Calculate the moment of Inertia of an annular disc of inner radius $R_{1}$ and outer radius $R_{2}$ about an axis perpendicular to the plane in which the disc lies and passing through O.

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B

Chose a ring at a distance X from O and of thickness dx.

Mass per unit Area of Annular

disc = $\frac{M}{π (R_{2}^{2} - R_{1}^{2})} = p$

Let p = mass per unit Area

Note that the disc in uniform and mass distribution is uniform too.

Now mass of area element considered= dm

dm = $\frac{M}{π (R_{2}^{2} - R_{1}^{2})}$ $\times$ $2 π \times x d x$ = $\frac{2 M x d x}{(R_{2}^{2} - R_{1}^{2})}$

Moment of inertia of this small ring element about $O = d m . x^{2}$

Total moments of inertia of annular disc = $R_{1} \int R_{2} x^{2} d m$

l = $R_{1} \int R_{2} \frac{x^{2} . 2 M x d x}{(R_{2}^{2} - R_{1}^{2})}$

I = $\frac{2 M}{(R_{2}^{2} - R_{1}^{2})}$ $R_{1} \int R_{2} x^{3} d x$

= $\frac{M (R_{2}^{2} - R_{1}^{2}) (R_{2}^{2} + R_{1}^{2})}{2 (R_{2} + R_{1}) (R_{2} - R_{1})}$

= $\frac{M (R_{2} - R_{1}) (R_{2} + R_{1}) (R_{2}^{2} + R_{1}^{2})}{2 (R_{2} - R_{1}) (R_{2} + R_{1})}$

I = $\frac{M}{2} (R_{1}^{2} + R_{2}^{2})$

Suggest Corrections

Similar questions

Calculate the moment of Inertia of an annular disc of inner radius $R_{1}$ and outer radius $R_{2}$ about an axis perpendicular to the plane in which the disc lies and passing through $O$ .

find the moment of inertia of a uniform disc of radius R₁ having an empty symmetric annular region of radius R₂ in between , about an axis passing through geometrical centre and perpendicular to the disc

Q. Find the moment of inertia of a uniform disc of radius

R_{1}

having an empty symmetric annular region of radius

R_{2}

in between, about an axis passing through the geometrical centre and perpendicular to the disc.

Join BYJU'S Learning Program

Calculate the moment of Inertia of an annular disc of inner radius R1 and outer radius R2 about an axis perpendicular to the plane in which the disc lies and passing through O.

Calculate the moment of Inertia of an annular disc of inner radius $R_{1}$ and outer radius $R_{2}$ about an axis perpendicular to the plane in which the disc lies and passing through O.