Question

The moment of inertia of a annular disc of mass M,the inner diameter R1 and outer diameter is R2 about perpendicular to the axis is

Answer 1

Answer on Question #51386-Physics-Mechanics-Kinematics-Dynamics

The moment of inertia of a annular disc of mass $M$ , the inner diameter $R_{1}$ and outer diameter is $R_{2}$ about a perpendicular axis through its center is

Solution

An annular disc is just a circular disc from which a smaller, concentric disc has been removed, leaving a hole in its place as shown in the figure.

Let $O$ be the center of annular disc. Then, clearly, face area of the disc is $\pi (R_2^2 - R_1^2)$ and, therefore,

mass per unit area of the disk $= \frac{M}{\pi\left(R_2^2 - R_1^2\right)}$

Now, consider a coaxial ring of radius $x$ and width $dx$ shown in the figure. We clearly have,

face area of the ring $= 2\pi xdx$ and, therefore, it mass $= \frac{M}{\pi(R_2^2 - R_1^2)} 2\pi xdx = \frac{M2x}{(R_2^2 - R_1^2)} dx.$

Hence, the moment of inertia of this ring about an axis through $O$ and perpendicular to its plane is

$$\frac {M 2 x}{\left(R _ {2} ^ {2} - R _ {1} ^ {2}\right)} d x \cdot x ^ {2} = \frac {M 2 x ^ {3}}{\left(R _ {2} ^ {2} - R _ {1} ^ {2}\right)} d x.$$

The moment of inertia of the whole annular disk about an axis through O and perpendicular to its plane is thus easily obtained by integrating the above expression for the moment of inertia of this ring, between the limits $x = R_{1}$ and $x = R_{2}$ , so that, we have

$$\begin{array}{l} I = \int_ {R _ {1}} ^ {R _ {2}} \frac {M 2 x ^ {3}}{\left(R _ {2} ^ {2} - R _ {1} ^ {2}\right)} d x = \frac {2 M}{\left(R _ {2} ^ {2} - R _ {1} ^ {2}\right)} \int_ {R _ {1}} ^ {R _ {2}} x ^ {3} d x = \frac {2 M}{\left(R _ {2} ^ {2} - R _ {1} ^ {2}\right)} \left[ \frac {x ^ {4}}{4} \right] _ {R _ {1}} ^ {R _ {2}} = \frac {2 M}{\left(R _ {2} ^ {2} - R _ {1} ^ {2}\right)} \left[ \frac {\left(R _ {2} ^ {4} - R _ {1} ^ {4}\right)}{4} \right] \\ = \frac {M (R _ {2} ^ {2} + R _ {1} ^ {2})}{2}. \\ \end{array}$$

Answer: $\frac{M(R_2^2 + R_1^2)}{2}$ .

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