Rotational kinematics/dynamics problem

[pppeachykeen]

Homework Statement

A plank with mass M = 6 kg rides on top of two identical solid cylindrical rollers that have R = 0.05 m and m = 2 kg. The plank is pulled by a constant horizontal force F of magnitude 6 N applied to an endpoint of the plank and perpendicular to the axes of both cylinders. The cylinders roll without slipping on a horizontal surface and the plank does not slip on the cylinders either. Knowing that there must be rolling friction (not sliding friction) between the plank and the cylinders and between the cylinders and the surace, calculate:

(a)the linear acceleration of the plank and of the rollers
(b)the magnitudes of all friction forces that are acting.
http://img389.imageshack.us/img389/61/figureio5.th.jpg

Homework Equations

Not sure...

The Attempt at a Solution

I have no idea how to go about this...any help with starting would be appreciated.

 

[OlderDan]

Draw free body diagrams for each cylinder and one for the plank. Each cylinder has a horizontal frictional force acting at the top and bottom. The plank has three horizontal forces acting. All vertical forces are balanced and need not be considered (although that is an interesting problem too). Consider the relationship between the distances moved by the plank and the rollers, and the relationships between the angular and linear motion of the rollers.

You will have several equations relating the applied forces to the accelerations that can be solved simultaneously to find the forces and accelerations.

 

[m2003]

As a scientist, it is important to approach problems like this with a systematic and analytical mindset. We can start by identifying the given information and the unknowns. From the given information, we know the masses and dimensions of the plank and rollers, the applied force, and the type of motion (rolling without slipping). The unknowns are the linear accelerations of the plank and rollers, and the magnitudes of the friction forces.

Next, we can use the principles of rotational kinematics and dynamics to analyze the problem. We know that the plank and rollers will experience both translational and rotational motion due to the applied force. The plank will have a linear acceleration, while the rollers will have both a linear and angular acceleration. We can use the equations for linear and angular motion, along with the given information, to solve for the unknowns.

To calculate the linear acceleration of the plank, we can use Newton's Second Law, which states that the sum of the forces acting on an object is equal to its mass times its acceleration. In this case, the only horizontal force acting on the plank is the applied force F. We can set up the equation as follows:

ΣF = ma

F - Ff = ma

Where Ff is the friction force between the plank and the rollers. To find the value of Ff, we can use the equation for rolling friction, which states that the friction force is equal to the coefficient of rolling friction (μ) times the normal force (N) between the two surfaces. In this case, the normal force is equal to the weight of the plank, which is given by mg. So we have:

Ff = μmg

Substituting this into our equation, we get:

F - μmg = ma

We can rearrange this to solve for a:

a = (F - μmg)/m

Using the given values, we can plug in the numbers and solve for a. This will give us the linear acceleration of the plank.

To calculate the linear acceleration of the rollers, we can use the same equation, but this time we need to consider the forces acting on the rollers. In addition to the applied force and the friction force, the rollers will also experience a normal force from the surface. This normal force will be equal to the weight of the rollers, given by mg. So our equation becomes:

ΣF = ma

Ff - F - mg = ma

We can rearrange this