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Step-by-Step Solution
Step 1: Understand the Physical Situation
We have a simple pendulum of length $\ell$ performing simple harmonic motion (SHM). Its maximum kinetic energy in the first scenario is $K_{1}$. Next, the length of the pendulum is doubled to $2\ell$, and it performs SHM with the same angular amplitude as before. We denote its maximum kinetic energy by $K_{2}$. We want to find the relation between $K_{2}$ and $K_{1}$.
Step 2: Express the Maximum Kinetic Energy
When a pendulum swings, its maximum kinetic energy occurs at the lowest point of its swing, where all the potential energy lost from the highest point converts into kinetic energy. If the angular amplitude is $\theta$, then the maximum kinetic energy for a given length $\ell$ is:
$$
K = mg\ell\bigl(1 - \cos\theta\bigr).
$$
Therefore, for the original pendulum:
$$
K_{1} = mg\ell \bigl(1 - \cos \theta\bigr).
$$
Step 3: Doubling the Length
Now, the length of the pendulum is doubled to $2\ell$, but the angular amplitude $\theta$ remains the same. Hence, the new maximum kinetic energy is:
$$
K_{2} = mg \cdot (2\ell)\bigl(1 - \cos\theta\bigr).
$$
Step 4: Compare the Two Maximum Kinetic Energies
Substitute $2\ell$ into the expression:
$$
K_{2} = 2\, mg\ell \bigl(1 - \cos\theta\bigr).
$$
Factor out the term corresponding to $K_{1}$:
$$
K_{2} = 2 \bigl[ mg\ell \bigl(1 - \cos\theta\bigr) \bigr] = 2 K_{1}.
$$
Step 5: Conclusion
Therefore, when the length of the pendulum is doubled while keeping the same angular amplitude, the maximum kinetic energy doubles:
$$
K_{2} = 2 K_{1}.
$$
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