CAIE A-Level Physics A2 17.1 Simple Harmonic Oscillations Questions

Practise defining SHM, interpreting phase and parameters, applying displacement, velocity and acceleration equations and analysing their graphs.

Exam points

interpret SHM parameters, equations and phase relationships
apply a = −ω²x, x = x₀ sinωt and the SHM velocity relationships
analyse displacement, velocity and acceleration graphs in SHM

Covered syllabus points

17.1.1—The terms displacement, amplitude, period, frequency, angular frequency and 17.1.2—Simple harmonic motion occurs when acceleration is proportional to 17.1.3—A = –ω2x and recall and use, as a solution to this equation, x = x0 sin ωt 17.1.4—The equations v = v0 cos ωt and v = ± ω ()xx0 22−17.1.5—Graphical representations of the variations of displacement, velocity and

[Maximum number: 3]

Fig. 4.1 shows the variation with time t of the height h above the ground of an object of mass 36 kg that is undergoing vertical simple harmonic motion.

Fig. 4.1

(a)

For the oscillations of the object:

[ 3 ]

(i)

determine the amplitude $x_{0}$ , in cm

x_{0}=

[ 1 ]

$x_{0}=8.0 \mathrm{~cm}$

(ii)

show that the angular frequency $\omega$ is $1.6 \mathrm{rad} \mathrm{s}^{-1}$

[ 2 ]

$\omega=2 \pi / T$ $=2 \pi / 4.0=1.6 \mathrm{rad} \mathrm{s}^{-1}$

[Maximum number: 6]

A pendulum consists of a bob (small metal sphere) attached to the end of a piece of string. The other end of the string is attached to a fixed point. The bob oscillates with small oscillations about its equilibrium position, as shown in Fig. 4.1.

Fig. 4.1 (not to scale)

The length L of the pendulum, measured from the fixed point to the centre of the bob, is 1.24 m .
The acceleration a of the bob varies with its displacement x from the equilibrium position as shown in Fig. 4.2.

Fig. 4.2

(a)

State how Fig. 4.2 shows that the motion of the pendulum is simple harmonic.

[ 2 ]

(b)

Use Fig. 4.2 to determine the angular frequency $\omega$ of the oscillations.

\omega=\ldots \ldots \ldots \ldots \ldots . \ldots \ldots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ r a d ~ s^{-1}

[ 2 ]

$a_{0}=\omega^{2} x_{0}$

or
$a=-\omega^{2} x$

or
$\omega^{2}=- \text { gradient }$ $\begin{aligned} \omega =\sqrt{ }(0.40 / 0.050) =2.8 \mathrm{rad} \mathrm{~s}^{-1} \end{aligned}$

(c)

The angular frequency $\omega$ is related to the length L of the pendulum by

\omega=\sqrt{\frac{k}{L}}

where k is a constant.

Use your answer in (b)(i) to determine k. Give a unit with your answer.

k =unit..............

[ 2 ]

$\begin{aligned} k =\omega^{2} L =2.8^{2} \times 1.24 \end{aligned}$ $=9.7 \mathrm{~m} \mathrm{~s}^{-2}$

[Maximum number: 4]

A small object of mass 24 g rests on a platform. The platform is attached to an oscillator, as shown in Fig. 3.1.

Fig. 3.1

The oscillator moves the platform up and down.

(a)

The frequency of the oscillator is fixed, and the amplitude of the oscillations is gradually increased.

[ 4 ]

(i)

Calculate the maximum amplitude of the oscillations so the object does not lose contact with the platform.
amplitude = m

[ 2 ]

use of acceleration $=9.81 \mathrm{~m} \mathrm{~s}^{-2}$ $x_{0}=9.81 / 39^{2}$ $=6.4 \times 10^{-3} \mathrm{~m}$

(ii)

The amplitude of the oscillations is increased so it is greater than the value in (b)(i).

State and explain the position in an oscillation where the object first loses contact with the platform.

[ 2 ]

[Maximum number: 2]

Fig. 4.1 (not to scale)

Fig. 4.2

State how Fig. 4.2 shows that the motion of the pendulum is simple harmonic.

[Maximum number: 2]

An object is suspended from a spring that is attached to a fixed point as shown in Fig. 3.1.

Fig. 3.1

The object oscillates vertically with simple harmonic motion about its equilibrium position.

The oscillations of the object are now lightly damped.

Assume that the damping does not change the angular frequency of the oscillations.

On Fig. 3.2, sketch the variation with x of v when the amplitude of the oscillations is 0.060 m .

closed loop surrounding the origin with maximum x at $\pm 0.060 \mathrm{~m}$ passing through v=0
maximum velocity shown as $\pm 0.10 \mathrm{~m} \mathrm{~s}^{-1}$ passing through x=0

[Maximum number: 2]

Fig. 4.1 shows a heavy pendulum and a light pendulum, both suspended from the same piece of string. This string is secured at each end to fixed points.

Fig. 4.1

Both pendulums have the same natural frequency.
The heavy pendulum is set oscillating perpendicular to the plane of the diagram. As it oscillates, it causes the light pendulum to oscillate.

Fig. 4.2 shows the variation with time t of the displacements of the two pendulums for three oscillations.

Fig. 4.2

The variation with t of the displacement x of the light pendulum is given by

x=0.25 \sin 5.0 \pi t

where x is in centimetres and t is in seconds.

On Fig. 4.2, label both of the axes with the correct scales. Use the space below for any additional working that you need.

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Covered syllabus points

Question 1

Question (a)

Question (i)

Question (ii)

Question 2

Question (a)

Question (b)

Question (c)

Question 3

Question (a)

Question (i)

Question (ii)

Question 4

Question 5

Question 6