Overview
This lecture covers Hooke's Law, the behavior of springs under force, calculations for spring problems, elastic potential energy, and the concept of work in stretching or compressing springs.
Hooke's Law and Spring Forces
- Hooke's Law states that the force needed to stretch or compress a spring is proportional to the displacement: ( F = kx ), where ( k ) is the spring constant.
- This proportionality holds only within the spring's elastic limit.
- The restoring force exerted by a spring is ( F_{restoring} = -kx ), directed opposite to the displacement.
- Stiffness of a spring increases with increasing ( k ); higher ( k ) means a stiffer spring._
Solving Spring Constant and Force Problems
- To find ( k ): ( k = F/x ); units are Newtons per meter (N/m).
- A stiffer spring (larger ( k )) requires more force to stretch the same distance.
- To find force: ( F = kx ), ensuring displacement ( x ) is in meters.
- To compare different forces and displacements, use ratios: ( F_1/x_1 = F_2/x_2 ).
Work Done on Springs
- The work required to stretch or compress a spring by distance ( x ) from its natural length: ( W = \frac{1}{2}kx^2 ).
- On a force-displacement graph, work equals the area under the curve, which is a triangle for Hooke’s Law.
- For stretching from position ( x_a ) to ( x_b ), work is ( W = \frac{1}{2}k(x_b^2 - x_a^2) ).
- Work done by the applied force is positive when force and displacement are in the same direction.
Elastic Potential Energy in Springs
- Elastic potential energy stored in a spring: ( PE = \frac{1}{2}kx^2 ), where ( x ) is the displacement from the natural length.
- Change in potential energy equals the work done if there’s no acceleration.
Key Terms & Definitions
- Hooke's Law — Principle stating force and displacement in a spring are proportional within elastic limits (( F = kx )).
- Spring Constant (k) — A measure of spring stiffness in N/m.
- Restoring Force — The force exerted by a spring to return to its natural length (( -kx )).
- Elastic Potential Energy — Energy stored in a spring, ( \frac{1}{2}kx^2 ).
Action Items / Next Steps
- Practice converting units, especially centimeters to meters, in all calculations.
- Solve additional problems using both Hooke's Law and energy methods.
- Review the difference between elastic potential energy and work done in stretching or compressing a spring.