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

A mass oscillating on a spring in a gravity-free vacuum has two sorts of energy: kinetic energy and elastic (potential) energy. Kinetic energy is given by:

$$E_k = \frac{1}{2}mv^2$$

Elastic energy, or elastic potential energy, is given by:

$$E_e = \frac{1}{2}kx^2$$

So, the total energy stored by the oscillator is:

$$\Sigma E = \frac{1}{2}(mv^2 + kx^2)$$

This total energy is constant. However, the proportions of this energy which are kinetic and elastic change over time, since v and x change with time. If we give a spring a displacement, it has no kinetic energy, and a certain amount of elastic energy. If we let it go, that elastic energy is all converted into kinetic energy, and so, when the mass reaches its initial position, it has no elastic energy, and all the elastic energy it did have has been converted into kinetic energy. As the mass continues to travel, it is slowed by the spring, and so the kinetic energy is converted back into elastic energy - the same amount of elastic energy as it started out. The nature of the energy oscillates back and forth, but the total energy is constant.

If the mass is oscillating vertically in a gravitational field, the situation gets more complicated since the spring now has gravitational potential energy, elastic potential energy and kinetic energy. However, it turns out (if you do the maths) that the total energy is still constant, although the equilibrium position will be lower.

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

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?

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?

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

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.

/Worked Solutions/