A bully is chasing little Billy through the playground. In an attempt to escape,

Question

A bully is chasing little Billy through the playground. In an attempt to escape, Billy runs toward a stationary merry-go-round. He leaps onto it tangentially with a horizontal velocity of 4 m/s and lands on the edge. The bully decides to spin the merry-go-round with a constant torque of 30 N?m so that Billy flies off. Billy has nothing to hang onto and only friction is keeping him on the merry-go-round, how long does it take for poor young Billy to fly off of the merry-go-round? Billy is 35 kg and the merry-go- round is 100 kg. The radius of the merry-go-round is 1 m and the coefficient of static friction between the rubber on the soles of Billy's shoes and the steel merry-go-round is 0.7.

Explanation / Answer

The initial angular momentum of the system is Billy's angular momentum,

L = r mBilly v = 140 kg *m^2/s

Now, we treat the merry go round as a disk, with

Imerry = 1/2 M R^2 = 50 kg*m^2

The I due to Billy is

Ibilly = mBilly R^2 = 35 kg*m^2

Thus,

Itot = 85 kg*m^2

Thus, the initial angular velocity after Billy jumped, by conservation of angular momentum, is

w0 = 1.647 rad/s

Now, the bully is putting in a torque of 30 Nm, which is

Torque = I(alpha) ---> alpha = Torque/I

Thus,

alpha = 0.3529 rad/s^2

Now, Billy flies off if

centripetal force = us m g = m w^2 r

Solving for the critical w,

w = 2.6192 rad/s

Thus as

w = w0 + alpha t

Solving for t,

t = 2.755 s   [ANSWER]

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