17.1. Simple harmonic oscillations
A subsection of Physics, 9702, through 17. Oscillations
Listing 10 of 70 questions
A body undergoes simple harmonic motion. The variation with displacement x of its velocity v is shown in . 0.1 – 0.1 – 0.02 0.02 0.04 0.06 x / m v / m s–1 – 0.04 – 0.06 – 0.2 – 0.3 – 0.4 0.2 0.3 0.4 State the amplitude xo of the oscillations. xo = m Calculate the period T of the oscillations. T = s On , label with a P a point where the body has maximum potential energy. A bar magnet is suspended from the free end of a spring, as shown in . spring magnet coil One pole of the magnet is situated in a coil of wire. The coil is connected in series with a switch and a resistor. The switch is open. The magnet is displaced vertically and then released. The magnet oscillates with simple harmonic motion. State Faraday’s law of electromagnetic induction. The switch is now closed. Explain why the oscillations of the magnet are damped.
9702_m20_qp_42
THEORY
2020
Paper 4, Variant 2
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The defining equation of simple harmonic motion is a = – ω 2x. State the significance of the minus (–) sign in the equation. A trolley rests on a bench. Two identical stretched springs are attached to the trolley as shown in . The other end of each spring is attached to a fixed support. 18.0 cm support support bench trolley spring spring The unstretched length of each spring is 12.0 cm. The spring constant of each spring is 8.0 N m–1. When the trolley is in equilibrium the length of each spring is 18.0 cm. The trolley is displaced 4.8 cm to one side and then released. Assume that resistive forces on the trolley are negligible. Show that the resultant force on the trolley at the moment of release is 0.77 N. The mass of the trolley is 250 g. Calculate the maximum acceleration a of the trolley. a = m s–2 Use your answer in to determine the period T of the subsequent oscillation. T = s The experiment is repeated with an initial displacement of the trolley of 2.4 cm. State and explain the effect, if any, this change has on the period of the oscillation of the trolley.
9702_m21_qp_42
THEORY
2021
Paper 4, Variant 2
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A small wooden block of mass m floats in water, as shown in . wooden block mass m water density ρ The top face of the block is horizontal and has area A. The density of the water is ρ. State the names of the two forces acting on the block when it is stationary. The block is now displaced downwards as shown in so that the surface of the water is higher up the block. new position of water surface original position of water surface State and explain the direction of the resultant force acting on the wooden block in this position. The block in is now released so that it oscillates vertically. The resultant force F acting on the block is given by F = –Agρx where g is the gravitational field strength and x is the vertical displacement of the block from the equilibrium position. Explain why the oscillations of the block are simple harmonic. Show that the angular frequency ω of the oscillations is given by ω = Aρ g m . The block is now placed in a liquid with a greater density. The block is displaced and released so that it oscillates vertically. The variation with displacement x of the acceleration a of the block is measured for the first half oscillation, as shown in . 0.01 –0.01 –0.02 0.02 x / m a / m s–2 –1 –2 Explain why the maximum negative displacement of the block is not equal to its maximum positive displacement. The mass of the block is 0.57 kg. Use to determine the decrease ΔE in energy of the oscillation for the first half oscillation. E = J
9702_m22_qp_42
THEORY
2022
Paper 4, Variant 2
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A tube, closed at one end, has a constant area of cross-section A. Some lead shot is placed in the tube so that the tube floats vertically in a liquid of density ρ, as shown in . The total mass of the tube and its contents is M. When the tube is given a small vertical displacement and then released, the vertical acceleration a of the tube is related to its vertical displacement y by the expression a = – y , where g is the acceleration of free fall. Define simple harmonic motion. Show that the tube is performing simple harmonic motion with a frequency f given by f =  . Aρg M 2π Aρg M tube, area of cross-section A lead shot liquid, density Use [T shows the variation with time t of the vertical displacement y of the tube in another liquid. The tube has an external diameter of 2.4 cm and is floating in a liquid of density 950 kg m–3. Assuming the equation in , calculate the mass of the tube and its contents. mass = kg State what feature of indicates that the oscillations are damped. –3 –2 –1 0.4 0.2 0.6 0.8 1.0 1.2 1.4 y / cm t / s
9702_s05_qp_4
THEORY
2005
Paper 4, Variant 0
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A piston moves vertically up and down in a cylinder, as illustrated in . The piston is connected to a wheel by means of a rod that is pivoted at the piston and at the wheel. As the piston moves up and down, the wheel is made to rotate. State the number of oscillations made by the piston during one complete rotation of the wheel. number = ………………………. The wheel makes 2400 revolutions per minute. Determine the frequency of oscillation of the piston. frequency = ………………………. Hz Use cylinder wheel pivot piston pivot P [T The amplitude of the oscillations of the piston is 42 mm. Assuming that these oscillations are simple harmonic, calculate the maximum values for the piston of the linear speed, speed = …………………………. m s–1 the acceleration. acceleration = …………………………. m s–2 On , mark a position of the pivot P for the piston to have maximum speed (mark this position S), maximum acceleration (mark this position A).
