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

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