Two spheres (one hollow, one solid) are placed side by side...

Two spheres (one hollow, one solid) are placed side by side on an inclined plane and released at the same time. Both spheres roll down the inclined plane without slipping. (a) Using a free body diagram, explain what force provides the torque allowing the spheres to roll down the inclined plane. (b) Which sphere reaches the bottom of the inclined plane first and why? (c) How do the kinetic energies of the two spheres compare at the bottom of the inclined plane?

Two spheres (one hollow, one solid) are placed side by side on an inclined plane and released at the same time. Both spheres roll down the inclined plane without slipping.
(a) Using a free body diagram, explain what force provides the torque allowing the spheres to roll down the inclined plane.
(b) Which sphere reaches the bottom of the inclined plane first and why?
(c) How do the kinetic energies of the two spheres compare at the bottom of the inclined plane?

Key Concepts

Energy Distribution in Rolling Motion

In rolling motion, gravitational potential energy is converted into both translational kinetic energy and rotational kinetic energy. The proportion of energy allocated to each form depends on both the kinematics of rolling without slipping and the object's moment of inertia, which influences how quickly the sphere accelerates down the plane.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion and depends on how the object's mass is distributed relative to its rotation axis. For spheres, differences in moment of inertia (as seen between hollow and solid spheres) directly affect their acceleration and rotational energy when rolling down an incline.

Rolling Without Slipping

Rolling without slipping is a condition where the sphere's translational motion and rotational motion are linked; the point of contact with the surface is momentarily at rest. This relationship means that the linear velocity of the sphere's center of mass is directly proportional to its angular velocity, ensuring a coupling between rotational and translational dynamics.

Static Friction as a Source of Torque

Static friction, acting at the point where the sphere contacts the inclined plane, provides the necessary torque for rotational motion. Without slip, this frictional force does not do work in the energy sense but is crucial in ensuring that the sphere rotates as it translates down the incline.

Free Body Diagram

A free body diagram is a visual tool used to depict all the forces acting on an object. In the context of a rolling sphere on an incline, it shows the gravitational force, normal force, and frictional force, which is essential when analyzing the forces and torques acting on the sphere.

Transcript

00:01 In the given problem there is inclined plane which is making an angle theta with the horizontal.

00:12 There are two spheres rolling over the same inclined plane, identical looking spheres having the same mass.

00:31 Only difference is that one of the sphere is solid and the other one is hollow so if we look for all the forces acting on these spheres here this first one is hollow the second one is solid both of them are having the same mass so weight of both of them will be acting vertically downward through their centers so the components of these weights will also be identical if this angle is theta is here this angle will also be theta so the component perpendicular to the inclined plane will be m g cost theta for each of the two spheres the whole one and the solid one and another component along the inclined plane m g sine theta and mg sine theta and then normal reaction of this inclined plane over the spheres that is equal for both of the spheres equal to m g cos theta m g cost theta so all these forces if we look carefully whether it is mg sine theta or it is m g cos theta or normal reaction all of these forces are passing through the center of the spheres so none of them will create our torque about the center so we can write as both the components of the weights of spheres both the spheres are passing through their centers of mass so they will not create a torque.

03:12 Now if we look for the force of friction acting on both of these spheres here this is the direction of force of friction and the force of friction is not passing through the center of mass it is having a perpendicular distance equal to radius of the sphere so it is clear that only the force of friction will create a torque so only the forces of friction acting on both the spheres will create a torque and this is the answer for the first part of the problem now in the second part of the problem we have to find which of the two spheres, whether hollow one or solid one, will reach to the ground earlier.

04:08 For which, first of all, we look for the expression for the torque.

04:14 And we know torque is given by the product of force with the perpendicular distance from the point of rotation, which is nothing but the radius of the sphere.

04:24 And force of friction is mu times of normal reaction, means mu m .g.

04:29 To cost theta into r.

04:32 This is the torque...

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