17. Oscillations
A section of Physics, 9702
Listing 10 of 100 questions
A uniform beam is clamped at one end. A metal block of mass m is fixed to the other end of the beam causing it to bend, as shown in . x clamp beam equilibrium position displaced position metal block mass m The block is given a small vertical displacement and then released so that it oscillates with simple harmonic motion. The acceleration a of the block is given by the expression a x m k =- where k is a constant for the beam and x is the vertical displacement of the block from its equilibrium position. Explain how it can be deduced from the expression that the block moves with simple harmonic motion. For the beam, k = 4.0 kg s–2. Show that the angular frequency ω of the oscillations is given by the expression ω . m 2 0 = . The initial amplitude of the oscillation of the block is 3.0 cm. Use the expression in to determine the maximum kinetic energy of the oscillations. maximum kinetic energy = J Over a certain interval of time, the maximum kinetic energy of the oscillations in is reduced by 50%. It may be assumed that there is negligible change in the angular frequency of the oscillations. Determine the amplitude of oscillation. amplitude = m Permanent magnets are now positioned so that the metal block oscillates between the poles, as shown in . beam metal block permanent magnets The block is made to oscillate with the same initial amplitude as in . Use energy conservation to explain why the energy of the oscillations decreases more rapidly than in .
9702_m17_qp_42
THEORY
2017
Paper 4, Variant 2
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Explain what is meant by the natural frequency of vibration of a system. A block of metal is fixed to one end of a vertical spring. The other end of the spring is attached to an oscillator, as shown in . oscillator metal block spring The amplitude of oscillation of the oscillator is constant. The variation of the amplitude x0 of the oscillations of the block with frequency f of the oscillations is shown in . f x0 Name the effect shown in . State and explain whether the block is undergoing damped oscillations. State one example in which the effect shown in is useful.
9702_m18_qp_42
THEORY
2018
Paper 4, Variant 2
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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 small object of mass 24 g rests on a platform. The platform is attached to an oscillator, as shown in . platform oscillator object The oscillator moves the platform up and down. The total energy of the oscillations of the object is 2.2 × 10–4 J. In one oscillation the object travels a total distance of 14 mm. Calculate the angular frequency ω of the oscillations. ω = rad s–1 The frequency of the oscillator is fixed, and the amplitude of the oscillations is gradually increased. Calculate the maximum amplitude of the oscillations so the object does not lose contact with the platform. amplitude = m The amplitude of the oscillations is increased so it is greater than the value in . State and explain the position in an oscillation where the object first loses contact with the platform.
9702_m24_qp_42
THEORY
2024
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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A tube, closed at one end, has a uniform area of cross-section. The tube contains some sand so that the tube floats upright in a liquid, as shown in . tube d sand liquid When the tube is at rest, the depth d of immersion of the base of the tube is 16 cm. The tube is displaced vertically and then released. The variation with time t of the depth d of the base of the tube is shown in . 1.0 2.0 3.0 d / cm t / s Use to determine, for the oscillations of the tube, the amplitude, amplitude = cm the period. period = s Calculate the vertical speed of the tube at a point where the depth d is 16.2 cm. speed = cm s–1 State one other depth d where the speed will be equal to that calculated in . d = cm Explain what is meant by damping. The liquid in is now cooled so that, although the density is unchanged, there is friction between the liquid and the tube as it oscillates. Having been displaced, the tube completes approximately 10 oscillations before coming to rest. On , draw a line to show the variation with time t of depth d for the first 2.5 s of the motion.
9702_s08_qp_4
THEORY
2008
Paper 4, Variant 0
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Questions Discovered
100