The restoring force

An oscillation is simple harmonic when the force pulling the object back to the middle is proportional to how far it has been displaced, and always points back towards that middle position.

\[F = -kx\]

The restoring force for a mass on a spring

A mass on a spring at rest, and the same mass pulled down with the restoring force arrow pointing back up. — Pull the mass down and the spring pulls back up. The further you pull, the harder it pulls.

The minus sign is the whole idea: push the mass to the right and the force acts to the left. Combine that with Newton's second law, $F = ma$, and you get an equation relating acceleration to displacement.

\[a = -\frac{k}{m}\,x = -\omega^2 x\]

Because acceleration is proportional to displacement and opposite in sign, the motion traces out a sinusoid in time.

Check yourself

Have a go. If you get it wrong you will get a hint and one more try; after that we will show you how it works.

Question 1 10 points

A mass on a spring is displaced to the right of its rest position. In which direction does the restoring force act?

To the left, back towards the rest position
To the right, away from the rest position
Downwards, with gravity
There is no force until the mass is released

2 attempts left

Question 2 10 points

A spring of stiffness $k = 40\ \mathrm{N\,m^{-1}}$ carries a mass of $0.10\ \mathrm{kg}$. What is the angular frequency $\omega$? Give your answer in rad s⁻¹ to two significant figures.

2 attempts left