A ball of mass m = 500g is being lifted vertically upward by a cable wrapped aro

Question

A ball of mass m = 500g is being lifted vertically upward by a cable wrapped around a pulley which has a radius r2 = 1m. This pulley is fastened to a larger pulley, which has a radius r1 = 2m, so that they form a "dual" pulley and turn as a single unit about the center axle, relative to which the combined moment of inertia is I = .5 kgm^2. The larger pulley rotates by means of the cable wrapped around it and attached to the motor that maintains a tension of magnitude T1 = 7.5 N in the cable. Find:

a.) The tension force T2 in the cable connected to the ball.

b.) The angular acceleration (alpha) of the "dual" pulley.

c.) The vertical acceleration ay of the ball.

Explanation / Answer

Mass of ball = 500 gram = 0.5 kg

R1 = larger radius = 2 m

R2 = small radius = 1 m

Moment of inertia of dual pulley = Ip = 0.5. kg.m2

Tension T1 = 7.5 N

(a). T2 - Mg = Ma

( 7.5×2 - T2 ×1= 0.5×(a/1)

Solving both the equations

T2 = 9.95 N

(b). Angular acceleration = (7.5×2- 9.95×1)/(0.5) = 10.1 rad /s2

(c). Vertical acceleration = 10.1 ×1 = 10.1 m/s2

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