The rotational kinctic energy of a rotating body is proportional to

Options

A. periodic time
B. `("periodic time")^(2)`
C. `("periodic time")^(-1)`
D. `("periodic time")^(-2)`

Answer

To solve the question regarding the proportionality of rotational kinetic energy, we can follow these steps:

### Step 1: Understand the formula for rotational kinetic energy
The rotational kinetic energy (K.E.) of a rotating body is given by the formula:
\[ K.E. = \frac{1}{2} I \omega^2 \]
where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.

### Step 2: Relate angular velocity to periodic time
The angular velocity \(\omega\) is related to the periodic time \(T\) by the formula:
\[ \omega = \frac{2\pi}{T} \]

### Step 3: Substitute \(\omega\) in the kinetic energy formula
Substituting \(\omega = \frac{2\pi}{T}\) into the kinetic energy formula gives:
\[ K.E. = \frac{1}{2} I \left(\frac{2\pi}{T}\right)^2 \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ K.E. = \frac{1}{2} I \cdot \frac{4\pi^2}{T^2} \]
\[ K.E. = \frac{2I\pi^2}{T^2} \]

### Step 5: Identify the proportionality
From the simplified expression, we can see that the rotational kinetic energy \(K.E.\) is proportional to \(\frac{1}{T^2}\):
\[ K.E. \propto \frac{1}{T^2} \]

### Conclusion
Thus, we conclude that the rotational kinetic energy of a rotating body is proportional to \(\frac{1}{T^2}\), where \(T\) is the periodic time.

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