What is the equilibrium position of a ball attached to two springs?

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The equilibrium position of a ball attached to two identical springs is determined by balancing the forces exerted by the springs against the weight of the ball. The springs exert upward forces that depend on their extension or compression, which can be analyzed using Newton's second law. At the equilibrium point, the net force on the ball must equal zero, leading to the equation provided in the book. The solution involves calculating the distance X below the midpoint where the forces from the springs equal the gravitational force acting on the ball. Understanding the mechanics of the system is crucial for deriving the correct equilibrium position.
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A small ball of mass M is attached t two identical springs of constant k, which are attached to the floor and the roof. The springs have unstretched lengh L/2. At what position will the ball remain at rest?

The back of the book gives the answer.
\frac{1}{2}L(1+\frac {Mg}{kL})

But the text doesn't do a good job explaining how to arrive at that answer. I've stared at it for 20 minutes. Any thoughts?
 
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When the ball is exactly in the middle (at L/2) what forces do the springs exert on it? And at a point a distance X beneath that middle point? What does X have to be so that the net spring force balances the ball's weight?
 
Okay draw the forces...Apply Newton's second law.And then use the fact that the body is equilibrium.

The answer in the book gives the body-ceiling distance...


Daniel.
 
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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