A drum that unwinds cable is attached to an electric motor which, as it is start

Question

A drum that unwinds cable is attached to an electric motor which, as it is starting, has an angular velocity of omega=t^2/3 where t is in seconds. The diameter of the drum is 0.5 m. When the motor is turned on at t = 0, 2 meters of cable have already been unwound from the drum. The diameter of the cable on the drum would decrease as the cable is unwound but this can be neglected. At t = 3 seconds determine the following three quantities (a) find the total length of cable that has been unwound, (b) the acceleration of the end of the cable, A, and (c) the magnitude of the acceleration of point P on the drum.

Answers: s_t=3=2.75m, a_A=0.5 m/s^2, a_p=2.3 m/s^2

Explanation / Answer

d(th) /dt = w = t^2/3

th = t^3/9

from 0 to 3

th = 1 rad

length = 2+1*0.25 = 2.25 m

b) alpha = 2/3 t

at t = 3

alpha = 2 rad/s2

a = alpha*r = 0.5 m/s2

c)ap = sqrt(0.5^2 + (3^2*0.25)^2)

aP = 2.3m/s2

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