Question

A simple pendulum of length L has a period of T seconds. What should be the length for the period to be 3Tseconds?

Answer 1

Given data:

Length of peldulum is,

`l_1=L`

Period of pendulum is,

`T_1=T`

New period of pendulum is,

`T_2=3T`

Formula:

Formula of period of pedulum is as follows:

`T=2\Pi\sqrt[]{(l)/(g)}`

For old period of pendulum above equation becomes as follows:

`T=2\Pi\sqrt[]{(L)/(g)}`

Taking square of above equation,

`\begin{gathered} T^2=4\Pi^2(L)/(g) \\ L=T^2g(1)/(4\Pi)\text{ ..}.(1) \end{gathered}`

Now, for new period of pendulum,

`T_2=2\Pi\sqrt[]{(L_2)/(g)}`

Taking square of above equation,

`T_2=3T^{}_{}_{}`

Hence,

`\begin{gathered} (3T)^2=4\Pi^2\frac{L_2}{\text{g}}\ldots(2) \\ L_2=9T^2g(1)/(4\Pi)\ldots(3) \end{gathered}`

Taking ratio of equation-(3) and equation-(1),

`\begin{gathered} (L_2)/(L)=9T^2g(1)/(4\Pi)*(1)/(T^2g)4\Pi \\ (L_2)/(L)=9 \\ L_2=9L \end{gathered}`

Therefore, Length of pendulum should be 9L for the period to be 3T seconds.