A thin circular metal disc of radius 500.0 mm is set rotating about a central axis normal to its plane. Upon raising its temperature gradually, the radius increases to 507.5 mm. The percentage change in the rotational kinetic energy will be

Options

A. `1.5%`
B. `-1.5%`
C. `3%`
D. `-3%`

Explanation

To solve the problem of finding the percentage change in the rotational kinetic energy of a thin circular metal disc when its radius increases, we can follow these steps:

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

### Step 2: Calculate the moment of inertia for the disc
The moment of inertia \( I \) for a thin circular disc about an axis through its center is given by:
\[ I = \frac{1}{2} m r^2 \]
where \( m \) is the mass of the disc and \( r \) is its radius.

### Step 3: Determine the initial and final moment of inertia
Initially, the radius \( r_1 = 500.0 \, \text{mm} = 0.5 \, \text{m} \):
\[ I_1 = \frac{1}{2} m (0.5)^2 = \frac{1}{2} m (0.25) = \frac{1}{8} m \]

After the radius increases to \( r_2 = 507.5 \, \text{mm} = 0.5075 \, \text{m} \):
\[ I_2 = \frac{1}{2} m (0.5075)^2 = \frac{1}{2} m (0.2575) = \frac{1}{8} m (0.2575 / 0.25) = \frac{1}{8} m (1.03) \]

### Step 4: Calculate the initial and final kinetic energy
Assuming the angular velocity \( \omega \) remains constant, the initial and final kinetic energies can be expressed as:
\[ KE_1 = \frac{1}{2} I_1 \omega^2 = \frac{1}{2} \left(\frac{1}{8} m\right) \omega^2 = \frac{1}{16} m \omega^2 \]
\[ KE_2 = \frac{1}{2} I_2 \omega^2 = \frac{1}{2} \left(\frac{1}{8} m (1.03)\right) \omega^2 = \frac{1.03}{16} m \omega^2 \]

### Step 5: Calculate the change in kinetic energy
The change in kinetic energy \( \Delta KE \) is:
\[ \Delta KE = KE_2 - KE_1 = \left(\frac{1.03}{16} m \omega^2 - \frac{1}{16} m \omega^2\right) = \frac{0.03}{16} m \omega^2 \]

### Step 6: Calculate the percentage change in kinetic energy
The percentage change in kinetic energy can be calculated as:
\[ \text{Percentage Change} = \left(\frac{\Delta KE}{KE_1}\right) \times 100 = \left(\frac{\frac{0.03}{16} m \omega^2}{\frac{1}{16} m \omega^2}\right) \times 100 = 0.03 \times 100 = 3\% \]

### Final Answer
The percentage change in the rotational kinetic energy of the disc is **3%**.

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