Question

a student wanted to make a pendulum whose period would be one second. He used a string of length "L" and found that the period was 1/2 sec. To get the desired period. He should use a string whose length equal?

Answer 1

Answer on Question #42586, Physics, Mechanics | Kinematics | Dynamics

A student wanted to make a pendulum whose period would be one second. He used a string of length "L" and found that the period was 1/2 sec. To get the desired period he should use a string whose length equal?

Solution:

Given:

$$\begin{array}{l} T_{1} = 0.5 \, \text{s}, \\ L_{1} = L, \\ T_{2} = 1 \, \text{s}, \\ L_{2} = ? \end{array}$$

For small amplitudes, the period of such a pendulum can be approximated by:

$$T = 2 \pi \sqrt{\frac{L}{g}}$$

Thus,

$$\begin{array}{l} T_{1} = 2 \pi \sqrt{\frac{L_{1}}{g}} \\ T_{2} = 2 \pi \sqrt{\frac{L_{2}}{g}} \\ T_{2} = 2 T_{1} \\ \end{array}$$ $$2 \pi \sqrt{\frac{L_{2}}{g}} = 4 \pi \sqrt{\frac{L_{1}}{g}}$$

Thus,

$$\sqrt{L_{2}} = 2 \sqrt{L_{1}}$$ $$L_{2} = 4 L_{1} = 4 L$$

Answer. $L_{2} = 4 L$

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