Question $3(10$ pts$)$

For the system shown below:

Use $x,{}_1$ and ${}_2$ as the coordinates and assume positive displacement, velocity, and

acceleration in the direction shown by the arrows. Assume the connections between each

mass are rigid.

The wheel $m_2$ rolls without slipping. You must include all Free Body Diagrams $($FBD$)$ and

Mass Acceleration Diagrams $($MAD$)$ in order to apply Newton's Laws.

a$)(1\mathrm{.}5$ points$)$ Divide the system into individual subsystems for each body. Explain how

thesesubsystems are interconnected and highlight the key parameters relevant to each

body.

b$)(2$ points$)$ Draw the appropriate Free Body Diagrams $($FBDs$)$ and Mass Acceleration

Diagrams $($MADs$)$ for each subsystem. For $m_2,$ make sure to incorporate the condition that

itrolls without slipping, implying that the tangential velocity of the wheel is equal to the

translational velocity of the mass $m_0.$

c$)(1\mathrm{.}5$ points$)$ Using the FBDs and MADs, sum all forces and moments$/$torques to derive

the equations of motion for each subsystem. Ensure that you account for both rotational

and translational dynamics.

d$)(1\mathrm{.}5$ points$)$ Based on the derived equations of motion, use Newton's laws to find the

naturalfrequency $w_n$ of the system.

e$)(2$ points$)$ Identify where a damper should be placed to maximize the damping ratio for the

system. Include the damping term in the equations of motion and explore how the

placementaffects the system's response.

f$)(1\mathrm{.}5$ points$)$ Use an energy$-$based approach $($kinetic and potential energy$)$ to verify the

naturalfrequency $w_n$ obtained from the Newton's law approach.