# Homework H6.E - Fa22

DISCUSSION and HINTS

Recall the following four-step plan outline in the lecture book and discussed in lecture:

Step 1: FBDs
Draw individual free body diagrams (FBDs) of the wheel and block A.   Don't forget the equal-and-opposite pair of reaction forces acting between the wheel and the block atO. It is important to temporarily define a coordinate that describes the rotation of the wheel. Let's call this θ,  and define it to be positive in the clockwise sense.

Step 2: Kinetics (Newton/Euler)
Using your FBDs from above, write down the Euler equation for the disk (about the no-slip point C), and the Newton equation for the block. Combine these two equations into a single equation through the elimination of the reaction forces at O for the equations.

Step 3: Kinematics
You need to relate the angular acceleration of the disk and the acceleration of the block. How is this done? What are the results?

Step 4: EOM
From your equations in Steps 2 and 3, derive the equation of motion (EOM) of the system in terms of x.

Once you have determined the EOM for the system, identify the natural frequency, damping ratio and the damped natural frequency from the EOM. Also, from the EOM we know that the response of the system in terms of x is given by: x(t) = e-ζωnt [C*cos(ωdt)+ S*sin(ωdt)]. How do you find the response coefficients C and S?

## 6 thoughts on “Homework H6.E - Fa22”

1. George West Wehmann says:

Does sign convention for internal forces at O matter?

1. Joel Mason Laudo says:

No. As long as they are written as equal and opposite forces on the two individual FBD's the direction doesn't matter.

2. perez353 says:

I don’t believe sign convention should matter, because if an error is made the math should correct itself

3. Somya Jain says:

Do we consider friction for this question, since it is no slip and smooth?

1. Aquinas says:

I think we should consider friction for moment equation around center O, but it's not needed for moment around contact point C.

4. Joel Mason Laudo says:

Yes, you should have friction as a part of your wheel FBD, but not for the block since it is smooth. For the wheel, if you do your moment equation about the contact point you can neglect friction, since it acts at that point.