Rolling Motion with slipping

[jbphys303]

Homework Statement

A uniform density sphere is released such that it has an angular speed of 10 rev/sec and no initial linear velocity. The angular velocity vector is perfectly perpendicular to the linear momentum vector. Initially the ball slips as it moves along the surface, but after time t pure rolling without slipping begins.

The coefficient of friction between the sphere and the surface is 0.21.
The radius of the ball is 0.08m
mass sphere= 7.3 kg
w= 10 rev/sec

#1: How fast is the ball rolling at time t?

#2: How long did it take to reach this speed?

#3: How much energy was lost between time t=0 and t=t?

Homework Equations

Rotational Inertia (I)=2/5mr^2
kinetic energy (rotational)= 1/2Iw^2
ke= 1/2mv^2
v=rw

The Attempt at a Solution

I think I can figure it out once I know how much energy is lost by friction
So far I set up the following equation
1/2Iw^2-(energy lost by friction)= 1/2Iw(final)^2+ 1/2mv^2
Any help would be appreciated

 

[ehild]

The motion of the ball is composed of translation of its CM and of rotation around the CM. The kinetic friction points forward, accelerates translation and its torque with respect to the CM decelerates rotation. At the time when v=rw, static friction takes place and the ball rolls.

Use both equation for linear acceleration ma = F and for angular acceleration I*dw/dt=-RF. Solve for v and w. Find the time when wR=v.

ehild

 

[kerimek]

I would approach this problem by first analyzing the given information and identifying any relevant equations and principles.

From the given information, we know that the sphere has an initial angular speed of 10 rev/sec and no initial linear velocity. This means that the sphere is rotating but not moving in a linear direction. We also know that the angular velocity vector is perpendicular to the linear momentum vector, which is a result of the sphere slipping along the surface.

At a certain time t, pure rolling without slipping begins, which means that the sphere is now both rotating and moving in a linear direction without any slipping. This indicates that the friction between the sphere and the surface has increased to the point where it can prevent slipping.

To answer the questions, we can use the equations for rotational inertia, kinetic energy, and velocity. We know the mass and radius of the sphere, so we can calculate its rotational inertia using the equation I = 2/5 * m * r^2.

To answer question #1, we can use the equation v = r * w, where v is the linear velocity, r is the radius, and w is the angular velocity. This equation relates the linear and angular velocities for an object in pure rolling motion. Therefore, we can calculate the linear velocity of the sphere at time t.

For question #2, we can use the equation for kinetic energy, K = 1/2 * I * w^2, where K is the kinetic energy, I is the rotational inertia, and w is the angular velocity. We know the initial and final angular velocities, so we can calculate the change in kinetic energy between time t=0 and t=t. This change in kinetic energy is equal to the energy lost by friction, as stated in the problem.

Finally, for question #3, we can simply calculate the energy lost by friction by using the previously calculated change in kinetic energy.

Overall, to solve this problem, we need to use the equations for rotational inertia, kinetic energy, and velocity, and take into account the information about pure rolling without slipping and the coefficient of friction.