Moment of inertia of a disc of mass \(M\) and radius \(R\) about any of its diameters is \(\frac{MR^2}{4}\). The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, \(\frac{x}{2}MR^2\). The value of \(x\) is _________.
Moment of inertia of a disc of mass \(M\) and radius \(R\) about any of its diameters is \(\frac{MR^2}{4}\). The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, \(\frac{x}{2}MR^2\). The value of \(x\) is _________.
Show Hint
Remember the perpendicular and parallel axis theorems. They are essential for calculating moments of inertia about different axes.
Correct Answer: 3
Solution and Explanation
Step 1: Apply the Perpendicular Axis Theorem
The moment of inertia of a disc about its diameter is given as \(I_d = \frac{MR^2}{4}\). According to the perpendicular axis theorem, the moment of inertia of a planar lamina about an axis perpendicular to the plane is equal to the sum of the moments of inertia about two perpendicular axes in the plane that intersect the perpendicular axis at its point of intersection with the lamina. So, for a disc, the moment of inertia about an axis through its center and perpendicular to the plane is:
\[ I_c = I_d + I_d = 2 \left(\frac{MR^2}{4}\right) = \frac{MR^2}{2}. \]
Step 2: Apply the Parallel Axis Theorem
The moment of inertia about an axis normal to the disc and passing through a point on its edge can be found using the parallel axis theorem:
\[ I_e = I_c + MR^2 \]
where \(I_e\) is the moment of inertia about the edge, \(I_c\) is the moment of inertia about the center, and \(R\) is the distance between the two parallel axes (which is the radius of the disc in this case).
Step 3: Calculate \(I_e\)
Substitute \(I_c = \frac{MR^2}{2}\):
\[ I_e = \frac{MR^2}{2} + MR^2 = \frac{3}{2}MR^2. \]
Step 4: Find the Value of \(x\)
The moment of inertia about the edge is given as \(\frac{x}{2}MR^2\). We have found that \(I_e = \frac{3}{2}MR^2\). Therefore, \(x = 3\).
Conclusion: The value of \(x\) is 3.
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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)