Physics 9702 · AS & A Level · Damped and forced oscillations, resonance

Damped and forced oscillations, resonance — practice question

One end of a uniform beam is clamped in position. A metal block of mass $m$ is attached to the free end, so the beam bends, as illustrated in Fig. 3.1. The block is then given a small vertical displacement and released, after which it oscillates with simple harmonic motion. The acceleration $a$ of the block is described by $a = -\frac{k}{m}x$, where $k$ is a constant for the beam and $x$ is the vertical displacement of the block from its equilibrium position.
(a)[2]

Explain how the expression can be used to deduce that the block undergoes simple harmonic motion.

(b)[2]

For the beam, $k = 4.0\,\text{kg s}^{-2}$. Demonstrate that the angular frequency $\omega$ of the oscillations has the form $\omega = \frac{2.0}{\sqrt{m}}$.

(c)[3]

The initial amplitude of the block’s oscillation is $3.0\,\text{cm}$. Use the expression in (b) to find the maximum kinetic energy of the oscillations.

(d)[2]

Over a certain time interval, the maximum kinetic energy of the oscillations in (c) falls by $50\%$. Assume that the angular frequency of the oscillations changes by a negligible amount. Determine the amplitude of oscillation.

(e)[3]

Permanent magnets are now arranged so that the metal block oscillates between the poles, as shown in Fig. 3.2. The block is set oscillating with the same initial amplitude as in (c). Use energy conservation to explain why the energy of the oscillations decreases more rapidly than in (d).

Worked solution & mark scheme

This 12-mark question has a full step-by-step worked solution and mark scheme. One marking point: “$m$ constant or $k/m$ constant, so $a \propto x$” …

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