Question

Starting from rest, a uniform circular disc rolls without slipping down a plane of inclination. The initial height of the center of the disc above the ground is h. The time taken by the disc to reach the bottom of the plane is:_________

Answer 1

The question seeks to calculate the time it would take for a disc to roll down an incline, but without specific measurements or angles, this calculation cannot be completed accurately. Concepts of conservation of energy and the moment of inertia are integral to understanding the motion of rolling objects.

Step-by-step explanation:

The student's question concerns the dynamics of a uniform circular disc rolling down an inclined plane under the influence of gravity. To determine the time taken for the disc to reach the bottom, we would apply the principles of rotational motion and energy conservation.

However, the question as stated is incomplete because it lacks the necessary information such as the angle of the incline or the distance to the bottom, which would be required to calculate the time taken for the disc to reach the bottom.

Considering the conceptual understanding, both a disc rolling without slipping and an object sliding without rolling would reach the original height on another incline due to conservation of energy unless energy is lost to friction or air resistance in the case of a rolling object.

This principle is illustrated by comparing two cylinders rolling and sliding down an incline and determining that they reach the same height on a second incline (conservation of mechanical energy). The moment of inertia is a factor that influences how an object rolls; this quantity relates the mass of a rolling object to its rotational motion and is expressed in terms of the object's mass (M) and radius (R).

A detailed calculation of the time taken or the moment of inertia would require more information about the incline or the object's final velocity after rolling down the incline.

The contrast between rolling and sliding can also be addressed by considering a ball that rolls up a hill without slipping, where the difference in energy distribution (rotational plus translational kinetic energy) compared to an object sliding (only translational kinetic energy) affects the vertical height achieved before the object comes to a stop.