A20 kg object is acted on by a conservative force given by f = -3x - 5.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 = 6.0 m? (b) if the object has a velocity of 3 m/s in the negative direction of the x axis when it is at x = 8 m, what is its speed when it passes through the origin? what are the answers to (c) (a) and (d) (b) if the potential energy of the system is taken to be -6 j when the object is at x = 0? (a) numberunits(b)numberunits(c)numberu nits(d)numberunits

a) U = 414 J   b)   v = 10.19 m / s

Explanation:

a) force and potential energy are related

      F = - dU / dx

The expression for strength is

      F = - 3x -5x²

Let's clear and integrate

     dU = - F dx

     ∫dU = - ∫ (-3x - 5x²) dx

     U = 3 x² /2 + 5 x³ / 3

Indicate U₀ = 0 for x = 0 this is the lower limit

     U - 0 = 3/2 x² + 5/3 x³

The potential energy is

     U = 3/2 x² + 5/3 x³

a) We evaluate for the point

x = 6.0 m

     U = 3/2 6² + 5/3 6³

     U = 54 + 360

     U = 414 J

b) the force is conservative, therefore the mechanical energy is conserved

Point x = 8 m

     Em₈ = K + U = ½ m v² + 3/2 x² + 5/3 x³

     Em₈ = ½ 20 (-3)² + 3/2 8² + 5/3 8³

     Em₈ = 90 + 96 + 853.33

     Em₈ = 1039.33 J

Point x = 0

    Em₀ = K + U

    Em₀ = ½ 20 v² + 0

    Em₈ = Em₀

    1039.33 = 10 v²

    v = √1039.33 / 10

    v = 10.19 m / s

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