Rotational dynamics (rolling friction)

[mxbob468]

problem 1:

Homework Statement

this isn't quite a problem but a general confusion about how to setup the problem. for example:

a cylinder of radius R with mass M has a force T applied at its center of mass, pointing horizontally to the right. assuming the cylinder only rolls without slipping what happens?

Homework Equations

a=r$\alpha$
v=r$\omega$
ƩF=ma
ƩT=I$\alpha$

The Attempt at a Solution

one trouble i have is with friction, with what i guess is called "rolling friction". first question: does rolling friction oppose angular acceleration? is that the assumption i need to use when deciding the "direction" of the rolling frictional force?

anyway i setup my equations as such:

because the tension is applied at the COM it causes no torque and therefore we only have sliding friction (taking +x as the positive direction in our inertial reference frame):

T-Ff = Ma

Now friction does apply a torque, a clockwise torque and hence a negative sign needs to be put in by hand:

-Ff*R=I$\alpha$

rolling without slipping implies a=r$\alpha$ but in my book it says that "linear and angular accelerations are in opposite directions" so in fact it's

a=r$(-\alpha)$

i don't understand what this means? what "linear and angular accelerations are in opposite directions" means. angular acceleration (if we're thinking terms of vectors) points into the page and linear acceleration points in the +x direction. hencen not opposed in anyway i can imagine. does it mean something like linear acceleration in the positive +x direction always induces a clockwise angular acceleration? and then an angular acceleration induced by a counterclockwise torque would be "opposite" to linear acceleration? but that wouldn't make sense considering what my book is saying for this problem (since in this case they'd be point in the same direction)? and though i understand that using this relation (a=r$-\alpha$) i can solve for T and a, and the Ff needed so that the cylinder rolls without slipping i do not understand the negative sign.

i have more problems but i'll post them later in the thread (or open up a new one?).

 

[Redbelly98]

Since we take the clockwise direction to be negative, and the angular acceleration is clockwise, it is negative. And this happens for a positive (i.e., towards the right) linear acceleration -- so a and α have opposite signs.

i have more problems but i'll post them later in the thread (or open up a new one?).

Please start new threads to post separate problems.