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Answer :
The angular velocity of the playground merry-go-round after the child gets onto it is 0.73 rad/s.
To find the angular velocity, we can use the conservation of angular momentum. The initial angular momentum of the merry-go-round is equal to the final angular momentum when the child gets on it.
The initial angular momentum of the merry-go-round is given by the equation:
L_initial = I_initial * ω_initial
where L_initial is the initial angular momentum,
I_initial is the initial moment of inertia, and
ω_initial is the initial angular velocity.
The final angular momentum of the merry-go-round with the child is given by
L_final = I_final * ω_final
where L_final is the final angular momentum,
I_final is the final moment of inertia, and
ω_final is the final angular velocity.
Since the child grabs the outer edge of the merry-go-round, the moment of inertia changes. We can calculate the final moment of inertia using the equation:
I_final = I_merry-go-round + I_child
where I_merry-go-round is the moment of inertia of the merry-go-round and
I_child is the moment of inertia of the child.
The moment of inertia of the merry-go-round is given by the equation:
I_merry-go-round = [tex](1/2)[/tex] * m_merry-go-round * [tex]r^2[/tex]
where m_merry-go-round is the mass of the merry-go-round and
r is the radius.[tex]r^2[/tex]
where m_child is the mass of the child.
Now, let's calculate the values:
Given:
m_merry-go-round = 115 kg
r = 1.4 m
m_child = 22.0 kg
ω_initial = 0.58 rev/s
First, calculate the initial moment of inertia of the merry-go-round:
I_initial = (1/2) * m_merry-go-round * [tex]r^2[/tex]
Substituting the given values:
I_initial = [tex](1/2) * 115 kg * (1.4 m)^2 = 142.1 kg·m^2[/tex]
Next, calculate the final moment of inertia:
I_final = I_merry-go-round + I_child
Substituting the given values:
I_final = [tex](1/2) * 115 kg * (1.4 m)^2 + 22.0 kg * (1.4 m)^2 = 256.0 kg·m^2[/tex]
Now, we can use the conservation of angular momentum to find the final angular velocity:
L_initial = L_final
I_initial * ω_initial = I_final * ω_final
Substituting the calculated values:
[tex]142.1 kg·m^2 * 0.58 rev/s = 256.0 kg·m^2[/tex] * ω_final
Solving for ω_final:
ω_final = ([tex]142.1 kg·m^2 * 0.58 rev/s) / (256.0 kg·m^2) = 0.73 rad/s[/tex]
Therefore, the angular velocity of the playground merry-go-round after the child gets onto it is approximately 0.73 rad/s.
To know more about angular momentum, visit
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