Kinda hard problem involving friction and a spool.

[ehild]

My bad I apologize! But how do we solve the case with linear acceleration since we can't use the same algebra to cancel everything we don't know?

In case of skidding, the friction is kinetic, instead of being static. You know that kinetic friction is μN. You can express N with the weight and T and the angle.

There are two equations, one for linear acceleration, and the other for angular acceleration. You can solve both in terms of tension, angle, mass and moment of inertia and see how they are related at different pulling angles.

ehild

 

[schaefera]

So, the case where it's static friction is the one case in which there is no acceleration and then we turn the cranks with kinetic friction there to find forward and backwards dependencies?

 

[ehild]

So, the case where it's static friction is the one case in which there is no acceleration and then we turn the cranks with kinetic friction there to find forward and backwards dependencies?

No, there is static friction in case of pure rolling. During rolling, there can be acceleration, both linear and angular, related by the equation a=R dω/dt.
The rolling can happen both forward and backwards.

When the spool skids, the friction is kinetic. There can be both linear and angular acceleration.

ehild.

 

[Infinitum]

No, there is static friction in case of pure rolling. During rolling, there can be acceleration, both linear and angular, related by the equation a=R dω/dt.
The rolling can happen both forward and backwards.

When the spool skids, the friction is kinetic. There can be both linear and angular acceleration.

ehild.

We're asked when the spool switches from rolling to skidding. So rolling with skidding shouldn't be considered...

 

[ehild]

calculate the critical value of θ (call it θc) at which the condition goes from rolling to skidding.

It is not the spool that switches from rolling to skidding.
Applying the same tension, the spool will not switch from rolling to skidding at the critical angle, but from rolling to standing...
But it can skid at the critical angle without rotating.

Do you only want to answer to the problem writer? Are you not interested in the phenomenon?

ehild

 

[schaefera]

See next post whoops

 

[schaefera]

Right, so another way to think of this:

If the frictional force is kinetic, then by definition it opposes the motion of the object, meaning the object must be skidding to the right.

If friction is static, then the tension has not yet pulled hard enough on the object to overcome the static frictional force to the left, meaning that the object will roll, but to the left (there wasn't enough force to cause it to accelerate right). Think of this as if you're pulling on the string, but the static friction is so strong that it causes the object to pivot (and begin to spin) toward the left.

Now clearly, as long as it's moving to the right, it's can't roll smoothly (that is, roll without skidding) because the tension to the right is stronger than the kinetic friction to the left... but if the tension is causing a torque that makes it want to rotate counter-clockwise, this will be stronger than the friction wanting it to rotate clockwise. Thus, w has a negative value while v has a positive value, so we can never meet the wR=v condition for smooth rolling.

Thus, the critical angle is the angle where the skidding to the right would switch to rolling to the left!

 

[Infinitum]

It is not the spool that switches from rolling to skidding.

Umm...what changes from rolling to skidding/standing, if not the spool?

But it can skid at the critical angle without rotating.

This is what I'm trying to say.

Do you only want to answer to the problem writer? Are you not interested in the phenomenon?

ehild

Very much so. I enjoy scientific phenomena much more than school-questions. But this being a homework thread, I guess charmedbeauty must be looking for a way to solve his problem

Would be really interesting to discuss this in the General science sub-forums!

If the frictional force is kinetic, then by definition it opposes the motion of the object, meaning the object must be skidding to the right.

If friction is static, then the tension has not yet pulled hard enough on the object to overcome the static frictional force to the left, meaning that the object will roll, but to the left. Think of this as if you're pulling on the string, but the static friction is so strong that it causes the object to pivot (and begin to spin) toward the left.

Thus, the critical angle is the angle where the skidding to the right would switch to rolling to the left!

Yep. That's how I think it is, too. At angles above the critical, you will have the spool moving backwards, and angles below the critical, it will move forward.

 

[schaefera]

It's a little hard to get into that mindset, because it almost sounds like it's asking when it begins to roll after it has already been sliding to the right... clearly, that can't happen though, which I didn't realize until I pictured it that way!

 

[ehild]

Hi, Charmedbeauty,
we almost forgot about you...

ohh I think I get it from the eqn above solve for f_s

f_s=ma+Tcosθ

Correct.

Nμ_s=ma+Tcosθ

That equation is valid at the limiting case when the spool starts to skip...

torque=f_sR-Tr=ma

The torque is equal to angular acceleration times moment of inertia: βI

In case of pure rolling, the spool moves with velocity V and rolls with angular velocity ω=V/R. A similar relation is valid between the linear acceleration of the spool and the angular acceleration of its rotation: a=Rβ. So you can write the torque equation in the form

f_sR-Tr=Ia/R.

From this equation and from the one you wrote for the acceleration,

f_s=ma+Tcosθ

you can cancel f_s and find the acceleration of the rolling spool. If you have the formula for a, you can discuss it: at what angles does the spool accelerate forward or backward, and which is the angle where the angular acceleration is zero so rolling can not start.
Substituting the acceleration back into the first equation for fs, you can check if fs is not greater than the maximum static friction μ_sN.


ehild