Topic 4 Simple Harmonic Motion1. The graph below shows the variation with time t of the displacement x of a particle undergoing simple harmonic motion.

Which graph correctly shows the variation with time t of the acceleration a of the particle?

(1)

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Topic 4 Simple Harmonic Motion

2.

A wooden block is at rest on a horizontal frictionless surface. A horizontal spring is attached between the block and a rigid support.

The block is displaced to the right by an amount X and is then released. The period of oscillations is T and the total energy of the system is E.

X For an initial displacement of 2 which of the following shows the best estimate for the periodof oscillations and the total energy of the system? Period A. B. C. D. T T Total energy

T 2 T 2

E 2 E 4 E 2 E 4(1)

3.

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. 1. ......................................................................................................................... ......................................................................................................................... 2. ......................................................................................................................... .........................................................................................................................(2)

(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.

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Topic 4 Simple Harmonic Motion

The mass of hydrogen atom is 1.7 10 (i)

27

kg. Use data from the graph above

to determine its amplitude of oscillation. .........................................................................................................................(1)

(ii)

to show that the frequency of its oscillation is 9.1 10 Hz. ......................................................................................................................... .........................................................................................................................(2)

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(iii)

to show that the maximum kinetic energy of the hydrogen atom is 6.2 10

18

J.

......................................................................................................................... ......................................................................................................................... .........................................................................................................................(2)

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Topic 4 Simple Harmonic Motion

(c)

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

(3)

(d)

Assuming that the motion of the hydrogen atom is simple harmonic, its frequency of oscillation f is given by the expression f= 1 2 k , 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)1

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Topic 4 Simple Harmonic Motion(ii) Estimate, using your answer to (d)(i), the maximum acceleration of the hydrogen atom. ......................................................................................................................... ......................................................................................................................... ......................................................................................................................... .........................................................................................................................(2)

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