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
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