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)

the independent variable, the one that is intentionally manipulated, is whether or not the car hits the barrier.

the dependent variable, the one that you measure the response in, is the distance that the washers travel.