show steps A disk (moment of inertia, I = 1/2 MR^2) has a string wound around it

Question

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A disk (moment of inertia, I = 1/2 MR^2) has a string wound around its circumference. The end of the string is held in place, and the disk is allowed to falls, unrolling the string as it does. The only forces on the disk the string tension acting at its edge, and gravity acting at center. The disk has mass M = 0.2kg radius R = 0.05m. a. Find the tension in the string, and the acceleration of the falling disk. You can assume the disk unrolls the string without sliding, so that v_c, = omega R and = alpha R can be used, as in rolling motion. b. The disk starts from rest at point "1" and descends through h = 0.3 meters to point "2". Find the velocity

Explanation / Answer

(a)Torque = I

=> (T)(R) = (MR2/2)()

=> = 2T/MR

Mg - T = Ma

=> a = g - (T/M)

a = R

=> g - (T/M) = R(2T/MR)

=> g = 3T/M

=> T = Mg/3 = 0.654 N

a = 2g/3 = 6.54 m/s2

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(b) Using energy conservation,

(Mgh) + W = (1/2)Mv2 +(1/2)I2

=> Mgh - (T)(h) = (1/2)M(R)2 +(1/2)I2

=> 0.589 - 0.196 = (1/2)M(R)2 +(1/2)(MR2/2)2

=> 0.393 = (3/4)MR22

=> = 1048 rad/s2

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