17.1. Simple harmonic oscillations
A subsection of Physics, 9702, through 17. Oscillations
Listing 10 of 70 questions
State, by reference to displacement, what is meant by simple harmonic motion. A mass is undergoing oscillations in a vertical plane. The variation with displacement x of the acceleration a of the mass is shown in . a x State two reasons why the motion of the mass is not simple harmonic. 1. 2. A block of wood is floating in a liquid, as shown in . oscillation of block liquid block The block is displaced vertically and then released. The variation with time t of the displacement y of the block from its equilibrium position is shown in . 0.4 0.8 1.2 1.6 2.0 2.4 2.0 1.5 1.0 0.5 –0.5 –1.0 –1.5 –2.0 t / s y / cm Use data from to determine the angular frequency ω of the oscillations, ω = rad s−1 the maximum vertical acceleration of the block. maximum acceleration = m s−2 The block has mass 120 g. The oscillations of the block are damped. Calculate the loss in energy of the oscillations of the block during the first three complete periods of its oscillations. energy loss = J
9702_s16_qp_41
THEORY
2016
Paper 4, Variant 1
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A metal block hangs vertically from one end of a spring. The other end of the spring is tied to a thread that passes over a pulley and is attached to a vibrator, as shown in . spring block pulley vibrator The vibrator is switched off. The metal block of mass 120 g is displaced vertically and then released. The variation with time t of the displacement y of the block from its equilibrium position is shown in . –3 –2 –1 y / cm 0.2 0.4 0.6 0.8 1.0 W / s For the vibrations of the block, calculate the angular frequency ω, ω = rad s−1 the energy of the vibrations. energy = J The vibrator is now switched on. The frequency of vibration is varied from 0.7f to 1.3f where f is the frequency of vibration of the block in . For the block, complete to show the variation with frequency of the amplitude of vibration. Label this line A. amplitude 0.7I I 1.3I frequency Some light feathers are now attached to the block in to increase air resistance. The frequency of vibration is once again varied from 0.7f to 1.3f. The new amplitude of vibration is measured for each frequency. On , draw a line to show the variation with frequency of the amplitude of vibration. Label this line B.
9702_s16_qp_42
THEORY
2016
Paper 4, Variant 2
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State, by reference to displacement, what is meant by simple harmonic motion. A mass is undergoing oscillations in a vertical plane. The variation with displacement x of the acceleration a of the mass is shown in . a x State two reasons why the motion of the mass is not simple harmonic. 1. 2. A block of wood is floating in a liquid, as shown in . oscillation of block liquid block The block is displaced vertically and then released. The variation with time t of the displacement y of the block from its equilibrium position is shown in . 0.4 0.8 1.2 1.6 2.0 2.4 2.0 1.5 1.0 0.5 –0.5 –1.0 –1.5 –2.0 t / s y / cm Use data from to determine the angular frequency ω of the oscillations, ω = rad s−1 the maximum vertical acceleration of the block. maximum acceleration = m s−2 The block has mass 120 g. The oscillations of the block are damped. Calculate the loss in energy of the oscillations of the block during the first three complete periods of its oscillations. energy loss = J
9702_s16_qp_43
THEORY
2016
Paper 4, Variant 3
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A metal plate is made to vibrate vertically by means of an oscillator, as shown in . direction of oscillations sand plate oscillator Some sand is sprinkled on to the plate. The variation with displacement y of the acceleration a of the sand on the plate is shown in . –10 y / mm a / m s–2 –5 –4 –3 –2 –1 –8 –6 –4 –2 Use to show how it can be deduced that the sand is undergoing simple harmonic motion. Calculate the frequency of oscillation of the sand. frequency = Hz The amplitude of oscillation of the plate is gradually increased beyond 8 mm. The frequency is constant. At one amplitude, the sand is seen to lose contact with the plate. For the plate when the sand first loses contact with the plate, state the position of the plate, calculate the amplitude of oscillation. amplitude = mm
9702_s18_qp_41
THEORY
2018
Paper 4, Variant 1
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State two conditions necessary for a mass to be undergoing simple harmonic motion. 1. 2. A trolley of mass 950 g is held on a horizontal surface by means of two springs attached to fixed points P and Q, as shown in . Q P trolley mass 950 g spring The springs, each having a spring constant k of 230 N m–1, are always extended. The trolley is displaced along the line of the springs and then released. The variation with time t of the displacement x of the trolley is shown in . x t1 t 1. State and explain whether the oscillations of the trolley are heavily damped, critically damped or lightly damped. 2. Suggest the cause of the damping. The acceleration a of the trolley of mass m may be assumed to be given by the expression a = – m k x d n . 1. Calculate the angular frequency ω of the oscillations of the trolley. ω = rad s–1 2. Determine the time t1 shown on . t1 = s
