IB SL Waves Exercize x2

1. This question is about oscillations and waves.

(a) A rectangular piece of wood of length l floats in water with its axis vertical as shown in diagram 1.

The length of wood below the surface is d. The wood is pushed vertically downwards a distance A such that a length of wood is still above the water surface as shown in diagram 2. The wood is then released and oscillates vertically. At the instant shown in diagram 3, the wood is moving downwards and the length of wood beneath the surface is d + x.

(i) On diagram 3, draw an arrow to show the direction of the acceleration of the wood.(1)

(ii) The acceleration a of the wood (in m s–2) is related to x (in m) by the following equation.

a = x

l14

Explain why this equation shows that the wood is executing simple harmonic motion.

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(iii) The period of oscillation of the wood is 1.4 s. Show that the length l of the wood is 0.70 m.

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(b) The wood in (a), as shown in diagram 2, is released at time t = 0. On the axes below, sketch a graph to show how the velocity v of the wood varies with time over one period of oscillation.

(1)

(c) The distance A that the wood is initially pushed down is 0.12 m.

(i) Calculate the magnitude of the maximum acceleration of the wood.

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(ii) On your sketch graph in (b) label with the letter P one point where the magnitude of the acceleration is a maximum.

(1)

(d) The oscillations of the wood generate waves in the water of wavelength 0.45 m.The graph shows how the displacement D, of the water surface at a particular distance from the wood varies with time t.

Using the graph, calculate the

(i) speed of the waves.

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(ii) ratio of the displacement at t = 1.75 s to the displacement at t = 0.35 s.

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(iii) ratio of the energy of the wave at t = 1.75 s to the energy at t = 0.35 s

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(Total 15 marks)

2. This question is about simple harmonic motion and waves.

(a) A particle of mass m that is attached to a light spring is executing simple harmonic motion in a horizontal direction.

State the condition relating to the net force acting on the particle that is necessary for it to execute simple harmonic motion.

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(b) The graph shows how the kinetic energy EK of the particle in (a) varies with the displacement x of the particle from equilibrium.

(i) Using the axes above, sketch a graph to show how the potential energy of the particle varies with the displacement x.

(2)

(ii) The mass of the particle is 0.30 kg. Use data from the graph to show that the frequency f of oscillation of the particle is 2.0 Hz.

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(c) The particles of a medium M1 through which a transverse wave is travelling, oscillate with the same frequency and amplitude as that of the particle in (b).

(i) Describe, with reference to the propagation of energy through the medium, what is meant by a transverse wave.

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(ii) The speed of the wave is 0.80 m s–1. Calculate the wavelength of the wave.

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(d) The diagram shows wavefronts of the waves in (c) incident on a boundary XY between medium M1 and another medium M2.

The angle between the normal, and the direction of travel of the wavefronts is 30°.

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(i) The speed of the wave in M1 is 0.80 m s–1. The speed of the waves in M2 is1.2 m s–1.Calculate the angle between the direction of travel of the wavefronts in M2 and the normal.

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(ii) On the diagram, sketch the wavefronts in M2.(1)

(Total 15 marks)

3. This question is about simple harmonic motion.

(a) In terms of the acceleration, state two conditions necessary for a system to perform simple harmonic motion.

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(b) A tuning fork is sounded and it is assumed that each tip vibrates with simple harmonic motion.

The extreme positions of the oscillating tip of one fork are separated by a distance d.

(i) State, in terms of d, the amplitude of vibration.

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(ii) On the axes below, sketch a graph to show how the displacement of one tip of the tuning fork varies with time.

(1)

(iii) On your graph, label the time period T and the amplitude a.(2)

(c) The frequency of oscillation of the tips is 440 Hz and the amplitude of oscillation of each tip is 1.2 mm. Determine the maximum

(i) linear speed of a tip.

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(ii) acceleration of a tip.

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(Total 10 marks)

4. Simple harmonic motion and the greenhouse effect

(a) A body is displaced from equilibrium. State the two conditions necessary for the body to execute simple harmonic motion.

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(b) In a simple model of a methane molecule, a hydrogen atom and the carbon atom can be regarded as two masses attached by a spring. A hydrogen atom is much less massive than the carbon atom such that any displacement of the carbon atom may be ignored.

The graph below shows the variation with time t of the displacement x from its equilibrium position of a hydrogen atom in a molecule of methane.

The mass of hydrogen atom is 1.7 10–27 kg. Use data from the graph above

(i) to determine its amplitude of oscillation.

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(ii) to show that the frequency of its oscillation is 9.1 1013 Hz.

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(iii) to show that the maximum kinetic energy of the hydrogen atom is 6.2 10–18 J.

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(c) On the grid below, sketch a graph to show the variation with time t of the velocity v of the hydrogen atom for one period of oscillation starting at t = 0. (There is no need to add values to the velocity axis.)

(3)

(d) Assuming that the motion of the hydrogen atom is simple harmonic, its frequency of oscillation f is given by the expression

,21

pmkf

where k is the force per unit displacement between a hydrogen atom and the carbon atom and mp is the mass of a proton.

(i) Show that the value of k is approximately 560 N m–1.

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(ii) Estimate, using your answer to (d)(i), the maximum acceleration of the hydrogen atom.

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(e) Methane is classified as a greenhouse gas.

(i) Describe what is meant by a greenhouse gas.

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(ii) Electromagnetic radiation of frequency 9.1 1013 Hz is in the infrared region of the electromagnetic spectrum. Suggest, based on the information given in (b)(ii), why methane is classified as a greenhouse gas.

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(Total 17 marks)

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