Angular acceleration of cylinder caused by descending mass

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Mass m= 8 kg is suspended from a cord which passes over a frictionless pulley then wraps tengentially on the surface of a uniform cylinder, mass M= 12 kg radius R = .66 m, on a plane inclined at angle b= 25 degrees from horizontal. As the suspended mass descends it gives angular acceleration A to the cylinder rolling it without slipping up the plane.
See attachment #1 for picture showing parameters.

a. Develop a relationship between acceleration am, of the descending mass, and the linear acceleration ac, of the c. m. of the cylinder up the plane.

b. Find the accelerations am, of the descending mass, and the acceleration ac, of the c.m of the cylinder.

c. Find the friction force F on the cylinder at contact point. State whether this friction force is up or down the plane.

part a.
As mass m descends, the cylinder rotates and also moves upward along the plane. If in time t, the cylinder rotates exactly one revolution, it moves distance 2 Pi R up the plane, and also unwraps length 2 Pi R from its surface. Therefore, the mass must move downward distance (2)(2 Pi R) in the same time t. From motion equations for constant acceleration, starting from rest, the acceleration of mass m is:
(1) '4 Pi R = .5 am t^2' and the acceleration ...

Solution Summary

With clear narrative and complete formulas, the problems are completed.

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