A 1.80 m radius playground merry-go-round has a mass of 120 kg and is rotating with an angular velocity of 0.300 rev/s. What is its angular velocity after a 19.0 kg child gets onto it by grabbing its outer edge? The child is initially at rest.
To find the angular velocity of the merry-go-round after the child gets onto it, we can use the principle of conservation of angular momentum.
The angular momentum of an object is given by the product of its moment of inertia and its angular velocity. The moment of inertia for a solid disk (merry-go-round) rotating about its center can be found using the formula:
I = (1/2) * m * r^2
Where I is the moment of inertia, m is the mass of the object, and r is the radius of the object.
For the initial system (merry-go-round), the total angular momentum is given by:
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.
For the final system (merry-go-round + child), the total angular momentum 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 there is no external torque acting on the system (no net external force that would cause the system to rotate faster or slower), the total angular momentum of the system is conserved:
L_initial = L_final
Now let's calculate the initial and final moment of inertia and use them to find the final angular velocity.
Given:
Radius of the merry-go-round (r) = 1.80 m
Mass of the merry-go-round (m_merry-go-round) = 120 kg
Mass of the child (m_child) = 19.0 kg
Initial angular velocity of the merry-go-round (ω_initial) = 0.300 rev/s
First, calculate the initial moment of inertia (I_initial) of the merry-go-round using the formula mentioned earlier:
I_initial = (1/2) * m_merry-go-round * r^2
I_initial = (1/2) * 120 kg * (1.80 m)^2
Next, calculate the final moment of inertia (I_final) of the system (merry-go-round + child). Since the child is grabbing the outer edge of the merry-go-round, the moment of inertia will change. The moment of inertia of a point mass rotating about a fixed axis is given by:
I_point mass = m * r^2
I_final = I_merry-go-round + I_child
I_final = (1/2) * m_merry-go-round * r^2 + m_child * r^2
I_final = (1/2) * 120 kg * (1.80 m)^2 + 19.0 kg * (1.80 m)^2
Now that we have the initial and final moment of inertia, we can solve for the final angular velocity by rearranging the conservation of angular momentum equation:
L_initial = L_final
I_initial * ω_initial = I_final * ω_final
Solving for ω_final:
ω_final = (I_initial * ω_initial) / I_final
Substituting the values:
ω_final = (I_initial * ω_initial) / I_final
Finally, substitute the calculated values for I_initial, I_final, and ω_initial into the equation to find ω_final.
Please note that the unit of the angular velocity will be in rad/s.