Pulley system with 2 inclined planes on either side, a block, and a cylinder

[khfrekek92]

Homework Statement

Two inclined planes on either side of a frictionless pulley have angles theta (30 degrees) and phi (60 degrees). On the theta plane is a circular solid cylinder of mass M (1.2 kg) and radius R (.2 m). It is connected by a weightless rope and over the pulley to the other plane to a black of mass m (3 kg). The coefficients of static and kinetic friction are .8 and .5 respectively. The system is released from rest at time t=0. What is the resulting linear and angular acceleration of the cylinder, and which direction is it moving?

Homework Equations

a=mv^2/r ??
F_f=uN
F=ma
I=MR^2

The Attempt at a Solution

Through intuition I'm fairly sure the cylinder will be pulled up the theta plane (as the block is on a much steeper incline, and it is more than double the mass). But I'm not sure how to calculate the accelerations of the cylinder. I've set up the fore-diagrams and I'm guessing you use f=ma to find to linear acceleration? And then how do I get the angular?
Thanks so much in advance!

 

[anathema]

Thank you for your question. To calculate the linear acceleration of the cylinder, you can use the equation F=ma, where F is the net force acting on the cylinder and m is its mass. The net force can be calculated by considering the forces acting on the cylinder: the weight of the cylinder, the tension in the rope, and the friction force. The weight of the cylinder is mg, where g is the acceleration due to gravity. The tension in the rope can be calculated using the angles of the inclined planes and the pulley, and the friction force can be calculated using the coefficient of friction and the normal force on the cylinder. Once you have calculated the net force, you can solve for the acceleration.

To calculate the angular acceleration, you can use the equation τ=Iα, where τ is the net torque acting on the cylinder, I is the moment of inertia of the cylinder, and α is the angular acceleration. The net torque can be calculated by considering the torque due to the weight of the cylinder and the friction force. The moment of inertia of the cylinder can be calculated using the equation I=MR^2. Once you have calculated the net torque, you can solve for the angular acceleration.

I hope this helps. Please let me know if you have any further questions.