Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?

**(a)** the rotation of earth about its axis.

**(b) **motion of an oscillating mercury column in a U-tube.

**(c)** motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower most point.

**(d)** general vibrations of a polyatomic molecule about its equilibrium position.

Asked by Pragya Singh | 1 year ago | 114

**(a)** Rotation of the earth is not to and fro motion about a fixed point. Therefore, it is periodic but not S.H.M.

**(b)** Simple harmonic motion

**(c)** Simple harmonic motion

**(d)** General vibrations of a polyatomic molecule about its equilibrium position is periodic but not SHM. A polyatomic molecule has a number of natural frequencies. Therefore its vibration is a superposition of simple harmonic motions of a number of different frequencies.

A mass attached to a spring is free to oscillate, with angular velocity ω, in a horizontal plane without friction or damping. It is pulled to a distance x_{0} and pushed towards the centre with a velocity v_{0} at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, x_{0} and v_{0}. [Hint: Start with the equation x = a cos (ωt+θ) and note that the initial velocity is negative.]

A body describes simple harmonic motion with an amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is

**(a)** 5 cm

**(b)** 3 cm

**(c)** 0 cm.

A circular disc of mass 10 kg is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be 1.5 s. The radius of the disc is 15 cm. Determine the torsional spring constant of the wire. (Torsional spring constant α is defined by the relation J = –α θ, where J is the restoring couple and θ the angle of twist).

Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.

You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of

**(a)** the spring constant k and

**(b)** the damping constant b for the spring and shock absorber system of one wheel, assuming that each wheel supports 750 kg.