Question

A 2.90-kg hoop 1.50 m in diameter is rolling to the right without slipping on a horizontal floor at a steady 2.50 rad/s .

Part A

How fast is its center moving?

Express your answer with the appropriate units.

Part B

What is the total kinetic energy of the hoop?

Part C

Find the magnitude of the velocity vector of each of the following points, as viewed by a person at rest on the ground: (i) the highest point on the hoop; (ii) the lowest point on the hoop, (iii) a point on the right side of the hoop, midway between the top and the bottom.

Find the magnitude of the velocity vector for each of the points in part C, except as viewed by someone moving along with same velocity as the hoop.

Answer 1

a) As the hoop is rolling without slip, the speed of its mass center is the tangential velocity.

and as the hoop is moving to the right (positive x direction) the velocity of the center of the mas is

b) the total kinetic energy of the hoop is the translational kinetic energy of the hoop   and the rotational kinetic energy

where   is the moment of inertia of the hoop. hence

c.i) At the highest point of the loop, the velocity relative to an observer on the ground is

where   is the velocity relative to the center of the mass. On the top, this velocity is in positive x direction.

then,

but, . Hence

so, at this point, the magnitude of the velocity relative to an observer on the ground is 3.76m/s

c.ii) At the lowest point of the loop, the velocity relative to the center of the mass is

therefore

so, at this point, the magnitude of the velocity relative to an observer on the ground is zero

c.iii) at a point on the right side of the hoop,  the velocity relative to the center of the mass is

hence,

and the magnitud of the velocity relative to an oberever on the ground is