What is the Angular Speed of a Toy Train's Wheel in a Rotating Track System?

AI Thread Summary
The discussion focuses on calculating the angular speed of a toy train's wheel in a rotating track system. The toy train track is mounted on a wheel with a mass of 3.59 kg and a radius of 1.79 m, while the train itself weighs 0.203 kg and moves at a steady speed of 0.551 m/s. The user initially sought clarification on which momentums to conserve for the calculations, specifically questioning the conservation of momentum between the train and the entire train-track system. After some calculations involving the moment of inertia and angular speed, the user ultimately resolved their query. The thread highlights the application of conservation principles in rotational dynamics.
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A toy train track is mounted on a large wheel that is free to turn with negligible friction about a vertical axis as shown in the figure above. The mass of the wheel plus track is 3.59 kg and the radius is 1.79 m. Ignore the mass of the spokes and hub. A toy train of mass 0.203 kg is placed on the track and, with the system initially at rest, the electric power is turned on. The train reaches a steady speed of 0.551 m/s with respect to the track. What is the angular speed of the wheel?

I found the momentum of the train and used conservation of momentum and applied it to the track. I have a question though. What two momentums should I be conserving? should it be the ( masses of the entire train-track system) * w =(mass of the train only) *(given velocity (converted to w))?
 
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w(t) = .30782 = (velocity/circumference) * 2*pi
I(t) = .65043 = m(t) * radius^2
I(system) = 12.15315 = (m(w)+m(t)) * (radius)^2
w(system) = w(t)*I(t) / I(system)
= .0165

not right! please help.
 
n/m i got it
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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