Question

Two disc made of same material and same thickness having radius R and Alpha R. their moment of inertia about their own axis are in ratio 1:16 calculate the value of Alpha​

Answer 1

it has given that, Two discs made of same material and same thickness having radius R and αR. ratio of their moment of inertia about their own axis is 1 : 16.

We have to calculate value of α

solution : let density of each disc is ρ

so, mass of disc , m = ρAx [ x is thickness of disc]

A = πr²

now m = ρπr²x = (ρπx)r²

moment of inertia of a disc about its own axis = 1/2 mr²

= 1/2 (ρπx)r² . r² = 1/2(ρπx) r⁴

here discs are made of same material. so density of disc must be same. and also thickness of discs are same.

so, ρ and x are constant terms.

therefore I is directly proportional to r⁴

i.e., (I₁/I₂) = (r₁/r₂)⁴

given , (I₁/I₂) = 1/16

r₁ = R , R₂ = αR

so, (1/16) = (R/αR)⁴ = 1/α⁴

⇒α = 2

Therefore the value of α is 2.