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

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

k = \frac{\Delta F}{\Delta x} = \frac{mg}{\Delta x} = \frac{0.01 \times 9.81}{0.05} = 1.962\mbox{ Nm}^{-1}

E = \frac{1}{2}kx^2 = 0.5 \times 1.962 \times 0.05^2 = 2.45\mbox{ mJ}

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}mv_{max}^2 = \frac{1}{2}kx_{max}^2

v_{max}^2 = \frac{kx_{max}^2}{m}

v_{max} = x_{max}\sqrt{\frac{k}{m}} = 0.2 \times \sqrt{\frac{100}{0.5}} = 2.83\mbox{ 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}mv_{max}^2 = \frac{1}{2}kx_{max}^2

k = \frac{mv_{max}^2}{x_{max}^2} = \frac{0.5 \times 15^2}{0.5^2} = 450\mbox{ 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

Energy SHM.svg

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:

\Sigma E = \frac{1}{2}(kx^2 + mv^2) = \frac{1}{2}(k(A\cos{\omega t})^2 + m(-A\omega\sin{\omega t})^2) = \frac{1}{2}(kA^2\cos^2{\omega t} + mA^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:

\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{kA^2}{2}(\cos^2{\omega t} + \sin^2{\omega t}) = \frac{kA^2}{2}