A 20 kg object is acted on by a conservative force given by F = -3.0 x - 5.0 x2,

Question

A 20 kg object is acted on by a conservative force given by F = -3.0 x - 5.0 x2, with F in newtons and x in meters. Take the potential energy associated with the force to be zero when the object is at x = 0.
(a) What is the potential energy of the system associated with the force when the object is at x = 3.0 m?
___ J
(b) If the object has a velocity of 2.0 m/s in the negative direction of the x axis when it is at x = 6.0 m, what is its speed when it passes through the origin?
___ m/s
(c) What are the answers to (a) and (b) if the potential energy of the system is taken to be -6.0 J when the object is at x = 0?
(a) ___ J
(b) ___ m/s

Explanation / Answer

potential energy U(x) = - F dx = 1.5x2 + 5x3/3 + U(0)
a) if U(0) = 0, U(3.0) = 1.5x2 + 5x3/3 = 58.5 J
b) at x = 5.0 m, energy E = mv2/2 + U(5.0) = 405.8 J
at x = 0, energy = mv'2/2 + U(0) = E = 405.8 J,
v' = (2E/m) = [v2 + 2U(5.0)/m] =6.37 m/s
c) if U(0) = -8.0 J,
U(x) = 1.5x2 + 5x3/3 - 8.0
U(2.0) = 11.3 J
v' is unchanged, v' = 6.37 m/s

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