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17.1. Simple harmonic oscillations
A subsection of Physics, 9702, through 17. Oscillations
Listing 10 of 70 questions
State what is meant by simple harmonic motion. A trolley of mass m is held on a horizontal surface by means of two springs. One spring is attached to a fixed point P. The other spring is connected to an oscillator, as shown in . P spring spring trolley oscillator The springs, each having spring constant k of 130 N m−1, are always extended. The oscillator is switched off. The trolley is displaced along the line of the springs and then released. The resulting oscillations of the trolley are simple harmonic. The acceleration a of the trolley is given by the expression a = − 2k m x where x is the displacement of the trolley from its equilibrium position. The mass of the trolley is 840 g. Calculate the frequency f of oscillation of the trolley. f = Hz ⎛ ⎞ ⎝ ⎠ The oscillator in is switched on. The frequency of oscillation of the oscillator is varied, keeping its amplitude of oscillation constant. The amplitude of oscillation of the trolley is seen to vary. The amplitude is a maximum at the frequency calculated in . State the name of the effect giving rise to this maximum. At any given frequency, the amplitude of oscillation of the trolley is constant. Explain how this indicates that there are resistive forces opposing the motion of the trolley.
9702_s21_qp_41
THEORY
2021
Paper 4, Variant 1
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A U-shaped tube contains some liquid. The liquid column in each half of the tube has length L, as shown in . L L The liquid columns are displaced vertically. The liquid then oscillates in the tube. The liquid levels are displaced from the equilibrium positions as shown in . The acceleration a of the liquid in the tube is related to the displacement x by the expression a = − ⎛ ⎞ ⎝ ⎠ g L x where g is the acceleration of free fall. Explain how the expression shows that the liquid in the tube is undergoing simple harmonic motion. The length L of each liquid column is 18 cm. Determine the period T of the oscillations. T = s x x The oscillations of the liquid in the tube are damped. In any one complete cycle of the oscillations, the amplitude decreases by 6.0% of its value at the beginning of the oscillation. Determine the ratio energy of oscillations after 3 cycles initial energy of oscillations . ratio =
9702_s21_qp_42
THEORY
2021
Paper 4, Variant 2
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State what is meant by simple harmonic motion. A trolley of mass m is held on a horizontal surface by means of two springs. One spring is attached to a fixed point P. The other spring is connected to an oscillator, as shown in . P spring spring trolley oscillator The springs, each having spring constant k of 130 N m−1, are always extended. The oscillator is switched off. The trolley is displaced along the line of the springs and then released. The resulting oscillations of the trolley are simple harmonic. The acceleration a of the trolley is given by the expression a = − 2k m x where x is the displacement of the trolley from its equilibrium position. The mass of the trolley is 840 g. Calculate the frequency f of oscillation of the trolley. f = Hz ⎛ ⎞ ⎝ ⎠ The oscillator in is switched on. The frequency of oscillation of the oscillator is varied, keeping its amplitude of oscillation constant. The amplitude of oscillation of the trolley is seen to vary. The amplitude is a maximum at the frequency calculated in . State the name of the effect giving rise to this maximum. At any given frequency, the amplitude of oscillation of the trolley is constant. Explain how this indicates that there are resistive forces opposing the motion of the trolley.
9702_s21_qp_43
THEORY
2021
Paper 4, Variant 3
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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 . L x string bob oscillations equilibrium position (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 . –0.4 –0.2 0.2 0 0 –0.02 0.02 0.04 0.06 x / m a / m s–2 –0.04 –0.06 0.4 State how shows that the motion of the pendulum is simple harmonic. Use to determine the angular frequency ω of the oscillations. ω = rad s–1 The angular frequency ω is related to the length L of the pendulum by ω = L k where k is a constant. Use your answer in to determine k. Give a unit with your answer. k = unit While the pendulum is oscillating, the length of the string is increased in such a way that the total energy of the oscillations remains constant. Suggest and explain the qualitative effect of this change on the amplitude of the oscillations.
9702_s22_qp_41
THEORY
2022
Paper 4, Variant 1
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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 . L x string bob oscillations equilibrium position (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 . –0.4 –0.2 0.2 0 0 –0.02 0.02 0.04 0.06 x / m a / m s–2 –0.04 –0.06 0.4 State how shows that the motion of the pendulum is simple harmonic. Use to determine the angular frequency ω of the oscillations. ω = rad s–1 The angular frequency ω is related to the length L of the pendulum by ω = L k where k is a constant. Use your answer in to determine k. Give a unit with your answer. k = unit While the pendulum is oscillating, the length of the string is increased in such a way that the total energy of the oscillations remains constant. Suggest and explain the qualitative effect of this change on the amplitude of the oscillations.
