A hoop of mass M and radius R rotates about an axle at the edge of the hoop. The

Question

A hoop of mass M and radius R rotates about an axle at the edge of the hoop. The hoop starts at its highest position and is given a very small push to start it rotating. (Use any variable or symbol stated above along with the following as necessary: g.)

A hoop of mass M and radius R rotates about an axle at the edge of the hoop. The hoop starts at its highest position and is given a very small push to start it rotating. (Use any variable or symbol stated above along with the following as necessary: g.) At its lowest position, what is the angular velocity of the hoop? At its lowest position, what is the speed of the lowest point on the hoop?

Explanation / Answer

R = radius of the hoop
M = mass of the hoop

Use the center of mass to determine the change in potential energy.

delta PE = M*g*2R

This will equal the rotational kinetic energy of the hoop.

The kinetic energy will be the sum of the rotational KE of the hoop about its center (which is the center of mass of the hoop) plus the rotational KE of the center of mass.

The kinetic energies of each of these depends on the moment of inertia and these happen to be the same for the center of mass as for the hoop plus therotational velocity of each will be the same.

We have I = M*R^2.

KE hoop = (1/2)*I*w^2 = (1/2)M*R^2*w^2

KE center of mass = (1/2)*I*w^2 = (1/2)M*R^2*w^2

KE Sum = M*R^2*w^2 = dPE = M*g*2R
R*w^2 = 2*g

a) w = SQRT(2*g/R)

The velocity of the lowest point is just the angular velocity times the radius of the motion of the outermost point from the axis of rotation which is just 2*R.
velocity = w*2R
b)

velocity = 2*SQRT(2*g*R)

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