2. A particle of mass m is subjected to a force given by F(x) = a-2bx^2. (a) Fin
ID: 2287054 • Letter: 2
Question
2. A particle of mass m is subjected to a force given by F(x) = a-2bx^2. (a) Find V(x) (b) Make sketches of F(x) and V(x) [Hint For a given function, identify the values of x that make the function zero (i.e., the roots), identify where the minima and maxima are using condition: dV(x)/dx = 0, and then evaluate the minimum and maximum values (i.e., Vmin and/or Vmax). Seminar Problem #3 should also provide some hints in regard to how this kind of problem may be tackled]. (e) Discuss the motion of the particle for different values of energy and point out any regions in which motion is: (i) bounded, unbounded, or forbidden, and (ii) places where the particle is at equilibrium (stable and unstable) .Explanation / Answer
F=a-bx^2
Dv/dx= a-bx^2
V= ?a-bx^2 dx
V=ax-(bx3/3)
To find maxima and minima, let us put v=0
ax-(bx3/3)=0
ax= bx3/3
a=bx2/3
x=(3a/b)1/2
let us put this in the given equation of F(x).
F=a-bx^2
F=a-b(3a/b)
F=a-3a
F=-2a
Thus,
Vmin= ax-(bx3/3)= a(3a/b)1/2-(b(3a/b)3/2/3)
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