Rotational Motion: Final Translational Speed

[hzx]

Homework Statement

A wheel with rotational inertia I = 1/2MR^2 about its horizontal central axle is set spinning with initial angular speed omega_0. It is then lowered, and at the instant its edge touches the ground the speed of the axle is zero. Initially the wheel slips when it touches the ground, but then begins to move forward and eventually rolls without slipping. What is the wheel's final translational speed?

Homework Equations

torque=I*alpha=FR

The Attempt at a Solution

I solved for friction force using the torque equation above, f=(1/2)(Ma)
But when I look at the force diagram, wouldn't it be just f that's causing the overall acceleration so f=Ma? I'm so confused. In addition, what other equations should I use to find the final translational speed?

 

[Andrew Mason]

Homework Statement

A wheel with rotational inertia I = 1/2MR^2 about its horizontal central axle is set spinning with initial angular speed omega_0. It is then lowered, and at the instant its edge touches the ground the speed of the axle is zero. Initially the wheel slips when it touches the ground, but then begins to move forward and eventually rolls without slipping. What is the wheel's final translational speed?

Homework Equations

torque=I*alpha=FR

The Attempt at a Solution

I solved for friction force using the torque equation above, f=(1/2)(Ma)
But when I look at the force diagram, wouldn't it be just f that's causing the overall acceleration so f=Ma? I'm so confused. In addition, what other equations should I use to find the final translational speed?

Can you determine the horizontal acceleration while slipping? Does the axle (i.e the wheel) accelerate after slippage ends?

AM

 

[ehild]

The friction accelerates the centre of mass and decelerates rotation. Write both equations, and solve them for the velocity of the of the CoM and angular velocity. Use the condition of pure rolling - what is the relation between the angular velocity and the velocity of the CoM?

 

[Andrew Mason]

A wheel with rotational inertia I = 1/2MR^2 about its horizontal central axle is set spinning with initial angular speed omega_0. It is then lowered, and at the instant its edge touches the ground the speed of the axle is zero. Initially the wheel slips when it touches the ground, but then begins to move forward and eventually rolls without slipping. What is the wheel's final translational speed?

By the way, hzx, welcome to PF!

The friction accelerates the centre of mass and decelerates rotation. Write both equations, and solve them for the velocity of the of the CoM and angular velocity. Use the condition of pure rolling - what is the relation between the angular velocity and the velocity of the CoM?

You appear to be speaking about the force from kinetic friction i.e. while the wheel is slipping, not the static friction once slippage ends. The thing that makes this readily solvable is the fact that kinetic friction force is basically the same regardless of the speed or amount of slippage, so long as there is some slippage. After slipping stops, the positive acceleration will end.

AM

 

[haruspex]

I solved for friction force using the torque equation above, f=(1/2)(Ma)

It would help if you were to show all your working, but I am guessing you used $R\alpha=a$. That is only true for rolling contact, so not valid while slipping.
There is a sneaky way to solve this problem without worrying about torques or forces: use conservation of angular momentum. The trick is to pick the right reference axis. The answer drops straight out.