Ncert Exercises & Solutions

14.16 Answer the following questions :

(a) Time period of a particle in SHM depends on the force constant k and mass m of the particle: $T = 2 π \sqrt{\frac{m}{k}}$ .

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 $2 π \sqrt{\frac{l}{g}}$ . 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 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?

$(a) The time period of a simple pendulum is given by, T = 2 π \sqrt{\frac{m}{k}}$

$For a simple pendulum : k \propto m \Rightarrow \frac{m}{k} = Constant$

$Hence, the time period T of a simple pendulum is independent of the mass of the bob .$

$(b) In the case of a simple pendulum, the restoring force acting on the bob$

$of the pendulum :$

$F = - mgsinθ$

$For small θ, \sin θ ≃ θ$

$For large θ, \sin θ is greater than θ .$

$This decreases the effective value of g .$

$Hence, the time period increases as : T \propto \sqrt{\frac{1}{g}}$

(c)

$Time period of the wristwatch = 2 π \sqrt{\frac{m}{k}}$

The time period of the wristwatch does not depend on the acceleration due to gravity. So the time shown by the wristwatch of a man falling from the top of a tower is not affected by the fall.

(d)

$When a simple pendulum mounted in a cabin falls freely under gravity, g_{eff} = 0 .$
$So the frequency of the simple pendulum, f = 0 .$

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