A solid disk is rotating about a frictionless axle with an initial angular veloc

Question

A solid disk is rotating about a frictionless axle with an initial angular velocity of 15 rad/s in a clockwise direction. The disk has a mass of 5.0 kg and a radius of 15 cm. A force of 20 N is applied to the disk for 0.5 s. This force is applied in a direction tangent to the edge of the disk that causes the disk to increase speed in the clockwise direction. I (disk) = ½ MR2
a. What are the initial magnitude and direction of the angular momentum about the center of the disk? (Assume the disk is rotating in the plane of the paper.)
b. What are the final magnitude and direction of the angular momentum about the center of the disk?
c. What are the final magnitude and direction of the angular velocity of the disk?

Explanation / Answer

solid disc initial angular velocity w1= 15 rad/s in clock wise direction

mass of disc m = 5.0 kg, radius r = 0.15 m,

force applied in the direction of the tnagent to the edge of the disc is 20 N about t = 0.5 s

we know that the moment of inertia of disc I = 1/2 mR^2

a) angular momentum L = I*w = 0.5*5*0.15^2*15 = 0.84375 kg m2/s. the direction is in to the plane

b) for final angular momentum , angular velocity required

   as force acts on the disk, torque acts on it so torque

T = I alpha = RF sin theta ==> alpha = RF sin theta/I
                   = 0.15*20*1/(0.5*5*0.15^2) = 53.33 rad/s2

angular acceleration aplha = dW/dt = W2-W1/dt = W2 = alpha*dt +W1 =53.33*0.4 + 15 = 36.33 rad/s
   angular momentum is L2 IW2 = (0.5*5*0.15^2)*36.332 =2.04 kg m2/s

direction is in to the plane

c) fina angular velocity is 53.33 rad/s clock wise direction

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