Oscillations

Question : 16

Total: 25

Answer the following questions:
(a) Time period of a particle in S.H.M. depends on the force constant k and mass m of the particle : T=2π√

. 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√

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

Solution:

(a) For a simple pendulum, force constant or spring factor k is proportional to mass m, therefore, m cancels out in denominator as well as in numerator. That is why the time period of simple pendulum is independent of the mass of the bob.
(b) The effective restoring force acting on the bob of simple pendulumin displaced position is F=– mg ‌sin‌θ. When q is small, sin‌θ≈θ. Then theexpression for time period of simple pendulum is given by
T=2π√

.
When θ is large sin θ<θ. If the restoring force mg ‌sin‌θ is replaced by mgθ, this amounts to effective reduction in the value of ‘g’ for large angles and hence an increase in the value of time period T.
(c) Yes, because the working of the wrist watch depends on spring action and it has nothing to do with gravity.
(d) We know that gravity disappears under free fall, so frequency is zero.

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