Pulling a plank across cylinders - Rotational Dynamics

[jkiddmagic]

Homework Statement

A plank with a mass M = 6.10 kg rides on top of two identical, solid, cylindrical rollers that have R = 4.10 cm and m = 2.00 kg. The plank is pulled by a constant horizontal force F of magnitude 4.80 N applied to the end of the plank and perpendicular to the axes of the cylinders (which are parallel). The cylinders roll without slipping on a flat surface. There is also no slipping between the cylinders and the plank.

(a) Find the initial acceleration of the plank at the moment the rollers are equidistant from the ends of the plank.

(b) Find the acceleration of the rollers at this moment.

(c) What friction forces are acting at this moment? (Let f_p be the frictional force exerted by each roller on the plank, and let f_g be the rolling friction exerted by the ground on each roller.)

f_p
f_g

http://www.webassign.net/serpse8/10-p-088.gif

Homework Equations

F = ma
T=I∂
I = (1/2)mR² (Moment of inertia for a cylinder)

The Attempt at a Solution

From f = ma, I understand that the net force pulling the system is equal to the net inertial mass multiplied by the acceleration of the system.

Net force would be equal to F pulling minus force of friction giving me the equation:
F - F_f = ma

In order to find F_f I used the T=I∂ equation
T=I∂
(f_p - f_g)R = (1/2 * mR²) * (a/R)

Simplifying this results in
(f_p - f_g) = (1/2)ma
which I can plug back into F = ma netting me

F - (1/2)ma = m_neta

So I plug in all my values:
4.8 - a = (6.1 + (2)(1/2)(2)(.041^2))a
and I get a = .676 for part (a)
and a = .338 for part (b)

Both of these answers are close, but wrong. The correct answers for a and b are .632 and .316, respectively. What am I doing wrong/missing?

Then, I'm unsure where to begin for part (c). If I plug in acceleration back into my net torque equations, I'm left with 2 unknowns and 1 equation.

 

[haruspex]

In order to find F_f I used the T=I∂ equation
T=I∂
(f_p - f_g)R = (1/2 * mR²) * (a/R)

A couple of problems there. In which rotational directions do the forces f_p and f_g act on the rollers? You haven't defined a. What is it the linear acceleration of?

which I can plug back into F = ma netting me

F - (1/2)ma = m_neta

What are you equating to F_f in order to do that? Is m in the above equation the given m (mass of one roller) or mass of the system?

 

[jkiddmagic]

Woops, I have f_p and f_g mixed up, so it should be f_g - f_p because friction with the ground acts in the direction of the applied force. Then a is the linear acceleration of one of the rollers. Since there is no slipping between the cylinders and the plank, linear acceleration should be equal to system acceleration, correct?

f_f = f_g - f_p = (1/2)ma where m is the mass of a roller.

So I realize that I should be doing F - 2Ff = ma because I have 2 rollers, and thus, 2 sources of friction. But that still yields me an answer of .59 for acceleration, which is still incorrect.

 

[haruspex]

Woops, I have f_p and f_g mixed up, so it should be f_g - f_p

I'll ask again: in which rotational directions do these two forces act on the rollers?

Then a is the linear acceleration of one of the rollers.

So what is the linear acceleration of the plank? Hint: think twice before answering.

 

[jkiddmagic]

Both of them act in the clockwise direction, I think.

The linear acceleration of the plank should be two times that of an individual roller.

 

[haruspex]

Both of them act in the clockwise direction, I think.

Right. So instead of f_g - f_p or f_p - f_g it should be?

The linear acceleration of the plank should be two times that of an individual roller.

Yes.

 

[haruspex]

Both of them act in the clockwise direction, I think.

One small correction/clarification to my earlier response.
It is not clear at the outset which way f_g acts. But that doesn't matter as long as it is treated consistently. We can take it as acting in the same direction as F, in which case it will oppose f_p in torque but work with it in linear acceleration of the rollers, or take it as being in the opposite direction. If we make the wrong guess we'll simply get a negative value. The important thing is that we'll have f_p+f_g in one equation and f_p-f_g in the other.
You may by now be aware of two other current threads on essentially the same problem, but perhaps with different numbers.