We know, \(\mathrm{T}=2 \pi \sqrt{\frac{l}{\mathrm{~g}}}\) The time period of seconds pendulum, \(\mathrm{T}_{1}=2 \mathrm{~s}\) The length of the second pendulum \(\ell_{1}=100 \mathrm{~cm}=1 \mathrm{~m}\) Now, the new length of the pendulum \(\ell_{2}=2 \ell_{1}\) \(\Rightarrow \ell_{2}=200 \mathrm{~cm}=2 \mathrm{~m}\) Let the new time period of the pendulum be \(=\mathrm{T}_{2}\) \(\Rightarrow \mathrm{T} \alpha \sqrt{l}\) \(\Rightarrow \frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}=\sqrt{\frac{l_{1}}{l_{2}}} \Rightarrow \mathrm{T}_{2}=\mathrm{T}_{1} \sqrt{\frac{l_{2}}{l_{1}}}\) \(\mathrm{T}_{2}=2 \times \sqrt{\frac{2 l_{1}}{l_{1}}}=2 \sqrt{2} \mathrm{~s}\) The time period becomes \(2 \sqrt{2}\) times the original one.

Step by step solution

01

Write down the formula for the time period of a pendulum

The formula for the time period of a simple pendulum is: \(T = 2\pi\sqrt{\frac{l}{g}}\)

02

Write down the given data

We are given: - The original time period, \(T_1 = 2s\) - The original length of the pendulum, \(l_1 = 100 cm = 1 m\) - The new length of the pendulum, \(l_2 = 2l_1 = 200 cm = 2 m\)

03

Find the relationship between the time period and length

We can use the proportionality relationship between the time period \(T\) and the length \(l\) of a pendulum: \(T \propto \sqrt{l} \Rightarrow \frac{T_1}{T_2}=\sqrt{\frac{l_1}{l_2}}\)

04

Find the new time period \(T_2\)

We can find the new time period \(T_2\) by rearranging our proportionality relationship and plugging the given values: \(T_2 = T_1 \sqrt{\frac{l_2}{l_1}} = 2 \times \sqrt{\frac{2}{1}} = 2\sqrt{2}~s\) The new time period \(T_2\) is \(2\sqrt{2}\) seconds.

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