17.1. Simple harmonic oscillations
A subsection of Physics, 9702, through 17. Oscillations
Listing 10 of 70 questions
Define the radian. State, by reference to simple harmonic motion, what is meant by angular frequency. A thin metal strip, clamped horizontally at one end, has a load of mass M attached to its free end, as shown in . L x load mass M metal strip oscillation of load clamp The metal strip bends, as shown in . When the free end of the strip is displaced vertically and then released, the mass oscillates in a vertical plane. Theory predicts that the variation of the acceleration a of the oscillating load with the displacement x from its equilibrium position is given by a = – ML c c m x where L is the effective length of the metal strip and c is a positive constant. Explain how the expression shows that the load is undergoing simple harmonic motion. For a metal strip of length L = 65 cm and a load of mass M = 240 g, the frequency of oscillation is 3.2 Hz. Calculate the constant c. c = kg m3 s–2
9702_w17_qp_42
THEORY
2017
Paper 4, Variant 2
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State, by reference to simple harmonic motion, what is meant by angular frequency. A thin metal strip is clamped at one end so that it is horizontal. A load of mass M is attached to its free end. The load causes a displacement s of the end of the strip, as shown in . s load mass M metal strip clamp The load is displaced vertically and then released. The load oscillates. The variation with the acceleration a of the displacement s of the load is shown in . –1.0 –0.6 –0.2 0.2 a / m s–2 0.6 1.0 –0.8 –0.4 0.4 0.8 1.0 2.0 3.0 4.0 s / cm Use to determine 1. the displacement of the load before it is made to oscillate, displacement = cm 2. the amplitude of the oscillations of the load. amplitude = cm Show that the load is undergoing simple harmonic motion. Calculate the frequency of oscillation of the load. frequency = Hz
9702_w17_qp_43
THEORY
2017
Paper 4, Variant 3
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A U-tube contains liquid, as shown in . liquid liquid L x x The total length of the column of liquid in the tube is L. The column of liquid is displaced so that the change in height of the liquid in each arm of the U-tube is x, as shown in . The liquid in the U-tube then oscillates with simple harmonic motion such that the acceleration a of the column is given by the expression a = – L g e o x where g is the acceleration of free fall. Calculate the period T of oscillation of the liquid column for a column length L of 19.0 cm. T = s The variation with time t of the displacement x is shown in . +2.0 +1.0 T 2T 3T t x / cm –1.0 –2.0 The period of oscillation of the liquid column of mass 18.0 g is T. The oscillations are damped. Suggest one cause of the damping. Calculate the loss in total energy of the oscillations during the first 2.5 periods of the oscillations. energy loss = J
9702_w18_qp_41
THEORY
2018
Paper 4, Variant 1
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A U-tube contains liquid, as shown in . liquid liquid L x x The total length of the column of liquid in the tube is L. The column of liquid is displaced so that the change in height of the liquid in each arm of the U-tube is x, as shown in . The liquid in the U-tube then oscillates with simple harmonic motion such that the acceleration a of the column is given by the expression a = – L g e o x where g is the acceleration of free fall. Calculate the period T of oscillation of the liquid column for a column length L of 19.0 cm. T = s The variation with time t of the displacement x is shown in . +2.0 +1.0 T 2T 3T t x / cm –1.0 –2.0 The period of oscillation of the liquid column of mass 18.0 g is T. The oscillations are damped. Suggest one cause of the damping. Calculate the loss in total energy of the oscillations during the first 2.5 periods of the oscillations. energy loss = J
9702_w18_qp_43
THEORY
2018
Paper 4, Variant 3
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A mass is suspended vertically from a fixed point by means of a spring, as illustrated in . mass spring The mass is oscillating vertically. The variation with displacement x of the acceleration a of the mass is shown in . 0.5 –0.5 –1.0 –1.5 1.0 1.5 1.0 1.5 0.5 –0.5 –1.0 –1.5 x / cm a / m s–2 State what is meant by the displacement of the mass on the spring. Suggest how shows that the mass is not performing simple harmonic motion. The amplitude of oscillation of the mass may be changed. State the maximum amplitude x0 for which the oscillations are simple harmonic. x0 = cm For the simple harmonic oscillations of the mass, use to determine the frequency of the oscillations. frequency = Hz The maximum speed of the mass when oscillating with simple harmonic motion of amplitude x0 is v0. On , show the variation with displacement x of the velocity v of the mass for displacements from +x0 to –x0. –x0 x0 v0 v –v0 x
9702_w19_qp_41
THEORY
2019
Paper 4, Variant 1
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A ball of mass M is held on a horizontal surface by two identical extended springs, as illustrated in . ball mass M oscillator fixed point One spring is attached to a fixed point. The other spring is attached to an oscillator. The oscillator is switched off. The ball is displaced sideways along the axis of the springs and is then released. The variation with time t of the displacement x of the ball is shown in . –1.5 –1.0 –0.5 0.5 1.0 1.5 0.4 0.8 1.2 1.6 2.0 2.4 t / s x / cm State: what is meant by damping the evidence provided by that the motion of the ball is damped. The acceleration a and the displacement x of the ball are related by the expression – 2 a M k x = c m where k is the spring constant of one of the springs. The mass M of the ball is 1.2 kg. Use data from to determine the angular frequency ω of the oscillations of the ball. ω = rad s–1 Use your answer in to determine the value of k. k = N m–1 The oscillator is switched on. The amplitude of oscillation of the oscillator is constant. The angular frequency of the oscillations is gradually increased from 0.7ω to 1.3ω, where ω is the angular frequency calculated in . On the axes of , show the variation with angular frequency of the amplitude A of oscillation of the ball. A 0.7ω 1.0 angular frequency ω 1.3ω Some sand is now sprinkled on the horizontal surface. The angular frequency of the oscillations is again gradually increased from 0.7ω to 1.3ω. State two changes that occur to the line you have drawn on . 1. 2.
