can you show me step by step solution for the problem above A particle experienc

Question

can you show me step by step solution for the problem above

A particle experiences a potential energy function U{x) = xA - 8æ2 , where U is in joules and x is in meters. Determine the conservative force function producing the potential energy function. Fx = -4x3 + 16x Graph one above the other: The potential energy function vs. position, [U vs. x] The force in the .x-direction vs. position, [Fx vs. x] Determine the x-position of every equilibrium point (write down the values), identify them on your graphs, and label each as being either stable or unstable, x = 2 m (stable), x = 0 m (unstable) If the particle has a mechanical energy of 9 J, where could it be located? Give a range of x-values. Indicate the particle's mechanical energy on your energy diagram with a horizontal line. -3 m le x le 3 m

Explanation / Answer

a) Fx = -dU/dx

Fx = - d(x^4-8x^2) /dx

Fx = -(4x^3 -16x )

Fx = -4x^3+16x

c) equilibrium points ar ethose where energy is minimum

dU/dx =0

4x^3-16x=0

x(x^2-4) = 0

x=0,2,-2

d^2U/dx^2 = 12x^2 -16

for x=0 only d^U/dx^2 <0

hence the function has minimum at x=0

d) particle has energy =9J

x^4-8x^2=9

x^4-8x^2-9=0

(x^2-9)(x^2+1) = 0

x^2+1 can't be 0

so x^2-9 = 0

x=+-3

so the particle can be located anywhere between -3<= x<=3

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