1. This Question Is About a Simple Pendulum

IB PHYSICS SL TOPIC 4: Assignment Simple Harmonic Motion

Name: ______Total mark: /48

72 minutes

1. This question is about a simple pendulum.

(a) A pendulum consists of a bob suspended by a light inextensible string from a rigid support. The pendulum bob is moved to one side and then released. The sketch graph shows how the displacement of the pendulum bob undergoing simple harmonic motion varies with time over one time period.

On the sketch graph above,

(i) label with the letter A a point at which the acceleration of the pendulum bob is a maximum. (1)

(ii) label with the letter V a point at which the speed of the pendulum bob is a maximum. (1)

(b) Explain why the magnitude of the tension in the string at the midpoint of the oscillation is greater than the weight of the pendulum bob.

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1 (c) The pendulum bob is moved to one side until its centre is 25 mm above its rest position and then released.

(i) Show that the speed of the pendulum bob at the midpoint of the oscillation is 0.70 m s–1.

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(ii) The mass of the pendulum bob is 0.057 kg. The centre of the pendulum bob is 0.80 m below the support. Calculate the magnitude of the tension in the string when the pendulum bob is vertically below the point of suspension.

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2 (d) The point of suspension of the pendulum bob is moved from side to side with a small amplitude and at a variable driving frequency f.

For each value of the driving frequency a steady constant amplitude A is reached. The oscillations of the pendulum bob are lightly damped.

(i) On the axes below, sketch a graph to show the variation of A with f.

(ii) Explain, with reference to the graph in (d)(i), what is meant by resonance.

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(e) The pendulum bob is now immersed in water and the variable frequency driving force in (d) is again applied. Suggest the effect this immersion of the pendulum bob will have on the shape of your graph in (d)(i).

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3 2. 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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4 (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.)

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

1 k f  , 2 mp

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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5 (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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...... (2) (Total 17 marks) 3. 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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6 (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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7 (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°.

–1 (i) The speed of the wave in M1 is 0.80 m s . The speed of the waves in M2 is 1.2 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)