1. The potentialenergy of an object constrained to the x axis is givenby U ( x )

Question

1.      The potentialenergy of an object constrained to the x axis is givenby

U(x) = 3x2 –2x3, where U is in joules andx is in meters. (a) Determine the force

Fx associated with this potential energyfunction. (b) Assuming no other forces

acton the object, at what positions is this object inequilibrium?      

2.      You are designinga game for small children and want to see if the ball’smaximum speed is sufficient to require the use of goggles. In yourgame, a 15.0-g ball is to be shot from a spring gun whose springhas a force constant of 600 N/m. The spring will be compressed 5.00cm when in use. How fast will the ball be moving as it leaves thegun and how high will the ball go if the gun is aimed verticallyupward? What would be your recommendation on the use ofgoggles?

Thank You.

Explanation / Answer

U(x) = 3x2 –2x3, Fx = -dU(x)/dx = -6x + 6x2 = 6x(x - 1) in equilibrium, U(x) is minimum, U' = 0, or F = 0, or x = 0, x =1 U" (x) > 0 U"(x) = 12x - 6 = 6(2x - 1) U"(0) < 0, U"(1) > 0 so answer: x = 1 m

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