If F = (2xy)i - (3x2)j Newtons and it makes a 4 kg object move along a curve y =

Question

If F = (2xy)i - (3x2)j Newtons and it makes a 4 kg object move along a curve y = 2x2 from (0,0) to (2, 8), what is the work done by this force? If the initial velocity is 15 m/s, what is the final velocity? Assume gravity is also doing work on the object? If the potential energy is given by U(x,y) = 3x2 + 5y - 2y2 + 8 Joules, find the associated conservative force and express it in unit notation. If U(x) = 4x4 - 2x2 - x + 7 Joules, find the points of Stable & Unstable Equilibrium and the potential energy of the particle at those points. What will happen to an object if it starts out at x = 0 or 2? Assume that its mass is 2 kg.

Explanation / Answer

a)

Work done by F = 10 J

work done by gravitational = -m g y = -4 * 9.8 * 8 = -313.6 J

net work = -313.6 + 10 = -303.6 J

net work = (1/2) m vf2 - (1/2) m vi2

-303.6 = 0.5* 4 * vf2 - 0.5 * 4 * (15*15)

-303.6 = 2 * vf2 - 450

>>> vf = 8.56 m/s

b)

Fx = -dU/dx = -6 x

Fy = -dU/dy = -(5 - 4 y) = 4 y - 5

>>> F = (-6 x) i + (4 y - 5) j

C)

dU/dx = 0

>>> 12 x2 - 4 x - 1 = 0

>>> x1 = 0.5 >>>> stable equilibrium >>>> U(0.5) = 6.5 J

>>> x2 = -0.167 >>>>> unstable equilibrium >>>> U(-0.167) = 7.09 J

If the object start out at x=0 or x=2, it will oscillate about x=0 point

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