Hi there!
![\boxed{\omega = 0.38 rad/sec}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Comega%20%3D%200.38%20rad%2Fsec%7D)
We can use the conservation of angular momentum to solve.
![\large\boxed{L_i = L_f}](https://tex.z-dn.net/?f=%5Clarge%5Cboxed%7BL_i%20%3D%20L_f%7D)
Recall the equation for angular momentum:
![L = I\omega](https://tex.z-dn.net/?f=L%20%3D%20I%5Comega)
We can begin by writing out the scenario as a conservation of angular momentum:
![I_m\omega_m + I_b\omega_b = \omega_f(I_m + I_b)](https://tex.z-dn.net/?f=I_m%5Comega_m%20%2B%20I_b%5Comega_b%20%3D%20%5Comega_f%28I_m%20%2B%20I_b%29)
= moment of inertia of the merry-go-round (kgm²)
= angular velocity of merry go round (rad/sec)
= final angular velocity of COMBINED objects (rad/sec)
= moment of inertia of boy (kgm²)
= angular velocity of the boy (rad/sec)
The only value not explicitly given is the moment of inertia of the boy.
Since he stands along the edge of the merry go round:
![I = MR^2](https://tex.z-dn.net/?f=I%20%3D%20MR%5E2)
We are given that he jumps on the merry-go-round at a speed of 5 m/s. Use the following relation:
![\omega = \frac{v}{r}](https://tex.z-dn.net/?f=%5Comega%20%3D%20%5Cfrac%7Bv%7D%7Br%7D)
![L_b = MR^2(\frac{v}{R}) = MRv](https://tex.z-dn.net/?f=L_b%20%3D%20MR%5E2%28%5Cfrac%7Bv%7D%7BR%7D%29%20%3D%20MRv)
Plug in the given values:
![L_b = (20)(3)(5) = 300 kgm^2/s](https://tex.z-dn.net/?f=L_b%20%3D%20%2820%29%283%29%285%29%20%3D%20300%20kgm%5E2%2Fs)
Now, we must solve for the boy's moment of inertia:
![I = MR^2\\I = 20(3^2) = 180 kgm^2](https://tex.z-dn.net/?f=I%20%3D%20MR%5E2%5C%5CI%20%3D%2020%283%5E2%29%20%3D%20180%20kgm%5E2)
Use the above equation for conservation of momentum:
![600(0) + 300 = \omega_f(180 + 600)\\\\300 = 780\omega_f\\\\\omega = \boxed{0.38 rad/sec}](https://tex.z-dn.net/?f=600%280%29%20%2B%20300%20%3D%20%5Comega_f%28180%20%2B%20600%29%5C%5C%5C%5C300%20%3D%20780%5Comega_f%5C%5C%5C%5C%5Comega%20%3D%20%5Cboxed%7B0.38%20rad%2Fsec%7D)