How Do You Calculate Tensions and Accelerations in a Modified Atwood Machine?

[hime]

Homework Statement

A modified atwood machine consists of a hoop hanging from a solid disk pulley whose other end is a string attached to massless axel through the center of a sphere on a flat surface. The sphere rolls without slipping. The system is released from rest.

What are the linear and angular acceleration of the sphere?
What are the two tensions between the disk and the hoop and the disk and the sphere?

Homework Equations

F=m*a
Torque=I*angular acceleration
Torque=Force*radius
a=angular acceleration*radius
v=omega*radius

The Attempt at a Solution

Mass of sphere=M1 Radius of sphere=R1
Mass of disk=M2 Radius of disk=R2
Mass of hoop=M3 Radius of hoop=R3
The forces acting on the hoop: T2-M3*g=0 => T2=M3*g
The forces acting on the sphere: T1*R1-Ffric*R1=Torque =Isphere*angular acceleration
The forces acting on the disk: T1=T2

 

[NateD]

+M2*gThe force Ffric can be calculated using the coefficient of kinetic friction.Ffric=μ*(T1-M2*g)Therefore, we have three equations and three unknowns, so we can solve for T1,T2 and angular acceleration.The linear acceleration of the sphere is given by a=angular acceleration*radius