Question

1.) A certain region is characterized by a force equation F_x (x)=8x^3-14x-6, where U is in joules and x is in meters.

a.) Find the proper units for the 8, 14, and 6.

b.) Find an equation for the potential energy U(x).

c.) Verify that x= -1, -1/2, 3/2 are each equilibrium points.

d.) Which of the equilibrium points are stable? Which are unstable? Which are neutral? Explain.

Answer 1

a) force has units N

8 has units N/m^3

14 has N/m

6 has N

b) f = -du/dx

u = -8x^4/4+14x^2/2+6x

u(x) = -2x^4+7x^2+6x

c) at x =-1

f(-1) = -8+14-6 = 0

f(-1/2) = -1+7-6 = 0

f(3/2) = 27-21-6 = 0

d) if it is stable then d^2u/dx^2>0

IF IT IS UNSTABLE d^2u/dx^2<0

du/dx = -8x^3+14x+6

d^2u/dx^2 = -24x^2+14

at x = -1

d^2u/dx^2 = -24+14 <0

at x = -1/2

d^2u/dx^2 = -24(1/4)+14 = -6+14 = 8 >0

at x = 3/2

d^2u/dx^2 = -24(9/4)+14 = -54+14 = -40<0

x = -1/2 is stable and x = -1 and 3/2 are unstable equilibrium