Chapter 2: Problem 68

A simple pendulum of length ' $\ell$ ' and time period ' $\mathrm{T}$ ' on earth is taken onto the surface of moon. How should the length of the simple pendulum be changed on the moon such that the time period is constant. (Take $\left.g_{e}=6 g_{i n}\right)$

Short Answer

Expert verified

Answer: $\ell_{in} = \frac{\ell}{6}$

Step by step solution

Write the formula for the time period of a simple pendulum on Earth

The formula for the time period of a simple pendulum is $T=2\pi\sqrt{\frac{\ell}{g}}$. On Earth, we can write this as $T=2\pi\sqrt{\frac{\ell}{g_e}}$.

Write the formula for the time period of a simple pendulum on the Moon

Using the same formula but considering the gravitational acceleration on the Moon, which is $g_{in} = \frac{g_e}{6}$, we can write the formula for the time period on the Moon as $T=2\pi\sqrt{\frac{\ell_{in}}{g_{in}}}$.

Make the time periods equal

Since we want the time period to remain constant, we can set the two time period expressions equal to each other: $$2\pi\sqrt{\frac{\ell}{g_e}}=2\pi\sqrt{\frac{\ell_{in}}{g_{in}}}$$

Solve for the new length $\ell_{in}$

Divide both sides by $2\pi$ and square both sides to eliminate the square root, we get: $$\frac{\ell}{g_e}=\frac{\ell_{in}}{g_{in}}$$ Now, plug in the values of $g_e$ and $g_{in}$ which is $g_{in} = \frac{g_e}{6}$, and solve for $\ell_{in}$: $$\ell_{in}=\frac{\ell}{g_e} \times \frac{g_e}{6} = \frac{\ell}{6}$$

Final answer

Therefore, in order for the time period to remain constant, the length of the simple pendulum on the Moon must be changed to $\ell_{in} = \frac{\ell}{6}$.

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A simple pendulum of length ' \(\ell\) ' and time period ' \(\mathrm{T}\) ' on earth is taken onto the surface of moon. How should the length of the simple pendulum be changed on the moon such that the time period is constant. (Take \(\left.g_{e}=6 g_{i n}\right)\)

Short Answer

Step by step solution

Write the formula for the time period of a simple pendulum on Earth

Write the formula for the time period of a simple pendulum on the Moon

Make the time periods equal

Solve for the new length \(\ell_{in}\)

Final answer

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