# Harmonic motion and Electrical Fields

1. Jun 13, 2015

### Qwurty2.0

1. The problem statement, all variables and given/known data
Two positive charges +Q are affixed rigidly to the x-axis, one at x = +d and the other at x = -d. A third charge +q of mass m, which is constrained to move only along the x-axis, is displaced from the origin by a small distance s << d and then released from rest. (a) Show that (to a good approximation) +q will execute simple harmonic motion and determine an expression for its oscillation period T. (b) If these three charges are each singly ionized sodium atoms (q = Q = +e) at the equilibrium spacing d = 3 x 10-10 m typical of the atomic spacing in a solid, find T in picoseconds.

2. Relevant equations
Fnet = F1 + F2
F = k * Q1 * Q2 / r2
T = 2 * π * √(m/k) (Simple Harmonic Motion)
Q = q = chargeelectron = 1.602 x 10-19 C

3. The attempt at a solution
I solved part (a) but am getting the wrong answer for part (b).

Answer part a: T = π * √((m * d3) / (k * Q * q))

(b):
Sodium Ion (Na+)
Molar MassNa+ = 22.9898 g/mol = 0.0229898 Kg/mol
Mass per Na+ molecule = 0.0229898 Kg/mol / 6.02 molecules/mol = 3.8189 x 10-26 Kg
dequilibrium = 3 x 10-10 m

T = π * √( (3.8189 x 10-26 Kg) * (3 x 10-10 m)3 / ((8.99 x 109 N⋅m2 / C) * (1.602 x 10-19 C)2))

= π * √( 1.031103 x 10-54 kg⋅m2 / 2.307 x 10-28 N⋅m2 )

= π * 6.685 x 10-14 Kg/N

= 2.1 x 10-13 seconds (?)

= 0.2 picoseconds (the correct answer is 2 picoseconds).

Are my units wrong? I found some mistakes while writing this post but after fixing them I am off by 1x101.

2. Jun 13, 2015

### Staff: Mentor

Your units do not match. Check the Coulombs.
Fixing that, I get the same answer as you got. If formula (a) is right, then the 2 picoseconds could be a mistake in the given solution.

3. Jun 13, 2015

### Qwurty2.0

I'll fix my units. The answer for question (a) is correct according to the book solutions.

EDIT: I misread the answer, it is 0.2 ps. So there is no problem (besides my units).

Last edited: Jun 13, 2015