Problem statementSolution video |

**DISCUSSION THREAD**

We encourage you to interact with your colleagues here in conversations about this homework problem.

*CLARIFICATION*: Consider the bar to be *parallel* to the incline. This is made possible by having half-thickness of the bar being *R, *thus keeping the centerline of the bar parallel to the incline.

**DISCUSSION**

**Four-step plan (for deriving EOM)**

**Step 1: FBD**

Draw *individual* free-body diagrams for the two cylinders and for bar B. Do not forget the friction forces at the no-slip contact points when drawing your FBDs of the cylinders and bar. Define rotation coordinates *β *and *ϕ* for the upper and lower cylinders, respectively. Let's say that you choose both of these rotation coordinates to be positive in the CW sense. Be sure that the direction of the spring force on the bar is consistent with the definition of *x*.

**Step 2: Kinetics - Newton/Euler**

Write down the Newton/Euler equations for the two cylinders and the bar. Be sure to abide by the sign conventions defined for *β**, ϕ* and *x *when writing down these equations.

* Step 3: Kinematics*The kinematics that you need here are to relate

*β*

*and*

*ϕ*to

*x*. Be reminded that the contact points of the cylinders with the fixed incline are the instant centers for the cylinders. As before, be sure to abide by the sign conventions defined for

*β*

*, ϕ*and

*x*. For

*β*and

*ϕ*defined positive in the CW sense, we have

*x*=

*+3Rβ*and

*x*=

*+2Rϕ*

*.*

**Step 4: Equation of motion**

Combine your equations from Steps 2 and 3 to end up with the differential equation of motion for the system in terms of *x*.

To define the EOM in terms of Z, do we have to make new equations for two separate states and combine them with Z?

I'm still not sure how to determine if the spring is initially stretched or un-stretched. The two pages in the book related to setting up the FBD's for these problems only confuse me more. Any help or explanation would be much appreciated.

I believe the variable "x" is used to denote how much the string has stretched, and since x=0, so the string should initially be unstretched.

*spring not string, it's been a long day.

Wouldn't x = +Rϕ instead of 2Rϕ?

It is 2R theta because you consider that the contact point is the instant center. The velocity at the top of the disk A is x dot, which is the same as the velocity at O for the bigger disk. These two point are connected through the rod

I am unable to get any unknown variable in my moment equation for the smaller disk except for ϕ double dot. This means I cannot incorporate it into my other equations. Can anyone help with this?

I figured it out. There's friction on the top and bottom of the disk.

Wouldn't you have to use a different direction coordinate for the bottom cylinder, as it moves at a different speed from the upper cylinder / bar?

I think you can use two variables, one to describe the rotation of the upper cylinder and one the lower cylinder. You can use the same coordinate direction as both would be in the k direction.