The oscillator consists of a molecule that has three carbons at the vertices of

Question

The oscillator consists of a molecule that has three carbons at the vertices of triangle. Two of the carbons form the base of the triangle and each have an excess of one electron. The third carbon with a deficit of 2 electrons is located along a line that is perpendicular to the line connecting the other two carbons and is the equidistant from them. You need to find the acceleration of the positively charged carbon atom as a function of the distance between the negatively charged carbons and the distance of the positively charged carbon from the line connecting the two negatively charged carbons as well as the other properties of the carbon ions

Explanation / Answer

let the distacne between the two excess 1 electron each at the base of the triangle be d
let the third carbon atom be on the perpendicular bisector of this side at distance x form this line

then distance of other carbon atoms form this one, r = sqroot(d^2/4 + x^2)

so because this third carbon atom ( having charge +2e, where e = 1.6*10^-19 C) is on the perpendicular bisector of theline joining other two atoms which have the same charge, the x component of the force on third carbon atom due to the other two carbon atoms will cancel out and only the y component will add up

so, net force on the third carbon atom, F = 2k*e*(2e)*x/(d^2/4 + x^2)^3/2 [ taking the cos component of the force along y direction]
F = 2k*e*(2e)*x/(d^2/4 + x^2)^3/2
F = 4k*e^2*x/(d^2/4 + x^2)^3/2

now from newtons second law
F = ma ( where m is mass of carbon atom and a is its acceleration)
so m*a = F = 2k*e*(2e)*x/(d^2/4 + x^2)^3/2
a = 2k*e*(2e)*x/m*(d^2/4 + x^2)^3/2

Now if x < < D
then x^2 < < D^2

so a = 2k*e*(2e)*x/m*(d^2/4)^3/2
a = 32k*e^2*x/m*d^3
as k, e, m and d are constant, so acceleration of the particle is directly proportional to its distance from the centreline joining other two atoms and this force is always poiting towards the equilibrium point, hence this force gives Simple Harmonic motion of the carbon atom

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