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Answer :
The angular velocity of the merry-go-round after the child gets on is 1.837 rad/s (in SI units).
To find the angular velocity of the merry-go-round after a 19 kg child gets on, we need to apply the principle of conservation of angular momentum.
The initial angular momentum of the merry-go-round is given by the formula:
[tex]L_{initial} = I_{initial} * w_{initial}[/tex]
Where [tex]L_{initial}[/tex] is the initial angular momentum, [tex]I_{initial}[/tex] is the initial moment of inertia, and [tex]w_{initial}[/tex] is the initial angular velocity.
The moment of inertia of a disk rotating about its axis is given by the formula:
I = (1/2) * m * r²
Where m is the mass of the disk and r is the radius.
Substituting the values, we have:
[tex]I_{initial}[/tex] = (1/2) * 112 kg * (1.91 m)² = 206.0276 kg*m²
The final angular momentum of the system (merry-go-round + child) is given by:
[tex]L_{initial} = I_{initial} * w_{initial}[/tex]
Since the child is on the merry-go-round, we need to consider the moment of inertia of both the disk and the child. The moment of inertia of the child can be considered negligible compared to the disk, so we can assume it to be zero.
Therefore, [tex]I_{final} = I_{initial} + I_{child} = I_{initial}[/tex]
Substituting the values, we have:
I_final = 206.0276 kg*m²
Now, we can rearrange the formula for angular momentum to solve for the final angular velocity:
[tex]w_{final} = L_{final} / I_{final}[/tex]
Substituting the given values, we have:
[tex]w_{final} = L_{final} / I_{final}[/tex]= (112 kg * 3.4 rad/s) / 206.0276 kg*m²
Simplifying the expression, we get:
[tex]w_{final }[/tex]= 1.837 rad/s
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