A uniform solid cylinder with radius \(R\) and length \(L\) has a moment of inertia \(I_1\) about the axis of the cylinder. A concentric solid cylinder of radius \(R' = \frac{R}{2}\) and length \(L' = \frac{L}{2}\) is carved out of the original cylinder. If \(I_2\) is the moment of inertia of the carved-out portion, then \(\frac{I_1}{I_2} =\)
(Both \(I_1\) and \(I_2\) are about the axis of the cylinder.)
A uniform solid cylinder with radius \(R\) and length \(L\) has a moment of inertia \(I_1\) about the axis of the cylinder. A concentric solid cylinder of radius \(R' = \frac{R}{2}\) and length \(L' = \frac{L}{2}\) is carved out of the original cylinder. If \(I_2\) is the moment of inertia of the carved-out portion, then \(\frac{I_1}{I_2} =\)
(Both \(I_1\) and \(I_2\) are about the axis of the cylinder.)
Correct Answer: 32
Approach Solution - 1
The moment of inertia for a uniform solid cylinder about its axis is:
\[ I = \frac{1}{2} m R^2 \]
Step 1: Moment of inertia of the original cylinder (\(I_1\)):
\[ I_1 = \frac{1}{2} m_1 R^2 \]
Step 2: Moment of inertia of the carved-out cylinder (\(I_2\)):
The mass of the carved-out portion (\(m_2\)) is proportional to its volume:
\[ m_2 = \rho \cdot \text{Volume} = \rho \cdot \pi \left(\frac{R}{2}\right)^2 \cdot \frac{L}{2} = \frac{\rho \pi R^2 L}{8} \]
Moment of inertia:
\[ I_2 = \frac{1}{2} m_2 \left(\frac{R}{2}\right)^2 = \frac{1}{2} \cdot \frac{\rho \pi R^2 L}{8} \cdot \frac{R^2}{4} = \frac{\rho \pi R^4 L}{64} \]
Step 3: Ratio of moments of inertia (\(I_1/I_2\)):
The mass of the original cylinder (\(m_1\)) is:
\[ m_1 = \rho \cdot \pi R^2 L \]
Moment of inertia of the original cylinder:
\[ I_1 = \frac{1}{2} m_1 R^2 = \frac{1}{2} \cdot \rho \pi R^2 L \cdot R^2 = \frac{\rho \pi R^4 L}{2} \]
The ratio is:
\[ \frac{I_1}{I_2} = \frac{\frac{\rho \pi R^4 L}{2}}{\frac{\rho \pi R^4 L}{64}} = \frac{64}{2} = 32 \]
Approach Solution -2
The correct answer is 32.
I1=2m1R2I2=2m2(R/2)2
I2I1=m24m1=ρ⋅4πR2×2ℓ4⋅ρπR2ℓ⇒I2I1=32
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Concepts Used:
Rotational Motion
Rotational motion can be defined as the motion of an object around a circular path, in a fixed orbit.
Rotational Motion Examples:
The wheel or rotor of a motor, which appears in rotation motion problems, is a common example of the rotational motion of a rigid body.
Other examples:
- Moving by Bus
- Sailing of Boat
- Dog walking
- A person shaking the plant.
- A stone falls straight at the surface of the earth.
- Movement of a coin over a carrom board
Types of Motion involving Rotation:
- Rotation about a fixed axis (Pure rotation)
- Rotation about an axis of rotation (Combined translational and rotational motion)
- Rotation about an axis in the rotation (rotating axis)