9702_s22_qp_43
THEORY
2022
Paper 4, Variant 3
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A small steel sphere is oscillating vertically on the end of a spring, as shown in . steel sphere spring oscillations The velocity v of the sphere varies with displacement x from its equilibrium position according to v = ± 9.7 ( . ) x 11 6 - where v is in cm s–1 and x is in cm. Calculate the frequency of the oscillations. frequency = Hz Show that the amplitude of the oscillations is 3.4 cm. Calculate the maximum acceleration a0 of the sphere. a0 = m s–2 On , sketch the variation with x of the acceleration a of the sphere. – 4 – 2a0 – a0 2 a0 a0 a – 2 x / cm Describe, without calculation, the interchange between the potential energy and the kinetic energy of the oscillations.
9702_s23_qp_42
THEORY
2023
Paper 4, Variant 2
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State what is meant by resonance. A small ball is held in place using a stretched string. One end of the string is fixed to a wall and the other end is attached to a vibration generator, as shown in . ball string wall vibration generator Initially, the vibration generator is switched off. A student displaces the ball vertically and then releases it. shows the variation of the displacement of the ball with time after it is released. 0.1 0.2 0.3 0.4 0.5 time / s displacement 0.6 State the name of the phenomenon illustrated by the decrease in the amplitude of the oscillations in . Explain the decrease with time of the amplitude of the oscillations of the ball. Determine the frequency of the oscillations of the ball. frequency = Hz The vibration generator in is switched on and its frequency f of vibration is gradually increased from 0 to 10 Hz. On , sketch the variation with f of the amplitude of the oscillations of the ball. amplitude 2.5 5.0 f / Hz 7.5 10.0
9702_s24_qp_41
THEORY
2024
Paper 4, Variant 1
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A block of mass m oscillates vertically on a spring, as shown in . oscillations spring block equilibrium position The acceleration a of the block varies with displacement x from its equilibrium position, as shown in . –2A –A A a x 2A –Y –2Y Y 2Y 3Y –3Y The amplitude of the oscillations is 3Y and the maximum acceleration is 2A. Explain how shows that the oscillations of the block are simple harmonic. Deduce expressions, in terms of some or all of m, A and Y, for: the angular frequency ω of the oscillations ω = the maximum speed v0 of the oscillations v0 = the energy E of the oscillations. E = The period of the oscillations is 0.75 s and the value of 3Y is 1.8 cm. Determine an expression for x in terms of time t, where x is in cm and t is in seconds. x = , ,
9702_s24_qp_42
THEORY
2024
Paper 4, Variant 2
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State what is meant by resonance. A small ball is held in place using a stretched string. One end of the string is fixed to a wall and the other end is attached to a vibration generator, as shown in . ball string wall vibration generator Initially, the vibration generator is switched off. A student displaces the ball vertically and then releases it. shows the variation of the displacement of the ball with time after it is released. 0.1 0.2 0.3 0.4 0.5 time / s displacement 0.6 State the name of the phenomenon illustrated by the decrease in the amplitude of the oscillations in . Explain the decrease with time of the amplitude of the oscillations of the ball. Determine the frequency of the oscillations of the ball. frequency = Hz The vibration generator in is switched on and its frequency f of vibration is gradually increased from 0 to 10 Hz. On , sketch the variation with f of the amplitude of the oscillations of the ball. amplitude 2.5 5.0 f / Hz 7.5 10.0
9702_s24_qp_43
THEORY
2024
Paper 4, Variant 3
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The centre of the cone of a loudspeaker is oscillating with simple harmonic motion of frequency 1400 Hz and amplitude 0.080 mm. Calculate, to two significant figures, the angular frequency ω of the oscillations, ω = ………………………………. rad s–1 the maximum acceleration, in m s–2, of the centre of the cone. acceleration = ……………………………… m s–2 On the axes of , sketch a graph to show the variation with displacement x of the acceleration a of the centre of the cone. Use x a [T State the value of the displacement x at which the speed of the centre of the cone is a maximum. x = ……………………………… mm Calculate, in m s–1, this maximum speed. speed = ……………………………. m s–1
9702_w05_qp_4
THEORY
2005
Paper 4, Variant 0
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Questions Discovered
70