Question

A circular disc of radius b has a hole of radius a at its centre (see figure). If the mass per unit area of the disc varies as (σ₀/r) then the radius of gyration of the disc about its axis passing through the centre is :
(A) (a + b)/3
(B) √[(a² + b² + ab)/3]
(C) (a + b)/2
(D)
√[(a² + b² + ab)/2]

Answer 1

Answer:

sorry I don't know because I am in class 5

Answer 2

The radius gyration of the disc about its axis passing through the centre is k =  √ ( a^3 + b^3 +ab / 3)

Option (B) is correct.

Explanation:

dI = (dm)r^2

dI = (σdA)r^2

dI = (  r σ /r   2πrdr)r^2

dI = (σ 0  2π)r^2  dr

I = ∫dI = ∫  a- b  σ 0  2πr^ 2  dr

I = =σ0 2π(   b^ 3  − a^ 3  / 3)

m= ∫ dm = ∫σdA

m = σ 0  2π∫   b-a  dr

m = σ 0  2π(b−a)

Radius of gyration

k = c I / m =  √ (b^3 - a^3 )/ 3 (b - a)

k =  √ ( a^3 + b^3 +ab / 3)

Thus the radius gyration of the disc about its axis passing through the centre is k =  √ ( a^3 + b^3 +ab / 3)