A-level Physics (Advancing Physics)/Energy in Simple Harmonic Motion/Worked Solutions

1. A 10g mass causes a spring to extend 5 cm. How much energy is stored by the spring?

Gravitational Potential Energy transferred completely to Elastic Potential Energy

Gravitational Potential Energy = mass x gravity x height

GPE $=0.01m\times 0.05kg\times 9.8Nk{g}^{-1}=4.9\times 10^{-3}J$

2. A 500g mass on a spring (k=100) is extended by 0.2m, and begins to oscillate in an otherwise empty universe. What is the maximum velocity which it reaches?

$\frac{1}{2}m{v}_{max}^2=\frac{1}{2}k{x}_{max}^2$

$v_{max}^2=\frac{k{x}_{max}^2}{m}$

$v_{max}=x_{max}\sqrt{\frac{k}{m}}=0.2\times \sqrt{\frac{100}{0.5}}=2.83{\text{ ms}}^{-1}$

3. Another 500g mass on another spring in another otherwise empty universe is extended by 0.5m, and begins to oscillate. If it reaches a maximum velocity of 15ms⁻¹, what is the spring constant of the spring?

$\frac{1}{2}m{v}_{max}^2=\frac{1}{2}k{x}_{max}^2$

$k=\frac{m{v}_{max}^2}{x_{max}^2}=\frac{0.5\times 15^2}{0.5^2}=450{\text{ Nm}}^{-1}$

4. Draw graphs of the kinetic and elastic energies of a mass on a spring (ignoring gravity).

$E_e\propto {\cos }^2\omega t$

$E_k\propto {\sin }^2\omega t$

Simple HArmonic Motion

5. Use the trigonometric formulae for x and v to derive an equation for the total energy stored by an oscillating mass on a spring, ignoring gravity and air resistance, which is constant with respect to time.

$x=A\cos \omega t$

$v=-A\omega \sin \omega t$

Substitute these into the equation for the total energy:

$\mathrm{\Sigma }E=\frac{1}{2}(k{x}^2+m{v}^2)=\frac{1}{2}(k(A\cos \omega t{)}^2+m(-A\omega \sin \omega t{)}^2)=\frac{1}{2}(k{A}^2{\cos }^2\omega t+m{A}^2{\omega }^2{\sin }^2\omega t)=\frac{A^2}{2}(k{\cos }^2\omega t+m{\omega }^2{\sin }^2\omega t)$

We know that:

$\omega =\sqrt{\frac{k}{m}}$

Therefore:

${\omega }^2=\frac{k}{m}$

By substitution:

$\mathrm{\Sigma }E=\frac{A^2}{2}(k{\cos }^2\omega t+\frac{mk}{m}{\sin }^2\omega t)=\frac{A^2}{2}(k{\cos }^2\omega t+k{\sin }^2\omega t)=\frac{k{A}^2}{2}({\cos }^2\omega t+{\sin }^2\omega t)=\frac{k{A}^2}{2}$