Simple Pulley System: Accelerations Equation Explained

AI Thread Summary
The discussion focuses on understanding the acceleration relationship in a simple pulley system, specifically the equation a1 = 2*a2. This relationship arises from the geometric constraints of the system, where the movement of one mass (m2) affects the movement of another (m1) due to the length of the rope. By double-differentiating the position equations x1 = 2x2, the acceleration relationship can be derived. The explanation emphasizes that this is a straightforward application of geometry in the context of the pulley system. Understanding this relationship is crucial for solving related physics problems effectively.
asi123
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Homework Statement



Hey guys.
I have the following pulley system (in the pic).
I wrote down all the equations.
The thing I don't get is the accelerations equation (a1 = 2*a2). I understand that I can get to this equation somehow by the length of the rope, but how?
Thanks.

Homework Equations





The Attempt at a Solution

 

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asi123 said:
The thing I don't get is the accelerations equation (a1 = 2*a2). I understand that I can get to this equation somehow by the length of the rope, but how?

Hi asi123! :smile:

You get it by double-differentiating x1 = 2x2 …

and that is simple geometry … when m2 moves a distance x2, m1 moves a distance x1 = 2x2. :smile:
 
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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