Question

We want to support a thin hoop by a horizontal nail and have the hoop make one complete small-angle oscillation each 2.0 s .

Part A

What must the hoop's radius be?

R= m

Answer 1

A physical pendulum problem which means the angular frequency is given by:

   =

where
I = moment of inertia
m = mass
g = grav. acceleration
d = distance from pivot to mass center

where, the moment of inertia is taken about the edge of the ring, and the distance from the pivot to the cg, is the radius R. From the parallel axis theorem, the moment of inertia of the hoop about the nail

I = MR² + MR² = 2MR², so

=

R = 1/2*(T/2)²*g

R = 1/2* (2.0 / 6.28)²*9.8

R = 0.496m