Need help on this question, thanks in advance A spool has an inner radius of r.

Question

Need help on this question, thanks in advance

A spool has an inner radius of r. an outer radius of R, and a radius of gyration k. The center of mass is located at the center of the spool. A wire of negligible mass is wrapped around the inner portion of the spool as shown below. Determine the range of possible values for theta for which it is possible to apply a force P to the end of the wire so that the center of mass remains stationary as the spool rotates. The coefficient of kinetic friction between the spool and the surface is mu k

Explanation / Answer

let normal reaction from the ground be N

force balance in vertical direction

P*sin(theta) + N = M*g

N = M*g - P*sin(theta)

force balance in horizontal direction (centre of mass remains stationary)

uk*(M*g - P*sin(theta)) = P*cos(theta)

for rotation

P*r > uk*(M*g - P*sin(theta)) * R

or

P*r > P*cos(theta) * R

cos(theta) < r/R

theta > cos inverse (r/R)

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