Derivation of I_M2.using translational and rotational dynamics and kinematics. Y

Question

Derivation of I_M2.using translational and rotational dynamics and kinematics. You can find an equation for I_M2 using dynamics or by using energy considerations. You will do both. In this section you will use dynamics (and kinematics), in the next section you will use conservation of energy land kinematics) Note that your result for I_M2 should he exactly the same Draw an extended FBD for the hoop Include coord system and label ill quantities Use your FBDs to set up your dynamics equations. Don't solve here: Sigma F_x = Sigma F_y = Sigma tau = In the last equation, indicate with subscripts whether you are considering the torques (and moment of inertia) about the center of mass or the point of contact between the hoop and the track. List the acceleration constraint (how are alpha and a_cm related?): Solve your equation for the moment of inertia about the center of mass. Call this quantity I_M2. Note: you should only have measured quantities on the right hand side, i.e., mass, acceleration (from experiment), radius, angle. Final answer: I_M2 = Note: put your final formula here, not the numerical result.

Explanation / Answer

Fx is given by

=> Mg * sin(theta) - Friction =    Ma

Fy is given by

=>   N = Mg * cos(theta)

torque , T   =   IM2 * (alpha)

=>   Friction * r2   = IM2 * (alpha)

Also, a = r2 * (alpha)

IM2   =   1/2 * M * (r12 + r22)

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