Spring Force and Hooke's Law

Overview

This lecture introduces the concept of spring force, Hooke's law, and the calculation of work done by a spring force, including worked examples.

A spring force is a variable force exerted by a spring when stretched or compressed from its relaxed state.
Springs have a relaxed (equilibrium) position where no force is applied.
The spring exerts a restoring force that tries to return the system to the relaxed state.
Hooke's law describes the spring force: ( F_s = -k \cdot x ), where ( k ) is the spring constant and ( x ) is displacement from equilibrium.
The negative sign indicates the force always opposes the displacement.
The spring constant ( k ) measures the stiffness of the spring: larger ( k ) means a stiffer spring.

The spring force is variable and depends linearly on displacement.
A slinky has a small ( k ), while a thick spring has a large ( k ).
The force increases the further you stretch or compress the spring from its relaxed state.

For variable forces like springs, work must be found using integration.
Work done by a spring force from ( x_i ) to ( x_f ):
( W_{spring} = \frac{1}{2}k x_i^2 - \frac{1}{2}k x_f^2 )
If the initial position is the relaxed state (( x_i = 0 )),
( W_{spring} = -\frac{1}{2}k x^2 )
Work by an applied force stretching or compressing the spring is equal in magnitude and opposite in sign to the work done by the spring.

If final position is closer to equilibrium than initial, work done by the spring is positive.
If moving across equilibrium (equal magnitudes), net work is zero.
If ending farther from equilibrium, work done by the spring is negative.

Spring Force — Force exerted by a spring when displaced from equilibrium.
Hooke's Law — The relationship ( F_s = -k \cdot x ) between force, displacement, and spring constant.
Spring Constant (k) — A measure of the stiffness of a spring.
Restoring Force — Force that acts to return a system to equilibrium.
Work Done by Spring — Energy transferred due to movement against or with the spring force.

Try additional practice problems on work done by spring force using the formulas provided.
Pause and answer concept checks or examples as prompted during study.