String and pulley physics help

AI Thread Summary
The discussion revolves around a physics problem involving a fixed pulley, a mass M, and a movable ring with mass m. The objective is to determine how far the ring descends before reaching equilibrium and to calculate the maximum velocities of both masses at that point. Key equations include the balance of forces involving tension and gravitational force, as well as energy conservation principles. The initial approach suggests creating a free body diagram to analyze the forces acting on the system. Understanding these dynamics is crucial for solving the problem effectively.
Princesscindy
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Homework Statement



there is a fixed pulley and a mass M is hanging from a string which goes through the pulley and the string it attached to a ring which is free to move up and down a vertical pole...mass of this ring is m. initially the string is horizontal. now we have to find the distance by which 'm' comes down before coming to rest for the first time.. we have to find the equilibrium position of the system and the max velocities of m and M at that instant...

Homework Equations



TcosØ=mg
Mgh=(mv^2)/2
(loss of PE in M=gain of KE of m)

The Attempt at a Solution



the idea i have is that horizontal tension in the string cannot balance weight of m so it moves down till tension components balance Mg...please help me out with this...
 
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First, create a free body diagram of the system.
 
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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