Question

A merry-go-round is a common piece of playground equipment. A 3.0-mm-diameter merry-go-round, which can be modeled as a disk with a mass of 220 kg, is spinning at 24 rpm. John runs tangent to the merry-go-round at 5.6 m/s, in the same direction that it is turning, and jumps onto the outer edge. John's mass is 30 kg.

Answer 1

(I believe you need final angular velocity of system, if you need anything else do let me know through comments, Also diameter of merry-go-round should be 3.0 m, and not 3.0 mm, please check and confirm that)

Using Angular momentum conservation on system before and after child moves to the center:

Li = Lf

Ii*wi + m*v*R = If*wf

wi = initial angular speed of ride = 24.0 rpm = 24.0*2*pi/60 = 2.513 rad/sec.

wf = final angular speed of ride = ? rpm

Ii = Initial angular momentum of ride = I_disk = 0.5*M*R^2

M = mass of merry-go-round = 220 kg

R = radius of merry-go-round = 3.0 m/2 = 1.5 m

Ii = 0.5*220*1.5^2 = 247.5 kg*m^2

m = mass of child = 30 kg

v = speed of child = 5.6 m/s

If = 0.5*M*R^2 + m*R^2

If = (M/2 + m)*R^2

If = (220/2 + 30)*1.5^2 = 315 kg-m^2

Using these values:

wf = (wi*Ii+m*v*R)/(If)

wf = (2.513*247.5 + 30*5.6*1.5)/315

wf = 2.7745 rad/sec. = 2.7745*60/(2*pi)

wf = 26.49 rpm = final angular velocity

In two significant figure:

wf = 26 rpm

"Let me know if you have any query."