9702_s18_qp_42
THEORY
2018
Paper 4, Variant 2
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A metal plate is made to vibrate vertically by means of an oscillator, as shown in . direction of oscillations sand plate oscillator Some sand is sprinkled on to the plate. The variation with displacement y of the acceleration a of the sand on the plate is shown in . –10 y / mm a / m s–2 –5 –4 –3 –2 –1 –8 –6 –4 –2 Use to show how it can be deduced that the sand is undergoing simple harmonic motion. Calculate the frequency of oscillation of the sand. frequency = Hz The amplitude of oscillation of the plate is gradually increased beyond 8 mm. The frequency is constant. At one amplitude, the sand is seen to lose contact with the plate. For the plate when the sand first loses contact with the plate, state the position of the plate, calculate the amplitude of oscillation. amplitude = mm
9702_s18_qp_43
THEORY
2018
Paper 4, Variant 3
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A spring is hung vertically from a fixed point. A mass M is hung from the other end of the spring, as illustrated in . L mass M spring The mass is displaced downwards and then released. The subsequent motion of the mass is simple harmonic. The variation with time t of the length L of the spring is shown in . 1.0 0.2 0.4 0.6 0.8 t / s L / cm State: one time at which the mass is moving with maximum speed time = s one time at which the spring has maximum elastic potential energy. time = s Use data from to determine, for the motion of the mass: the angular frequency ω ω = rad s–1 the maximum speed maximum speed = m s–1 the magnitude of the maximum acceleration. maximum acceleration = m s–2 The mass M is now suspended from two springs, each identical to that in , as shown in . mass M Suggest and explain the change, if any, in the period of oscillation of the mass. A numerical answer is not required.
9702_s19_qp_42
THEORY
2019
Paper 4, Variant 2
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The piston in the cylinder of a car engine moves in the cylinder with simple harmonic motion. The piston moves between a position of maximum height in the cylinder to a position of minimum height, as illustrated in . 9.8 cm cylinder maximum height minimum height piston cylinder piston The distance moved by the piston between the positions shown in is 9.8 cm. The mass of the piston is 640 g. At one particular speed of the engine, the piston completes 2700 oscillations in 1.0 minute. For the oscillations of the piston in the cylinder, determine: the amplitude amplitude = cm the frequency frequency = Hz the maximum speed maximum speed = m s–1 the speed when the top of the piston is 2.3 cm below its maximum height. speed = m s–1 The acceleration of the piston varies. Determine the resultant force on the piston that gives rise to its maximum acceleration. force = N
9702_s20_qp_41
THEORY
2020
Paper 4, Variant 1
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A dish is made from a section of a hollow glass sphere. The dish, fixed to a horizontal table, contains a small solid ball of mass 45 g, as shown in . C ball mass 45 g surface of dish x The horizontal displacement of the ball from the centre C of the dish is x. Initially, the ball is held at rest with distance x = 3.0 cm. The ball is then released. The variation with time t of the horizontal displacement x of the ball from point C is shown in . –4 –2 –3 –1 x / cm t / s The motion of the ball in the dish is simple harmonic with its acceleration a given by the expression a = –(g R) x where g is the acceleration of free fall and R is a constant that depends on the dimensions of the dish and the ball. Use to show that the angular frequency ω of oscillation of the ball in the dish is 2.9 rad s–1. Use the information in to: determine R R = m calculate the speed of the ball as it passes over the centre C of the dish. speed = m s–1 Some moisture collects on the surface of the dish so that the motion of the ball becomes lightly damped. On the axes of , draw a line to show the lightly damped motion of the ball for the first 5.0 s after the release of the ball.
9702_s20_qp_42
THEORY
2020
Paper 4, Variant 2
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The piston in the cylinder of a car engine moves in the cylinder with simple harmonic motion. The piston moves between a position of maximum height in the cylinder to a position of minimum height, as illustrated in . 9.8 cm cylinder maximum height minimum height piston cylinder piston The distance moved by the piston between the positions shown in is 9.8 cm. The mass of the piston is 640 g. At one particular speed of the engine, the piston completes 2700 oscillations in 1.0 minute. For the oscillations of the piston in the cylinder, determine: the amplitude amplitude = cm the frequency frequency = Hz the maximum speed maximum speed = m s–1 the speed when the top of the piston is 2.3 cm below its maximum height. speed = m s–1 The acceleration of the piston varies. Determine the resultant force on the piston that gives rise to its maximum acceleration. force = N
9702_s20_qp_43
THEORY
2020
Paper 4, Variant 3
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Questions Discovered
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