What is the required speed to successfully complete the spinning cylinder game?

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The discussion centers on calculating the required rotational speed of a hollow cylinder in a game where an object must fall through a hole as the cylinder spins. The diameter of the cylinder is 0.5 meters, and participants are trying to determine the necessary angular velocity in radians per second to achieve this. One user calculated a speed of 2.45 rad/s, while the textbook suggests a lower value of 1.6 rad/s. Another participant emphasizes the importance of considering the variable acceleration of the falling object, suggesting a relationship between distance traveled and instantaneous acceleration. The conversation encourages further exploration of the physics involved in the problem.
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the hollow cylinder shown in fig 2.2 (see attachment) is free to rotate on a horizontal axis. one hole is cut in the side of the cylinder. the object of the game is to spin the cylinder so fast that the object dropped through the hole when it is in the uppermost position will fall through the same hole when it has rotated to the bottom position. if the diameter of the cylinder is .5 meters how fast must the hole be moving (hint: first calculate the time for the object to fall the appropriate distance, then use that result to determine the number of rad/s.) I get 2..45 rad/s but the book gets 1.6? Try it and see what you get. Thanks

Stevo
 

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Although I haven't see your attachments yet, you have to remember that the acceleration of the ball isn't constant (it temporarily becomes 0 when the cylinder is horizintal and is "g" when vertical).
Try to relate the distance traveled with the acceleration at some instant of time and the corresponding acceleration at that instant.
Hint: Use S = 0.5at^2, a = gsin(theta) where theta = ...

Can you go from here ?

Arun
 
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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