A thin hoop is supported in a vertical plane by a nail.What should the radius of the hoop be in order for it to have a period of oscillation of 1.00 s? The moment of inertia of a hoop of mass M and radius R about a point on its rim is 2MR². A)0.0154 m B)0.0621 m C)0.124 m D)0.1876 m E)0.248 m

Question

Quizplus · Accepted Answer

The period of oscillation for a hoop rotating about a fixed axis perpendicular to the plane of the hoop is given by:

T = 2π√(I/mgd)

where I is the moment of inertia of the hoop, m is its mass, g is acceleration due to gravity, and d is the distance between the axis of rotation and the center of mass of the hoop.

In this case, the hoop is supported by a nail, which means that the axis of rotation passes through its center of mass. Therefore, d = R, where R is the radius of the hoop.

Rearranging the equation for T, we get:

I = (mgR^2)/(4π^2T^2)

Substituting the given values, we get:

I = (2MR^2)/5

Setting T = 1.00 s, we can solve for R:

R = √[(5IT^2)/(2M)]

Substituting the given values, we get:

R = √[(5)(1.00 s)^2/(2)(M)] = √(5/2)R

Solving for R, we get:

R = 0.124 m

Therefore, the best choice is C.

A Thin Hoop Is Supported in a Vertical Plane by a Nail.What