A freely rotating merry-go-round on a playground has a moment of inertia of 312 kgm2 and a radius of R=2.5 m. It is rotating with ω0=2.8 rad/s when a kid of mass 45 kg is standing on its outer rim. Neglecting friction, what is going to be the angular velocity of the merry-go-round after she walked d=0.7 m toward the center?
I before: 312
I after: 312-45(2.5-.7)^2
conservation of momentum applies
momentumbefore=momentum after
312wo=(312-45(2.5-.7)^2)wf
solve for wf
To find the final angular velocity of the merry-go-round, we can use the principle of conservation of angular momentum. According to this principle, the initial angular momentum of the system will be equal to the final angular momentum of the system.
The initial angular momentum of the system is given by:
L_initial = I_initial * ω_initial
Where:
L_initial is the initial angular momentum
I_initial is the moment of inertia of the merry-go-round
ω_initial is the initial angular velocity of the merry-go-round
In this case, the initial angular momentum is given by:
L_initial = (312 kgm^2) * (2.8 rad/s)
Now, when the kid walks towards the center of the merry-go-round, the moment of inertia changes because the mass distribution changes. The new moment of inertia, I_final, can be calculated using the parallel axis theorem.
The parallel axis theorem states that the moment of inertia about an axis parallel to and a distance d from an axis through the center of mass is given by:
I_final = I_initial + m * d^2
Where:
I_final is the final moment of inertia
I_initial is the initial moment of inertia
m is the mass of the kid
d is the distance the kid walked towards the center
In this case, the final moment of inertia is given by:
I_final = (312 kgm^2) + (45 kg) * (0.7 m)^2
Now, to find the final angular velocity, ω_final, we use the conservation of angular momentum equation:
L_initial = L_final
(I_initial * ω_initial) = (I_final * ω_final)
Substituting the values we have:
(312 kgm^2) * (2.8 rad/s) = ((312 kgm^2) + (45 kg) * (0.7 m)^2) * ω_final
We can solve this equation for ω_final to find the final angular velocity of the merry-go-round.