Answer :
To solve this problem, we can use the principle of conservation of angular momentum. The total angular momentum of the system remains constant before and after the child moves to the center of the merry-go-round.
The initial angular momentum of the system is given by:
L_initial = I_initial * ω_initial
where I_initial is the moment of inertia of the system and ω_initial is the initial angular velocity.
The final angular momentum of the system is given by:
L_final = I_final * ω_final
where I_final is the moment of inertia of the system after the child moves to the center and ω_final is the final angular velocity.
Since angular momentum is conserved, we have:
L_initial = L_final
I_initial * ω_initial = I_final * ω_final
The moment of inertia of the system is given by:
I = I_merry-go-round + I_child1 + I_child2 + I_child3
Where I_merry-go-round is the moment of inertia of the merry-go-round and I_child1, I_child2, and I_child3 are the moments of inertia of the children.
The moment of inertia of a point mass is given by:
I = m * r^2
where m is the mass and r is the distance from the rotation axis.
Initially, the child with a mass of 38.8 kg is at the edge of the merry-go-round, so its distance from the center is the radius of the merry-go-round (1.60 m). When the child moves to the center, its distance from the center becomes zero.
The moment of inertia of the merry-go-round is given by:
I_merry-go-round = M * R^2
where M is the mass of the merry-go-round and R is its radius.
The final moment of inertia of the system is:
I_final = I_merry-go-round + I_child1 + I_child2 + I_child3'
where I_child3' is the moment of inertia of the child with a mass of 38.8 kg after moving to the center.
To find the new angular velocity, we can rearrange the equation:
ω_final = (I_initial * ω_initial) / I_final
Now we can plug in the given values and calculate the new angular velocity.
Merry-go-round mass (M) = 142 kg
Merry-go-round radius (R) = 1.60 m
Child 1 mass (m1) = 19.9 kg
Child 2 mass (m2) = 29.0 kg
Child 3 mass (m3) = 38.8 kg
Initial angular velocity (ω_initial) = 19.3 rpm
Initial moment of inertia (I_initial) = I_merry-go-round + I_child1 + I_child2 + I_child3
Final moment of inertia (I_final) = I_merry-go-round + I_child1 + I_child2 + I_child3'
Calculate the moment of inertia for each component:
I_merry-go-round = M * R^2
I_child1 = m1 * r^2
I_child2 = m2 * r^2
I_child3 = m3 * r^2
Finally, calculate the new angular velocity using the formula:
ω_final = (I_initial * ω_initial) / I_final
Plug in the values and calculate ω_final.
The new angular velocity in rpm will be the result.
Learn more about momentum here
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