JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Rotational Mechanics

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Consider a badminton racket with length scales as shown in the figure. If the mass of the linear and circular portions of the badminton racket are same (M) and the mass of the threads are negligible, the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at,

\frac{r}{2}

distance from the end A of the handle will ______

Mr^{2}

Answer: 52

Solution

The handle is a uniform rod of mass

M

and length

6 r

with its centre

3 r

from

A

, and the ring has mass

M

and radius

r

with its centre at

6 r + r = 7 r

from

A

. The axis is perpendicular to the handle, lies in the plane of the ring, and passes through the point

r /2

from

A

. For the rod,

I_{1} = \frac{M ( 6 r ) ^{2}}{12} + M (3 r - \frac{r}{2})^{2} = 3 M r^{2} + \frac{25}{4} M r^{2} = 9.25 M r^{2}

The axis is parallel to a diameter of the ring, about which

I = \frac{M r ^{2}}{2}

, and the ring's centre is

7 r - \frac{r}{2} = \frac{13 r}{2}

from it:

I_{2} = \frac{M r ^{2}}{2} + M (\frac{13 r}{2})^{2} = 0.5 M r^{2} + 42.25 M r^{2} = 42.75 M r^{2}

I = I_{1} + I_{2} = 52 M r^{2}

JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Integer answer

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JEE Main · 2021 · 26 Aug 2021 · Shift 1 · Rotational Mechanics

Solution

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