Question

A bicycle wheel is mounted on a fixed, frictionless axle, with alight string wound around its rim. The wheel has moment of inertia I=kmr², where m is its mass, r is its radius, and k is a dimensionless constant between zero and one. The wheel is rotating counterclockwise with angular velocity w₀, when at time t=0 someone starts pulling the string with a force of magnitude F. Assume that the string does not slip on the wheel.

Part A

Suppose that after a certain time tL, the string has been pulled through a distance L. What is the final rotational speed w_final of the wheel?

Express your answer in terms of L, F, I, and w0.

Part B

What is the instantaneous power P delivered to the wheel via the force F at time t=0?

Express the power in terms of some or all of the variables given in the problem introduction.

Answer 1

Concepts and reason

The required concepts to solve these questions are rotational kinetic energy, work energy theorem, torque and power.

Firstly, calculate the initial and final kinetic energy and use work-energy theorem to equate with change in rotational kinetic energies of the wheel. Solve the equation to find final angular velocity. After that, calculate torque on wheel due to force and then calculate the power delivered to the wheel.

Fundamentals

Rotational kinetic energy of an object is the energy by the virtue of its rotational motion. If is the moment of inertia of the object and is its angular velocity, then its rotational kinetic energy is,

Work energy theorem states that work done on an object appears in the form of change in its kinetic energy. If is the initial kinetic energy of the object, is its final kinetic energy and then the amount of work done is,

Torque is the force which is responsible for change in rotational state of motion. It is defined as the cross product between force applied and the position vector from origin at center of rotation to point at which acts.

The expression for magnitude of torque is,

Here, is the angle between force and position vector.

Work is defined as product of force and corresponding displacement in the direction of force.

Here, is the distance.

Work done by torque in rotating wheel by angle is,

For rotational motion, the expression for power is,

Here, P is the power, is the torque, and is angular velocity.

(A)

The expression for initial rotational kinetic energy of the wheel is,

Here, is the initial angular velocity.

The expression for final kinetic energy of the wheel is,

The expression for work energy theorem is,

The expression for work is,

Substitute for in equation for work energy theorem.

Rearrange the equation for final kinetic energy.

Substitute for , for in above equation.

Divide both sides by

(B)

The expression for magnitude of the torque is,

Substitute for in above equation.

For rotational motion, the expression for power is,

Substitute for in above equation.

At ,

Substitute for in equation.

Ans: Part A

Final angular velocity of the wheel is .