First slide

Periodic Motion

Question

A pendulum has period T for small oscillations. An obstacle is placed directly beneath the pivot, so that only the lowest one quarter of the string can follow the pendulum bob when it swings in the left of its resting position as shown in fig. (4). The pendulum is released from rest at a certain point A. The time taken by it to return to that point is

Moderate

A

T
B

T /2
C

3T / 4
D

T / 4

Solution

Time taken by pendulum to move from 4 to .B and back from B to A =T / 2 (where T = time period). Let T₁ be the time period of one small oscillation of the pendulum of length $l/ 4.$ We know that $Γ \propto \sqrt{l}$ Hence
$\frac{T_{1}}{T} = \sqrt{\frac{(l / 4)}{l}} = \frac{1}{2} or T_{1} = \frac{T}{2} As this pendulurn completes ouly half oscillation and hence the time taken will be T / 4 . Time period of oscillation = \frac{T}{2} + \frac{T}{4} = \frac{3 T}{4}$

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