A hollow sphere is released from the top of an inclined plane of inclination θ. (a) What should

A hollow sphere is released from the top of an inclined...

Key Concepts

Work?Energy Principle

The work–energy principle states that the net work done on an object is equal to its change in kinetic energy. In problems involving rolling motion, gravitational work is partitioned into translational kinetic energy, rotational kinetic energy, and, in cases of slipping, energy dissipated by friction. This principle is crucial for linking force analysis with energy analysis in both pure rolling and slipping conditions.

Kinetic Friction and Slipping Dynamics

When the friction is insufficient to enforce the no-slip, pure rolling condition, kinetic friction takes over and results in slipping. This modifies the dynamics because the usual kinematic constraint between translation and rotation no longer holds, and energy is partially dissipated as heat. The overall kinetic energy acquired by the object is then determined by both the net work done by gravity and the work lost to kinetic friction.

Moment of Inertia

The moment of inertia is a measure of an object’s resistance to angular acceleration about a given axis and depends on how the mass is distributed relative to that axis. In analyzing rolling bodies, the moment of inertia directly influences the rotational kinetic energy and the acceleration of the sphere as it rolls down an incline. It appears in the equations that couple translational and rotational motion.

Static Friction

Static friction is the force that prevents two contacting surfaces from sliding relative to each other. In the context of rolling motion, static friction provides the necessary torque that initiates and maintains rotation without relative slipping. The minimum coefficient of static friction required is determined by balancing the gravitational component along the incline against the frictional force needed to generate the torque for rotation.

Rolling Motion Dynamics

Rolling motion dynamics deals with the interplay between the translational and rotational movement of a body. In pure rolling (without slipping), there is a fixed relationship between the linear velocity of the center of mass and the angular velocity about the center (v = ?R). This constraint is maintained by friction and is essential for determining how gravitational forces lead to both linear acceleration and rotation.

Transcript

00:01 It is told that a holosphere is released from the top of an inclined plane.

00:05 So first of all i am drawing the diagram for this given cushion.

00:10 So this is an inclined plate and let the angle of inclination is theta and a hollow sphere is released from this point.

00:26 Let this is our holosphere.

00:28 Now in the first part of the question we have to find the minimum coefficient of friction between the sphere and the plane to prevent sliding so we have to find coefficient of friction that is mu so i am going to show the forces in this system the mass of this sphere let this that is m and radius is r so one is the weight of this system with this sphere that is mg that will act downward so the component of this mass in weight in this direction is m g sine theta and let the friction force between these two surfaces is that is f and the component of this weight in this direction is given by m g cos theta and one is the normal force let this is n so from force balance we can see that for sphere we can write that m z sine theta minus f that is net force is equal to let the acceleration of this sphere is in this direction that is a so this will equal to m a this is our equation first and second is the torque equation that is net torque is given by i alpha so here torque is due to this friction force that is f into r is equal to i is 2 by 3 mr square into alpha from here we know that in pure rolling a is equal to r alpha so alpha can be written as a by r so put alpha in this equation and we get f is equal to 2 by 3 m a this is our equation second now put the value of f in equation first so put the value of f in equation first so we get m g sine theta minus this is 2 by 3 m a is equal to m into a from here we get the acceleration that is 3 by 5g sine theta this is the acceleration of this sphere now we have to find the coefficient of friction so from equation to we can write that f is equal to this is 2 by 3m into now put the value of a this is 3 by 5g sine theta we know that friction force is given by mu into normal force and normal force is equal to mg cost theta is equal to 2 by 3m this is 3 by 5g sine theta from here we get the value of mu that is 2 by 5 10 theta so this is the answer of our first part of the question that we have to find the coefficient of friction now in the second part it is told that the coefficient of the coefficient of friction of friction is the half of the value that we have calculated and now we have to find the kinetic energy of the ball.

04:07 So we have given that in this tuition mu is half of this.

04:11 So this is 1 by 5 of 10 theta.

04:14 This is given to us.

04:21 Now again i am using torque equation that is net torque is equal to i alpha.

04:26 So here not torque is due to the friction force that is f into r is equal to i into alpha...

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