Uniform Discs Moment of Inertia – Detailed Solution

Moment of Inertia Ratio of Uniform Discs

Question: Consider two uniform discs of the same thickness and made of the same material. Their radii are:

• For disc 1: \( R_1 = R \)
• For disc 2: \( R_2 = \frac{\alpha R}{2} \)

If the ratio of their moments of inertia about their axes is \[ I_1 : I_2 = 1 : 16, \] then the value of \(\alpha\) is:

• (1) 16
• (2) 8
• (3) 4
• (4) 2

Select the Correct Option

(1) 16

(2) 8

(3) 4

(4) 2

Detailed Step‑by‑Step Explanation

Step 1: Moment of Inertia of a Uniform Disc
The moment of inertia \(I\) of a uniform disc about its central axis is given by: \[ I = \frac{1}{2} M R^2. \]

Step 2: Mass of the Disc
Since the discs are made of the same material and have the same thickness, their masses are proportional to their areas: \[ M \propto R^2. \] Thus, if disc 1 has mass \(M_1\) and radius \(R_1 = R\), then \[ I_1 \propto \frac{1}{2} M_1 R^2 \propto \frac{1}{2}R^4. \] For disc 2: \[ R_2 = \frac{\alpha R}{2} \quad \Rightarrow \quad M_2 \propto R_2^2 \propto \left(\frac{\alpha R}{2}\right)^2 = \frac{\alpha^2 R^2}{4}. \] And its moment of inertia is: \[ I_2 \propto \frac{1}{2} M_2 R_2^2 \propto \frac{1}{2} \times \frac{\alpha^2 R^2}{4} \times \left(\frac{\alpha R}{2}\right)^2. \]

Step 3: Express \(I_2\) in Terms of \(R\)
Calculate \(R_2^2\): \[ R_2^2 = \left(\frac{\alpha R}{2}\right)^2 = \frac{\alpha^2 R^2}{4}. \] Then, \[ I_2 \propto \frac{1}{2} \times \frac{\alpha^2 R^2}{4} \times \frac{\alpha^2 R^2}{4} = \frac{1}{2} \times \frac{\alpha^4 R^4}{16} = \frac{\alpha^4 R^4}{32}. \]

Step 4: Form the Ratio
We have: \[ I_1 \propto \frac{1}{2} R^4 \quad \text{and} \quad I_2 \propto \frac{\alpha^4 R^4}{32}. \] The ratio is: \[ \frac{I_1}{I_2} = \frac{\frac{1}{2} R^4}{\frac{\alpha^4 R^4}{32}} = \frac{1}{2} \times \frac{32}{\alpha^4} = \frac{16}{\alpha^4}. \]

Given \(I_1 : I_2 = 1 : 16\), we have: \[ \frac{16}{\alpha^4} = \frac{1}{16}. \]

Step 5: Solve for \(\alpha\)
Multiply both sides by \(\alpha^4\): \[ 16 = \frac{\alpha^4}{16} \quad \Rightarrow \quad \alpha^4 = 256. \] Taking the fourth root: \[ \alpha = 4. \]

Final Answer: The value of \(\alpha\) is 4, which corresponds to option (3).

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