things didn\'t go so well with Bobo, so after a brief stint in intensive care yo

Question

things didn't go so well with Bobo, so after a brief stint in intensive care you have been reassigned to the penguin park. bitter and confused, you decide to indulge in your newfound interest in curling by using your new tuxedoed friends as stones. as such, you slide a penguin across the level, frictionless surface of their play area. the coordinates of the penguin are x and you. he is acted on by a conservative force described by the potential energy function
U (x,y)=1/2k (x^2y+xy^2)
Derive a vector expression for the force acting on the penguin, and find an expression for magnitude of the force as a function of position.

Explanation / Answer

U = (k / 2) (x^2 y + x y^2)

Fx = - dU/dx (partial differnetiation)

Fx = - (k/2) (2xy + y^2)

Fy = - dU/dy = - (k/2) (x^2 + 2x y )

Vector form of force

F = Fxi + Fy j

F = (- k / 2) [ (2xy + y^2) i + (2x y + x^2) j ]

magnitude = sqrt[ Fx^2 + Fy^2]

= (k / 2) sqrt[ x^4 + 4 x^2 y^2 + 4 x^3 y + y^4 + 4 x^2y^2 + 4 x y^3]

= (k / 2) sqrt[ x^4 + y^4 + 8 x^2 y^2 + 4 x^3 y + 4 x y^3 ]

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