Question

Moment of inertia of a uniform quarter disc of radius \(\mathrm{R}\) and mass \(\mathrm{M}\) about an axis through its centre of mass and perpendicular to its plane is : (1) \(\frac{\mathrm{MR}^{2}}{2}-\mathrm{M}\left(\frac{4 \mathrm{R}}{3 \pi}\right)^{2}\) (2) \(\frac{\mathrm{MR}^{2}}{2}-\mathrm{M}\left(\sqrt{2} \frac{4 \mathrm{R}}{3 \pi}\right)^{2}\) (3) \(\frac{\mathrm{MR}^{2}}{2}+\mathrm{M}\left(\frac{4 \mathrm{R}}{3 \pi}\right)^{2}\) (4) \(\frac{\mathrm{MR}^{2}}{2}+\mathrm{M}\left(\sqrt{2} \frac{4 \mathrm{R}}{3 \pi}\right)^{2}\)

Short answer

Option (3): \(\frac{MR^2}{2} + M \left(\frac{4R}{3\pi}\right)^2\)

Step-by-step solution

Identify the Geometry and Divide the Problem

Given a uniform quarter disc of radius \(R\) and mass \(M\), consider its symmetry and distribution of mass. The moment of inertia of the whole disc about an axis through its center and perpendicular to its plane is known to be \(I = \frac{MR^2}{2}\). Since the quarter disc is a fraction of the whole disc, it can be treated as \(\frac{1}{4}\) of the whole disc plus a shift to its center of mass.

Calculate the Distance from the Center of Mass

For a quarter disc, the center of mass is not at the geometric center. The center of mass of a quarter disc is at a distance \(d = \frac{4R}{3\pi}\) from the geometric center of the full disc.

Use the Parallel Axis Theorem

The parallel axis theorem states that for any object, the moment of inertia about any axis parallel to the one through the center of mass can be found using: \[I = I_{cm} + Md^2\] where \(I_{cm}\) is the moment of inertia about the center of mass, \(M\) is the mass of the object, and \(d\) is the distance from the center of mass to the new axis.

Calculate Moment of Inertia about the Center of Mass

For a quarter disc, \(I_{cm}\) is given by \(I_{cm} = \frac{1}{4} \times \frac{MR^2}{2} = \frac{MR^2}{8}\).

Apply the Parallel Axis Theorem

Substitute the values into the parallel axis theorem formula: \[I = I_{cm} + Md^2 = \frac{MR^2}{8} + M \left(\frac{4R}{3\pi}\right)^2\]

Simplify the Expression

Calculate \(d^2\) and complete the simplification: \[d = \frac{4R}{3\pi}\] Thus, \[d^2 = \left(\frac{4R}{3\pi}\right)^2\] Now, \[I = \frac{MR^2}{8} + M \left(\frac{4R}{3\pi}\right)^2 = \frac{MR^2}{8} + M \frac{16R^2}{9\pi^2}\] Finally, we get:\[I = \left(\frac{MR^2}{2}\right) + M\left(\frac{4R}{3\pi}\right)^2\]

Compare with Given Options

After validating with the given options, it matches option (3): \(\frac{MR^2}{2} + M \left(\frac{4R}{3\pi}\right)^2\).

Key concepts

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It's often called the rotational equivalent of mass in linear motion. For different shapes and axes, moment of inertia varies.
For the problem of a quarter disc, the whole disc's moment of inertia about an axis through its center is given by: \( I = \frac{MR^2}{2} \).
Since we only have a quarter of the disc, this moment of inertia needs to be adjusted for the reduced mass and altered center of mass.

Center of Mass

The center of mass of an object is the point at which the entire mass of the object can be thought to be concentrated for the purpose of calculating mechanical properties like its moment of inertia.
For a quarter disc, this point is not at the geometric center. Instead, it lies at a distance of \( d = \frac{4R}{3\pi} \) from the geometric center.
Knowing the center of mass is crucial because it allows the use of the parallel axis theorem to shift the axis of rotation to this point.

Parallel Axis Theorem

The parallel axis theorem is used to calculate the moment of inertia of an object about any axis that is parallel to the object's center of mass.\br> It states that: \[ I = I_{cm} + Md^2 \]
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