For the same total mass, which of the following will have the largest moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of the body a) A disc of radius a b) A ring of radius a c) A square lamina of side a d) Four identical rods forming square of side a

Options

A. A disc of radius a
B. A ring of radius a
C. A square lamina of side a
D. Four identical rods forming square of side a

Answer

To determine which of the given shapes has the largest moment of inertia about an axis passing through the center of mass and perpendicular to the plane of the body, we will calculate the moment of inertia for each shape.

### Step 1: Moment of Inertia of a Disc
For a disc of radius \( a \) and mass \( m \), the moment of inertia \( I \) about an axis through its center and perpendicular to its plane is given by the formula:
\[ I_{\text{disc}} = \frac{1}{2} m a^2 \]

### Step 2: Moment of Inertia of a Ring
For a ring of radius \( a \) and mass \( m \), the moment of inertia about an axis through its center and perpendicular to its plane is given by:
\[ I_{\text{ring}} = m a^2 \]

### Step 3: Moment of Inertia of a Square Lamina
For a square lamina of side \( a \) and mass \( m \), the moment of inertia about an axis through its center and perpendicular to its plane is given by:
\[ I_{\text{square}} = \frac{1}{6} m a^2 \]

### Step 4: Moment of Inertia of Four Identical Rods Forming a Square
For four identical rods forming a square of side \( a \), the moment of inertia of one rod about its center is \( \frac{1}{12} m a^2 \). Since there are four rods, the total moment of inertia for the square configuration is:
\[ I_{\text{rods}} = 4 \times \frac{1}{12} m a^2 = \frac{1}{3} m a^2 \]

### Step 5: Compare the Moments of Inertia
Now we can summarize the moments of inertia calculated:
- For the disc: \( I_{\text{disc}} = \frac{1}{2} m a^2 \)
- For the ring: \( I_{\text{ring}} = m a^2 \)
- For the square lamina: \( I_{\text{square}} = \frac{1}{6} m a^2 \)
- For the four rods: \( I_{\text{rods}} = \frac{1}{3} m a^2 \)

### Step 6: Determine the Largest Moment of Inertia
Now we compare the values:
- \( \frac{1}{2} m a^2 \) (disc)
- \( m a^2 \) (ring)
- \( \frac{1}{6} m a^2 \) (square)
- \( \frac{1}{3} m a^2 \) (four rods)

Clearly, \( I_{\text{ring}} = m a^2 \) is the largest among these values.

### Final Answer
Thus, the shape with the largest moment of inertia about an axis passing through the center of mass and perpendicular to the plane of the body is:
**Option b) A ring of radius a.**

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