How can I find the acceleration of the plank

[kash25]

Hi,

If I have a plank (mass M) resting on two identical parallel cylinders (mass m, radius R) and I pull the plank with a horizontal force F (no slipping between cylinders and floor nor between plank and cylinders), what happens? How can I find the acceleration of the plank and the rolling cylinders? Why DOESN'T a = F/M give the acceleration of the plank AND the tangential acceleration of the cylinders?

Thank you!

 

[Bob Smith]

So by saying no slipping, the cylinders must begin to roll as the plank is pulled?

If I'm right here, you need to consider the system as a whole. That means in f=ma, you need to be using the total translational inertia of the system, rather than just the component masses. So of course the masses are present, but as the cylinders are rolling you need to convert their moment of inertia to translational inertia by using I/R², so a = F/(M+2m+2I/R²). Where I is the moment of inertia of the rollers, which for a cylinder is mR²/2, so this conveniently reduces to a=F/(M+3m).

Unless I'm mistaken, this should give the acceleration of both the plank, and the tangential acceleration of the cylinders.

 

[Doc Al]

Unless I'm mistaken...

I'm afraid you are.

If I have a plank (mass M) resting on two identical parallel cylinders (mass m, radius R) and I pull the plank with a horizontal force F (no slipping between cylinders and floor nor between plank and cylinders), what happens? How can I find the acceleration of the plank and the rolling cylinders?

Analyze the problem in the usual manner. Start by identifying the forces acting on the plank and the cylinders. (F is not the only force!) Draw free body diagrams for each object.

Apply Newton's 2nd law for translation and rotation to plank and cylinders. (How does the acceleration of the plank relate to the acceleration of the cylinders?)

Why DOESN'T a = F/M give the acceleration of the plank AND the tangential acceleration of the cylinders?

Why should it? To find the acceleration of the plank (or any object) you need the net force acting on it. Besides the applied force F, what other forces act on the plank?

 

[Bob Smith]

I'm afraid you are.

Ah yes, I'm thinking as if the plank was connected to the cylinders as if they were axles, in the situation described the plank would eventually roll off the top of the cylinders as they would only be traveling half as fast?

Actually I've also assumed that the cylinders are on the ground, but as the OP said rolling and not rotating, I'm hoping that was a correct assumption to make.

Although you are correct, for the cylinders to roll, there must be friction between the cylinders and the plank, and the ground.

My excuse is I was tired. :p