Question

A thin hoop and a solid disk having the same mass and outer radii of 1.3 g and 43 mm, respectively, are released from rest as shown. Each rolls without slipping. D28Determine the kinetic energy in J and the angular velocity in radian/s of each having travelled a distance of 2.1 m down the 6 deg incline: (a) thin hoop and (b) solid cylilnder.

Answer 1

The total kinetic energy is given by

.............EQ1

where, is the mass of the obeject,

, is the rotational moment of inertia,

is the outher radii

is the angular velocity,

, is the linear velocity for pure rolling,

Thus EQ1 changes to:

................EQ2

The potential energy loss will be:

,..........EQ3

where, is the angle of inclination,

is the distance travelled on the inclined plane.

Since objects are rolling down an inclined plane, the gain the kinetic energy (rotational + translational) will be equal to the loss of the potential energy. The kinetic energy will be same for the obects as their mass is equal. Thus we equate EQ2 and EQ3, we get

Also,

...........EQ4

A. For thin hoop, , substituting values in EQ4, we get

B. For solid cylinder (cross-section is solid disc), , substituting values in EQ4, we get