The Mass-Spring System as a Simple Harmonic Oscillator
A mass on a string
- A mass on a spring is a simple harmonic oscillator (SHO). When the mass is pushed or pulled either side of the equilibrium position, there's a restoring force exerted on it.
- The size and direction of this restoring force is given by Hooke's Law
\(\(F=k\Delta L\)\) - For a displacement, \(x\), the restoring force becomes \(F=-kx\) (negative because the force acts in the opposite direction to displacement)
- Newton's second law states that the resultant force on an object equals the mass of the object times its acceleration, F = ma. Inserting this into the equation \(F = -kx\), and replacing a with the acceleration for an object oscillating with SHM, \(a = -\omega^2x = -(2\pi f)^2x\), gives: \(-(2\pi f)^2mx=-kx\)
- The \(x\)'s cancel and we are left with:
\(f=\frac{1}{2\pi}\sqrt{ \frac{k}{m} }\) - Subbing in \(f=\frac{1}{T}\), gives us the period of a mass oscillating on a spring.
- \(T=2\pi \sqrt{ \frac{m}{k} }\)
Springs in parallel or series
- When you place springs in parallel, they add normally: \(k_{\text{total}}=k_{1}+k_{2}+k_{3}\dots\)
- When springs are placed in series, the spring constants add inversely: \(\frac{1}{k_{\text{total}}}=\frac{1}{k_{1}}+\frac{1}{k_{2}}+\frac{1}{k_{3}}\dots\)