Two Circular loops of radius R and nR are made from the same wire . The moment of inertia about the axis passing through centre and perpendicular to the plane of the larger loop is 8 times that of smaller loop. What is value of n

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Solution

Circular loops (rings) of the wire:
Radii: r₁ = r , r₂ = n * r

Mass of the wire per unit length = d
Mass of loop₁ = m₁ = 2 π r₁ * d = 2π r d
Mass of loop₂ = m₂ = 2 π r₂ * d = 2π n r d

Moment of Inertia of a circular loop about an axis perpendicular to the plane of the loop and passing through the center of the loop :
I = M R²,
M = Mass, R = radius

MOI of given loops:
I₁ = (2π r d) r² = 2 π d r³
I₂ = (2π n r d) (n r)² = 2 π d n³ r³

Given I₂ = 8 I₁
=> 2π d n³ r³ = 8 * 2 π d r³
=> n³ = 8
=> n = 2.

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