A merrygoround is a common piece of playground equipment A 4

A merry-go-round is a common piece of playground equipment. A 4.0-m-diameter merry-go-round with a mass of 280 kg is spinning at 22 rpm. John runs tangent to the merry-go-round at 5.0 m/s, in the same direction that it is turning, and jumps onto the outer edge. John\'s mass is 37 kg. What is the merry-go-round\'s angular velocity, in rpm, after John jumps on?

Solution

R = 1.4 m, M = 260 kg, v = 5 m/s

Using conservation of angular momentum.

L_i = I_i * w_i + m V R   

Final momentum L_f = I_f * w_f

Moment of inertia for a disk: I_i = 1/2 * M * R²

=1/2 * 260kg * (1.4 m)² = 254.8 kg.m²

Final moment of inertia can be determined using parallel axis theorem:

I_f = I_i + mR²

= 254.8 kg.m² + 39 kg * (1.4 m)²

= 331.24 kg.m²

Converting revolutions per minute to radians per second:

(22 rpm) * ( 2? rad / rev) * (1 min / 60s) = 2.30 rad/s

L_i = L_f

I_i * w_i + mVR=  I_f * w_f   

(254.8 kg*m²) * 2.30 rad/s + 39kg * 1.4 m * 5m/s = 331.24 kg.m² * ?_f

?_f = 2.59 rad/s

?_f = 24.73 rpm