A 44.8- child is standing on the outer edge of a merry-go-round that has moment of inertia 981 and radius 2.40 . The entire system is initially rotating at 0.180 . Find the angular velocity if the child moves to a final position 1.10 from the center of the merry-go-round.
To find the final angular velocity when the child moves to a new position, we can use the conservation of angular momentum. The initial angular momentum of the system is equal to the final angular momentum.
The formula for angular momentum is given by:
L = Iω
where:
L is the angular momentum,
I is the moment of inertia,
and ω is the angular velocity.
Initially, the total angular momentum of the system is:
L_initial = I_initial * ω_initial
When the child moves to the new position, the moment of inertia changes to a new value, I_final, and we need to find the final angular velocity, ω_final.
The angular momentum at the final position is:
L_final = I_final * ω_final
Since angular momentum is conserved, we have:
L_initial = L_final
Substituting the expressions for the initial and final angular momenta:
I_initial * ω_initial = I_final * ω_final
Now we can solve for ω_final.
The moment of inertia, I, of a point mass rotating around an axis a distance r from the center is given by:
I = m * r^2
where m is the mass.
Given that the child's mass is 44.8 kg, the initial moment of inertia (I_initial) can be calculated as:
I_initial = m_child * r_initial^2
Similarly, the final moment of inertia (I_final) can be calculated using the new distance from the center (r_final):
I_final = m_child * r_final^2
Substituting the expressions for the initial and final moment of inertias:
m_child * r_initial^2 * ω_initial = m_child * r_final^2 * ω_final
Simplifying the equation:
r_initial^2 * ω_initial = r_final^2 * ω_final
Now we can solve for ω_final:
ω_final = (r_initial^2 * ω_initial) / r_final^2
Substituting the given values:
ω_final = (2.40^2 * 0.180) / 1.10^2
Calculating the numerator:
2.40^2 * 0.180 = 1.0368
Calculating the denominator:
1.10^2 = 1.21
Now we can find ω_final:
ω_final = 1.0368 / 1.21
Calculating ω_final:
ω_final ≈ 0.856 rad/s
Therefore, the final angular velocity when the child moves to a new position is approximately 0.856 rad/s.
It can't be answered very well without units given.
conservation of momentum
Ii*wi=(If)*Wf
where If=Ii+masskid*r^2
initiall, the moment of inertia for the child is zero, because r=0
watch units on angular rotation initial...