compression) from the equilibrium position, provided the limit of proportionality is not exceeded.
- $F_H$: The restoring force exerted by the spring (N)
- $k$: The spring constant, a measure of the stiffness of the spring (N/m)
- $x$: The displacement from the equilibrium (unstretched) position (m)
The Negative Sign: The minus sign is mathematically crucial. It indicates that the restoring force $F_H$ always acts in the opposite direction to the displacement $x$. If you pull a spring to the right (positive $x$), the spring pulls back to the left (negative $F_H$).
Example 1: Compressing a Spring
Problem: A car's suspension spring has a spring constant of 40000 N/m. How much force must be applied to compress the spring by 5.0 cm?
Solution:
- Step 1: Convert units to SI.
$x = 5.0\text{ cm} = 0.05\text{ m}$ - Step 2: Apply Hooke's Law.
Since we are looking for the magnitude of the applied force (which is equal and opposite to the restoring force), we can drop the negative sign for the calculation:
$$F = kx = (40000)(0.05) = \mathbf{2000\text{ N}}$$