9702_s06_qp_4
THEORY
2006
Paper 4, Variant 0
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Define simple harmonic motion. A tube, sealed at one end, has a total mass m and a uniform area of cross-section A. The tube floats upright in a liquid of density ρ with length L submerged, as shown in a. tube L x liquid density ρ L L + + x L + x a b The tube is displaced vertically and then released. The tube oscillates vertically in the liquid. At one time, the displacement is x, as shown in b. Theory shows that the acceleration a of the tube is given by the expression a = – A ρg m x. Explain how it can be deduced from the expression that the tube is moving with simple harmonic motion. The tube, of area of cross-section 4.5 cm2, is floating in water of density 1.0 × 103 kg m–3. Calculate the mass of the tube that would give rise to oscillations of frequency 1.5 Hz. mass = g
9702_s11_qp_41
THEORY
2011
Paper 4, Variant 1
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A ball is held between two fixed points A and B by means of two stretched springs, as shown in . ball A B The ball is free to oscillate along the straight line AB. The springs remain stretched and the motion of the ball is simple harmonic. The variation with time t of the displacement x of the ball from its equilibrium position is shown in . 2.0 1.0 –1.0 –2.0 0.2 0.4 0.6 1.2 t / s x / cm 0.8 1.0 Use to determine, for the oscillations of the ball, 1. the amplitude, amplitude = cm 2. the frequency. frequency = Hz Show that the maximum acceleration of the ball is 5.2 m s–2. Use your answers in to plot, on the axes of , the variation with displacement x of the acceleration a of the ball. a / m s–2 x / 10–2 m Calculate the displacement of the ball at which its kinetic energy is equal to one half of the maximum kinetic energy. displacement = cm
9702_s13_qp_41
THEORY
2013
Paper 4, Variant 1
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A ball is held between two fixed points A and B by means of two stretched springs, as shown in . ball A B The ball is free to oscillate along the straight line AB. The springs remain stretched and the motion of the ball is simple harmonic. The variation with time t of the displacement x of the ball from its equilibrium position is shown in . 2.0 1.0 –1.0 –2.0 0.2 0.4 0.6 1.2 t / s x / cm 0.8 1.0 Use to determine, for the oscillations of the ball, 1. the amplitude, amplitude = cm 2. the frequency. frequency = Hz Show that the maximum acceleration of the ball is 5.2 m s–2. Use your answers in to plot, on the axes of , the variation with displacement x of the acceleration a of the ball. a / m s–2 x / 10–2 m Calculate the displacement of the ball at which its kinetic energy is equal to one half of the maximum kinetic energy. displacement = cm
9702_s13_qp_43
THEORY
2013
Paper 4, Variant 3
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State what is meant by simple harmonic motion. The variation with time t of the displacement x of two oscillators P and Q is shown in . x / cm −4 −2 −1 −3 2.8 2.4 2.0 1.6 1.2 0.8 0.4 t / s oscillator P oscillator Q The two oscillators each have the same mass. Use to determine the phase difference between the two oscillators, phase difference = rad the maximum acceleration of oscillator Q, maximum acceleration = m s–2 the ratio maximum kinetic energy of oscillations of Q maximum kinetic energy of oscillations of P . ratio = Use data from to sketch, on the axes of , the variation with displacement x of the acceleration a of oscillator Q. –3 –2 –1 a –4 x / cm
9702_s15_qp_42
THEORY
2015
Paper 4, Variant 2
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Questions Discovered
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