Rotational work and forces like static friction and tension

[Soren4]

I have a doubt regarding the definition of rotational work, which is as follows

$W = \int \tau_z d \theta$

Where $\tau_z$ is the component of the torque parallel to the axis of rotation $z$.

My doubt concerns the fact that, looking at this definition, it seems that any torque which has an axial component and causes a rotation, do rotational work.

However, consider a problem of pure rotational motion (i.e. rolling with no slipping), for example a disk rolling down an inclined plane and take as pivot point for calculate torque the center of mass of the disk. The static friction force generates a torque which is totally axial and is the cause of the rotation of the disk (if there was no friction the disk would just slip). But the static friction is a classic example of a force that does not do work because it does not cause any displacement.

A similar situation is the one with rigid bodies similar to yo-yos, i.e. falling pulleys that carry a wire (on which they roll without slipping). Again, the tension of the rope exerts an axial torque and causes the rotation of the yo-yo, but it seems very strange to me that tension can do work.

Do these forces perform rotational work? I probably misunderstood the definition but I do not see what is wrong. Thanks in advance for your suggestions.

 

[BvU]

You left out the fixed axis ...
Rolling example: if the axis is the point where the friction acts (the 'fixed axis'), there is no torque and no work.
yoyo example: if the axis is the point where the tension acts (the axis of rotation) there is torque from gravity, which is doing work.
(note that friction isn't required to keep a cylinder rolling, only to get it to roll)

 

[Soren4]

You left out the fixed axis ...
Rolling example: if the axis is the point where the friction acts (the 'fixed axis'), there is no torque and no work.
yoyo example: if the axis is the point where the tension acts (the axis of rotation) there is torque from gravity, which is doing work.
(note that friction isn't required to keep a cylinder rolling, only to get it to roll)

Thanks for the answer! Sorry if I expressed the condition of pure rotational motion incorrectly.

In the Wikipedia page it is claimed that the definition of rotational work is for a "rotation about a fixed axis through the center of mass". Now, considering the example of the disk on the incline, I'm totally ok with the fact that, if I consider the contact point of the disk with the incline to be the instantaneous axis of rotation, then static friction exerts zero torque while gravity does work, but, is there any reason why I cannot interpret the motion of the disk as a rotation about the center of mass, plus a traslational motion of the center of mass? If I do so, gravity still do ("traslational") work, but I don't see the reason why friction does no (rotational) work, according to the definition (it does exert a axial torque if the pivot point is in the center of mass). Am I missing something?

 

[BvU]

I don't think you are missing something. If you let something rotate, you also make use of friction. But it's you doing the work, not the friction.

 

[Soren4]

Thanks, forgive me but I do not get how I am doing work instead of friction, I mean I just let the disk roll down the incline, I do not exert any force on it

 

[BvU]

Thanks, forgive me but I do not get how I am doing work instead of friction, I mean I just let the disk roll down the incline, I do not exert any force on it

That was gravity doing the work.

I was referring to letting e.g. a ball spin: you need friction to do that,

 

[Soren4]

I don't think you are missing something. If you let something rotate, you also make use of friction. But it's you doing the work, not the friction.

That was gravity doing the work.
I was referring to letting e.g. a ball spin: you need friction to do that,

I hope I understood what you said (the fact that friction is not responsible by itself for the rotation, because it exists just if a force is exerted) but I still do not see why friction does no rotational work, just looking at the definition above, since fricition does exert an axial torque and that formula tells me that it means that friction does rotational work. Suppose to consider the motion of the disk as a combination of a traslational motion of CM and a rotation about the CM. Then if a force is exerted in the center of mass of the disk (like gravity), that force cannot exert torque, and I do not get how it can do the rotational work instead of friction

 

[BvU]

Other example: beam supported at two ends. One of the two supports disappears and the beam starts to fall and rotate. Does the other support do work ?

I admit I find it difficult to oppose your last paragraph. But you are not looking in an inertial frame and gravity does exert torque. Torque appears when moving an action point of a force in a direction perpendicular to the force. In this case gravitational force is moved sideways to directly above the support point. Torque and angular acceleration result.