Answer the following questions :

**(a)** Time period of a particle in SHM depends on the force constant k and mass m of the particle:

T = 2π (\( \sqrt{\dfrac{m}{k}}\))

A simple pendulum executes SHM approximately. Why then is the time period of a pendulum independent of the mass of the pendulum?

**(b)** The motion of a simple pendulum is approximately simple harmonic for small-angle oscillations. For larger angles of oscillation, a more involved analysis shows that T is greater than 2π(\( \sqrt{\dfrac{I}{g}}\)). Think of a qualitative argument to appreciate this result.

**(c)** A man with a wristwatch on his hand falls from the top of a tower. Does the watch give the correct time during the free fall?

**(d)** What is the frequency of oscillation of a simple pendulum mounted in a cabin that is freely falling under gravity?

Asked by Pragya Singh | 1 year ago | 94

**(a)** In the case of a simple pendulum, the spring constant k is proportional to the mass. The m is the numerator and the denominator will cancel each other. Therefore, the time period of the simple pendulum is independent of the mass of the bob.

**(b)** The restoring force acting on the bob of a simple pendulum is given by the expression

F=−mgsinθ

F is the restoring force

m is the mass of the bob

g is the acceleration due to gravity

θ is the angle of displacement

When θ is small, sinθ≈θ.

Then the expression for the time period of a simple pendulum is given by T=2π(\( \sqrt{\dfrac{I}{g}}\))

When θ is large, sinθ<θ. Therefore, the above equation will not be valid. There will be an increase in the time period T.

**(c)** Wristwatch works on spring action and does not depend on the acceleration due to gravity. Therefore, the watch will show the correct time.

**(d)** During the free fall of the cabin, the acceleration due to gravity will be zero. Therefore the frequency of oscillation of the simple pendulum will also be zero.

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.