A 8 kg disk with a radius of 2 meters has several forces applied to it as shown

Question

A 8 kg disk with a radius of 2 meters has several forces applied to it as shown in the sketch. F1 is 37 N and is applied at the edge of the disk, F2 is 6 N is applied at 1/4 the radius from the center of the disk, F3 is 10 N is applied at 3/4 the radius from the center, F4is 24 N is applied at the edge of the disk, and, lastly, F5 is 12 N and is applied at the edge of the disk at an angle of 21 degrees. If the moment of inertia about the center for the disk is I = 1/2 M R2, and the disk is initially spinning at 42 rad/s counter-clockwise its center,  how long, in seconds, does it take the disk to be spinning at 7 rad/s counter-clockwise about its center?

Explanation / Answer

Net torque in the disc about X =(37*2)-(6*2/4)-(10*2*3/4)+(12cos 21*2)=78.4 Nm(clockwise)

Mass moment of inertia of disc=0.5*8*2*2=16 m/s2

Angular acceleration of disc =78.4/16=4.9 rad/s2 (clockwise)

Time taken by disc to reach to 7rad/s from 42 rad/s

=(42-7)/4.9=7.49 seconds

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