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

Damped and forced oscillations, resonance — practice question

A ball of mass $M$ is supported on a horizontal surface by two identical extended springs, as shown in Fig. 4.1. One spring is connected to a fixed point. The other spring is connected to an oscillator. The oscillator is switched off. The ball is displaced sideways along the axis of the springs and is then released. Figure 4.2 shows how the displacement $x$ of the ball varies with time $t$.
(a(i))[1]

State what damping means.

(a(ii))[1]

State the evidence in Fig. 4.2 that the ball's motion is damped.

(b(i))[2]

The acceleration $a$ of the ball and its displacement $x$ are connected by $a = -\left(\frac{2k}{M}\right)x$, where $k$ is the spring constant of one spring. The mass $M$ of the ball is $1.2\,\text{kg}$. Use information from Fig. 4.2 to determine the angular frequency $\omega$ of the ball's oscillations.

(b(ii))[2]

Use your answer in (i) to find the value of $k$.

(c(i))[2]

The oscillator is now switched on. Its amplitude of oscillation remains constant. The angular frequency of the oscillations is then increased gradually from $0.7\omega$ to $1.3\omega$, where $\omega$ is the angular frequency found in (b)(i). On the axes in Fig. 4.3, show how the amplitude $A$ of the ball's oscillation varies with angular frequency.

(c(ii))[2]

Some sand is now spread on the horizontal surface. The angular frequency of the oscillations is again increased gradually from $0.7\omega$ to $1.3\omega$. State two changes that happen to the line you have drawn on Fig. 4.3.

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