Question

Two discs A and B of same material and thickness have radii R and 3R respectively. Their moments of inertia about their axis will be in the ratio ______.

Options

• 3:1

• 1:9

• 1:81

• 1:27

Answer 1

Two discs A and B of same material and thickness have radii R and 3R respectively. Their moments of inertia about their axis will be in the ratio 1.81.

Explanation:

The moment of inertia I of a uniform solid disc of mass M and radius R about its central axis (perpendicular to its plane) is given by:

`I = 1/2 MR^2`

Since both discs have the same material (density p) and thickness (t) the mass M can be expressed as:

M = Volume × p = (πR²t)p

Substituting this into the moment of inertia formula:

`I = 1/2 (piR^2tp) R^2 = 1/2 pitpR^4`

Since the material (p) and thickness (t) are constant for both discs, the moment of inertia is directly proportional to the fourth power of the radius:

I ∝ R⁴

Let the radius of disc A be R_A = R and the radius of disc B be R_B

= 3R. The ratio of their moments of inertia is:

`(I_A)/(I_B) = ((R_A)/(R_B))^4`

`(I_A)/(I_B) = (R/(3R))^4 = (1/3)^4`

`(I_A)/(I_B) = 1/81`