Formula for the Time Period of a Simple Pendulum
The time period \( T \) of a simple pendulum is given by:
$$ T = 2\pi \sqrt{\frac{L}{g}} $$
Where:
\( L \) = Length of the pendulum
\( g \) = Acceleration due to gravity
Step 1: Calculate the Original Time Period
For a pendulum with mass \( m \) and length \( L_0 \), the time period is:
$$ T_0 = 2\pi \sqrt{\frac{L_0}{g}} $$
Step 2: Analyze the Changes
According to the problem:
Step 3: Compute the New Time Period
The time period of the new pendulum is:
$$ T_{\text{new}} = 2\pi \sqrt{\frac{L_0/2}{g}} $$
Rewriting:
$$ T_{\text{new}} = 2\pi \sqrt{\frac{L_0}{2g}} $$
Comparing with \( T_0 \):
$$ T_{\text{new}} = T_0 \frac{1}{\sqrt{2}} $$
Step 4: Solve for \( x \)
According to the question:
$$ \frac{x}{2} = \frac{1}{\sqrt{2}} $$
Solving for \( x \):
$$ x = \sqrt{2} $$
Conclusion
The value of \( x \) is \(\sqrt{2}\).
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