17.3. Damped and forced oscillations, resonance
A subsection of Physics, 9702, through 17. Oscillations
Listing 10 of 21 questions
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
View
A bar magnet of mass 180 g is suspended from the free end of a spring, as illustrated in . spring magnet coil The magnet hangs so that one pole is near the centre of a coil of wire. The coil is connected in series with a resistor and a switch. The switch is open. The magnet is displaced vertically and then allowed to oscillate with one pole remaining inside the coil. The other pole remains outside the coil. At time t = 0, the magnet is oscillating freely as it passes through its equilibrium position. At time t = 3.0 s, the switch in the circuit is closed. The variation with time t of the vertical displacement y of the magnet is shown in . –2.0 –1.5 –1.0 –0.5 0.5 1.0 y / cm 1.5 2.0 t / s Determine, to two significant figures, the frequency of oscillation of the magnet. frequency = Hz State whether the closing of the switch gives rise to light, heavy or critical damping. Calculate the change in the energy ΔE of oscillation of the magnet between time t = 2.7 s and time t = 7.5 s. Explain your working. ΔE = J
9702_s17_qp_41
THEORY
2017
Paper 4, Variant 1
View
A bar magnet of mass 180 g is suspended from the free end of a spring, as illustrated in . spring magnet coil The magnet hangs so that one pole is near the centre of a coil of wire. The coil is connected in series with a resistor and a switch. The switch is open. The magnet is displaced vertically and then allowed to oscillate with one pole remaining inside the coil. The other pole remains outside the coil. At time t = 0, the magnet is oscillating freely as it passes through its equilibrium position. At time t = 3.0 s, the switch in the circuit is closed. The variation with time t of the vertical displacement y of the magnet is shown in . –2.0 –1.5 –1.0 –0.5 0.5 1.0 y / cm 1.5 2.0 t / s Determine, to two significant figures, the frequency of oscillation of the magnet. frequency = Hz State whether the closing of the switch gives rise to light, heavy or critical damping. Calculate the change in the energy ΔE of oscillation of the magnet between time t = 2.7 s and time t = 7.5 s. Explain your working. ΔE = J
9702_s17_qp_43
THEORY
2017
Paper 4, Variant 3
View
A hollow tube, sealed at one end, has a cross-sectional area A of 24 cm2. The tube contains sand so that the total mass M of the tube and sand is 0.23 kg. The tube floats upright in a liquid of density t, as illustrated in . liquid, density t tube, area of cross-section A h sand The depth of the bottom of the tube below the liquid surface is h. The tube is displaced vertically and then released. The variation with time t of the depth h is shown in . 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t / s h / cm Determine: the amplitude, in metres, of the oscillations amplitude = m the frequency of oscillation of the tube in the liquid frequency = Hz the acceleration of the tube when h is a maximum. acceleration = m s–2 The frequency f of oscillation of the tube is given by the expression f = 1 2π M A g c m t where g is the acceleration of free fall. Calculate the density t of the liquid in which the tube is floating. t = kg m–3 The oscillations illustrated in are undamped. In practice, the liquid does cause light damping. On , draw a line to show light damping of the oscillations for time t = 0 to time t = 1.4 s.
9702_s19_qp_41
THEORY
2019
Paper 4, Variant 1
View
A hollow tube, sealed at one end, has a cross-sectional area A of 24 cm2. The tube contains sand so that the total mass M of the tube and sand is 0.23 kg. The tube floats upright in a liquid of density t, as illustrated in . liquid, density t tube, area of cross-section A h sand The depth of the bottom of the tube below the liquid surface is h. The tube is displaced vertically and then released. The variation with time t of the depth h is shown in . 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t / s h / cm Determine: the amplitude, in metres, of the oscillations amplitude = m the frequency of oscillation of the tube in the liquid frequency = Hz the acceleration of the tube when h is a maximum. acceleration = m s–2 The frequency f of oscillation of the tube is given by the expression f = 1 2π M A g c m t where g is the acceleration of free fall. Calculate the density t of the liquid in which the tube is floating. t = kg m–3 The oscillations illustrated in are undamped. In practice, the liquid does cause light damping. On , draw a line to show light damping of the oscillations for time t = 0 to time t = 1.4 s.
