Acircular hoop of diameter d hangs on a nail. what is the period of its oscillations at small amplitude?

m = mass of the circular hoop

r = radius of the hoop

I = moment of inertia of the hoop

moment of inertia of the hoop about the center of hoop is given as

I = m r²

k = distance of the point of suspension from center of mass = r

using parallel axis theorem

I' = moment of inertia of hoop about the point of suspension

I' = I + m k²

I' = m r² + m k²

I' = m r² + m r²

I' = 2 m r²

Time period of oscillation for the hoop is given as

T = 2π sqrt(I'/mgk)

T = 2π sqrt(2 m r²/(mgr))

T = 2π sqrt(2 r/g)

since 2r = diameter = d

T = 2π sqrt(d/g)

its c its is a mixture of gases

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