question_answer 16)

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 \[{{g}_{e}}=9\cdot 8m{{s}^{-2}};\] A simple pendulum executes S.H.M. approximately. Why then is the time-period of a pendulum independent of the mass of the pendulum? (b) The motion of simple pendulum is approximately simple harmonic for small angles of osculation. For large angle of oscillation, a more involved analysis (beyond the scope of this book) shows that T is greater than \[{{T}_{m}}=?;{{T}_{e}}=3\cdot 5{{s}^{-1}}\] Think of a qualitative argument to appreciate this result. (c) A man with a wrist watch on his hand falls from the top of a tower. Does the watch give 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?

Answer:

(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 pendulum in displaced position is \[{{\text{T}}_{\text{e}}}\text{=2 }\!\!\pi\!\!\text{ }\sqrt{\frac{\text{l}}{{{\text{g}}_{\text{e}}}}}\text{ and }{{\text{T}}_{\text{m}}}\text{=2 }\!\!\pi\!\!\text{ }\sqrt{\frac{\text{l}}{{{\text{g}}_{\text{m}}}}}\]When \[\therefore \] is small, \[\sin \theta \approx \theta \]. Then the expression for time period of simple pendulum is given by \[{{\text{T}}_{\text{m}}}\text{=}{{\text{T}}_{\text{e}}}\sqrt{\frac{{{\text{g}}_{\text{e}}}}{{{\text{g}}_{\text{m}}}}}\] When \[=3\cdot 5\sqrt{\frac{9\cdot 8}{1\cdot 7}}=8\cdot 4s.\] is large, \[\sin \theta