Gradiant question involving potenetial energy

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To determine the force vector from the given potential energy function, the negative gradient of the potential energy must be calculated. The potential energy function provided is PE(x,y) = -5.2x + 10.9y + 3.9. The force vector can be derived by taking the partial derivatives of the potential energy with respect to x and y, resulting in the force components. The discussion emphasizes that the force is indeed the negative of the gradient of the potential energy, which simplifies the calculation. Understanding the gradient vector is crucial for solving this problem effectively.
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Homework Statement


Given the following 2D-position Vector and potential energy function determine the force vector for the object.


Homework Equations


vector r = <45.0, -20.4> m
PE(x,y) = -5.2x + 10.9y + 3.9 (j)


The Attempt at a Solution


I've attempted to make a solution but i can can't come up with an equation for solving this, the only guideline i have is an equation on a 1D problem and it has epsilon involved. I am not really sure what are the steps used to solve this equation.


any and all help would be appreciated.
Thanks!
 
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Isn't the force just the negative of the gradient of the potential energy? Look up the definition of the gradient vector. I think in this case it's doesn't even depend on position.
 
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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