9702_s19_qp_43
THEORY
2019
Paper 4, Variant 3
View
State what is meant by resonance. shows a heavy pendulum and a light pendulum, both suspended from the same piece of string. This string is secured at each end to fixed points. fixed points string heavy pendulum light pendulum Both pendulums have the same natural frequency. The heavy pendulum is set oscillating perpendicular to the plane of the diagram. As it oscillates, it causes the light pendulum to oscillate. shows the variation with time t of the displacements of the two pendulums for three oscillations. t / s displacement / cm heavy light The variation with t of the displacement x of the light pendulum is given by x = 0.25 sin 5.0rt where x is in centimetres and t is in seconds. Calculate the period T of the oscillations. T = s On , label both of the axes with the correct scales. Use the space below for any additional working that you need. Determine the magnitude of the phase difference φ between the oscillations of the light and heavy pendulums. Give a unit with your answer. φ = unit
9702_s22_qp_42
THEORY
2022
Paper 4, Variant 2
View
The variation with time t of the displacement x of the cone of a loudspeaker is shown in . 0.2 0.3 x / mm 0.1 – 0.2 – 0.1 – 0.3 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 t / ms Use to determine, for these oscillations, the amplitude, amplitude = mm the frequency. frequency = Hz State two times at which the speed of the cone is maximum, time ms and time ms the acceleration of the cone is maximum. time ms and time ms The effective mass of the cone is 2.5 g. Use your answers in to determine the maximum kinetic energy of the cone. kinetic energy = J The loudspeaker must be designed so that resonance of the cone is avoided. State what is meant by resonance. State and briefly explain one other situation in which resonance should be avoided.
9702_w09_qp_41
THEORY
2009
Paper 4, Variant 1
View
A bar magnet is suspended from the free end of a helical spring, as illustrated in . coil magnet helical spring 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. As the magnet passes through its rest position, a timer is started. The variation with time t of the vertical displacement y of the magnet from its rest position is shown in . 1.0 –1.0 –2.0 2.0 y / cm 4.0 3.0 2.0 1.0 5.0 6.0 7.0 8.0 9.0 10.0 t / s At time t = 4.0 s, the switch is closed. Use to state the evidence for the magnet to be undergoing free oscillations during the period t = 0 to t = 4.0 s, state, with a reason, whether the damping after time t = 4.0 s is light, critical or heavy, determine the natural frequency of vibration of the magnet on the spring. frequency = Hz State Faraday’s law of electromagnetic induction. Explain why, after time t = 4.0 s, the amplitude of vibration of the magnet is seen to decrease.
9702_w11_qp_41
THEORY
2011
Paper 4, Variant 1
View
A bar magnet is suspended from the free end of a helical spring, as illustrated in . coil magnet helical spring 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. As the magnet passes through its rest position, a timer is started. The variation with time t of the vertical displacement y of the magnet from its rest position is shown in . 1.0 –1.0 –2.0 2.0 y / cm 4.0 3.0 2.0 1.0 5.0 6.0 7.0 8.0 9.0 10.0 t / s At time t = 4.0 s, the switch is closed. Use to state the evidence for the magnet to be undergoing free oscillations during the period t = 0 to t = 4.0 s, state, with a reason, whether the damping after time t = 4.0 s is light, critical or heavy, determine the natural frequency of vibration of the magnet on the spring. frequency = Hz State Faraday’s law of electromagnetic induction. Explain why, after time t = 4.0 s, the amplitude of vibration of the magnet is seen to decrease.
9702_w11_qp_42
THEORY
2011
Paper 4, Variant 2
View
An object is suspended from a spring that is attached to a fixed point as shown in . oscillations object spring equilibrium position fixed point The object oscillates vertically with simple harmonic motion about its equilibrium position. State the defining equation for simple harmonic motion. Identify the meaning of each of the symbols used to represent physical quantities. The variation with displacement x from the equilibrium position of the velocity v of the object is shown in . – 0.04 0.04 0.08 0.12 – 0.08 – 0.12 – 0.2 – 0.2 – 0.1 0.1 0.2 v / m s–1 x / m The variation with x of the potential energy EP of the oscillations of the object is shown in . – 0.04 0.04 0.08 0.12 – 0.08 – 0.12 0.025 0.050 x / m EP / J Use and to: determine the amplitude x0 of the oscillations x0 = m show that the angular frequency of the oscillations is 1.7 rad s–1 determine the mass M of the object. M = kg The oscillations of the object are now lightly damped. State what is meant by damping. Assume that the damping does not change the angular frequency of the oscillations. On , sketch the variation with x of v when the amplitude of the oscillations is 0.060 m.
9702_w22_qp_41
THEORY
2022
Paper 4, Variant 1
View
Questions Discovered
21