A kid is standing on the edge of a merry-go-round that is spinning counter-clock

Question

A kid is standing on the edge of a merry-go-round that is spinning counter-clockwise at 2 rad/s. He has a mass of 50 kg and the merry-go round has a mass of 175 kg. The merry-go-round has a radius of 1.5 m. His identical twin jumps onto the edge of the merry-go-round when he is running 4 m/s in the other direction. a.)How fast is the merry-go-round going after he jumps on? b.)Determine how much of the total kinetic energy of the system (consiting of the kid, the twin and the merry-go-round) will change when the twin jumps on. c.) Include an angular momentum bar chart and an enery bar chart

Explanation / Answer

Angular momentum of system is conserved.

So, Initial angular momentum = final angular momentum

angular momentum of the merry-go-round + angular momentum of boy riding on it + angular momentum of boy running in opposite direction = final angular momentum of both boys + final angular momentum of the merry go round

Iw+ m(wr)r + mvr = Iw`+ 2m(w`r)r where I = moment of inertia of merry go round

m =mass of boy v = velocity of the boy running opposite to merry go round w= angular velocity of the merry go round w` = final angular velocity of merry go round

I= M(r^2) where M=mass of merry go round

so, M(r^2)*w + 50(2*1.5)*1.5 - 50*4*1.5 = 2 [ 50*w`*1.5*1.5] + M(r^2)*w`

175*1.5*1.5*2 + 50(2*1.5)*1.5 - 50*4*1.5 = 618.75w`

so, w`= 1.1515 rad/s

b) Initial kinetic energy(IKE) = rotational kinetic energy of merry go round+ rotational kinetic energy of boy riding it + rotationa kinertic energy of boy running opposite

IKE = [1/2 * I * w^2] + [1/2 * m* (wr)^2 ] + [ 1/2 * m* v^2]

= 1412.5J

Final Kinetic Energy (FKE) = rotational kinetic energy of merry go round + rotation kinetic energy of both the boys riding on it

FKE= [1/2 * I* (w`)^2] + 2* [1/2 * m* (w`*r)^2]

= 410.216 J

Change in total kinetic energy = FKE - IKE

= 410.216- ( 1412.5)

= - 1002.284 J

The bar charts are on this link : - https://www.dropbox.com/sh/9herugus60grxba/AAC_isXSkkDUXkImFP1xvo60a?dl=0

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