Chapter 14.1, Problem 12SSC

To determine

The way, the pendulum length must be changed to double its period and to halve the period.

Explanation of Solution

Introduction:

The time period of a pendulum is:

  $T=2\pi \sqrt{\frac{l}{g}}$

Here, l is the length of the pendulum and g is the acceleration due to gravity.

  $T=2\pi \sqrt{\frac{l}{g}}$

  $\frac{T_2}{T_1}=\sqrt{\frac{l_2}{l_1}}$

To double the period:

  $\frac{T_2}{T_1}=\sqrt{\frac{l_2}{l_1}}=2,$

Therefore, $\frac{l_2}{l_1}=4$ , the length must be quadrupled.

To halve the period,

  $\frac{T_2}{T_1}=\sqrt{\frac{l_2}{l_1}}=\frac{1}{2}$

  $\frac{l_2}{l_1}=\frac{1}{4}$

The length must be decreased to one by fourth its actual length.

Conclusion:

Therefore, $\frac{l_2}{l_1}=4$ , the length must be quadrupled and $\frac{l_2}{l_1}=\frac{1}{4}$ the length must be reduced to one-fourth its original length.