9702_w19_qp_42
THEORY
2019
Paper 4, Variant 2
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A mass is suspended vertically from a fixed point by means of a spring, as illustrated in . mass spring The mass is oscillating vertically. The variation with displacement x of the acceleration a of the mass is shown in . 0.5 –0.5 –1.0 –1.5 1.0 1.5 1.0 1.5 0.5 –0.5 –1.0 –1.5 x / cm a / m s–2 State what is meant by the displacement of the mass on the spring. Suggest how shows that the mass is not performing simple harmonic motion. The amplitude of oscillation of the mass may be changed. State the maximum amplitude x0 for which the oscillations are simple harmonic. x0 = cm For the simple harmonic oscillations of the mass, use to determine the frequency of the oscillations. frequency = Hz The maximum speed of the mass when oscillating with simple harmonic motion of amplitude x0 is v0. On , show the variation with displacement x of the velocity v of the mass for displacements from +x0 to –x0. –x0 x0 v0 v –v0 x
9702_w19_qp_43
THEORY
2019
Paper 4, Variant 3
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A pendulum consists of a metal sphere P suspended from a fixed point by means of a thread, as illustrated in . metal sphere P thread L x The centre of gravity of sphere P is a distance L from the fixed point. The sphere is pulled to one side and then released so that it oscillates. The sphere may be assumed to oscillate with simple harmonic motion. State what is meant by simple harmonic motion. The variation of the velocity v of sphere P with the displacement x from its mean position is shown in . 0.1 0.2 0.3 –0.3 –0.2 –0.1 –2 –4 –6 –8 –10 v / m s–1 x / cm Use to determine the frequency f of the oscillations of sphere P. f = Hz The period T of the oscillations of sphere P is given by the expression T = 2π g L c m where g is the acceleration of free fall. Use your answer in to determine the length L. L = m Another pendulum consists of a sphere Q suspended by a thread. Spheres P and Q are identical. The thread attached to sphere Q is longer than the thread attached to sphere P. Sphere Q is displaced and then released. The oscillations of sphere Q have the same amplitude as the oscillations of sphere P. On , sketch the variation of the velocity v with displacement x for sphere Q.
9702_w20_qp_41
THEORY
2020
Paper 4, Variant 1
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A simple pendulum consists of a metal sphere suspended from a fixed point by means of a thread, as illustrated in . thread sphere mass 94.0 g 0.90 cm 12.7 cm L (not to scale) The sphere of mass 94.0 g is displaced to one side through a horizontal distance of 12.7 cm. The centre of gravity of the sphere rises vertically by 0.90 cm. The sphere is released so that it oscillates. The sphere may be assumed to oscillate with simple harmonic motion. State what is meant by simple harmonic motion. State the kinetic energy of the sphere when the sphere returns to the displaced position shown in . kinetic energy = J Calculate the total energy ET of the oscillations. ET = J Use your answer in to show that the angular frequency ω of the oscillations of the pendulum is 3.3 rad s–1. The period T of oscillation of the pendulum is given by the expression T = 2π  L g where g is the acceleration of free fall and L is the length of the pendulum. Use data from to determine L. L = m
9702_w20_qp_42
THEORY
2020
Paper 4, Variant 2
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A pendulum consists of a metal sphere P suspended from a fixed point by means of a thread, as illustrated in . metal sphere P thread L x The centre of gravity of sphere P is a distance L from the fixed point. The sphere is pulled to one side and then released so that it oscillates. The sphere may be assumed to oscillate with simple harmonic motion. State what is meant by simple harmonic motion. The variation of the velocity v of sphere P with the displacement x from its mean position is shown in . 0.1 0.2 0.3 –0.3 –0.2 –0.1 –2 –4 –6 –8 –10 v / m s–1 x / cm Use to determine the frequency f of the oscillations of sphere P. f = Hz The period T of the oscillations of sphere P is given by the expression T = 2π g L c m where g is the acceleration of free fall. Use your answer in to determine the length L. L = m Another pendulum consists of a sphere Q suspended by a thread. Spheres P and Q are identical. The thread attached to sphere Q is longer than the thread attached to sphere P. Sphere Q is displaced and then released. The oscillations of sphere Q have the same amplitude as the oscillations of sphere P. On , sketch the variation of the velocity v with displacement x for sphere Q.
9702_w20_qp_43
THEORY
2020
Paper 4, Variant 3
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