In-class exercise 2

The potential energy (in J) due to an external field as a function of a particle’s coordinate is given by

Remove["Global`*"]

uf[x_]:=2Exp[((-(((x-3))^2))/2)]+1.5 Exp[((-(((x+4))^2))/2)]+Exp[((-(((x+0.1))^2))/2)]+(10^(-5)) (x^6)

Plot[uf[x],{x,-8,8}]

a) What is the force due to this external field on a particle at position x? What is the value of the force at x=0.3? [1 mark]

Fuf[x_]=

b) Two particles are both affected by this field, and also are connected by a spring with potential energy

us[r_]:= 3 (((r-4))^2)

Plot[us[r],{r,0,10}]

What is the spring force on particle 2 if the particle coordinates are given by x1[t] and x2[t] and particle 2 is always to the right of particle 1?  (Inspect your answer qualitatively to make sure that the direction of the force is correct.)  [2 marks]

r12:=

c) Solve for the trajectories of the two particles if their masses are m1=1kg and m2=2kg, initial positions are x1=0 and x2=2m and they start from rest.  Plot their trajectories (position versus time) for the first 5s.  [5 marks]