A certain conservative force is associated with potential energy U(x) = -x^3 +2x

ID: 2096122 • Letter: A

Question

A certain conservative force is associated with potential energy

U(x) = -x^3 +2x^2 + 3x.

A graph of this potential for a small region of the x-axis is shown below.

a) Identify the equilibrium locations and specify for each whether it is stable or unstable.

b) What range of motion is possible (i.e. limits of x position ) if the total energy of the system is 5 J?

c) How much kinetic energy does the system have when it is at x = 0 m if the total energy is 5 J?

d) Compute an expression for the force (as a function of position) associated with this potential energy.

The more potential energy something has, the more unstable it is.

soequilibrium locations are the ones at which U(X) =0 and for all the regions where U(X)>0 it is unstable and for all the regions U(X)<) it is stable

so -2<x<-1 and 0<x<3 it is unstable

-1<x<0 it is stable

Total energy = 5J

2 cases we have one in which total PE=5J and KE=0J

andthe other total KE=5J andPE=0J

so -x^3 +2x^2 + 3x.=0 x= 3 , -1 , -5 * 10^-11

-x^3 +2x^2 + 3x=5 x= 2.377 , -.1.651 , 1.27389

so range of X is -1.651<x<3

At x=0 PE=0 and we have TE=5J but PE+KE=TE so KE=5J

Force = -(dU/dx) = -( -3x^2+4x+3)

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