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## Homework Statement

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.

## Homework Equations

F

_{net}= F

_{1}+ F

_{2}

F = k * Q

_{1}* Q

_{2}/ r

^{2}

T = 2 * π * √(m/k) (Simple Harmonic Motion)

Q = q = charge

_{electron}= 1.602 x 10

^{-19}C

## The Attempt at a Solution

I solved part (a) but am getting the wrong answer for part (b).

Answer part a: T = π * √((m * d

^{3}) / (k * Q * q))

**(b):**Sodium Ion (Na+)

Molar Mass

_{Na+}= 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

d

_{equilibrium}= 3 x 10

^{-10}m

T = π * √( (3.8189 x 10

^{-26}Kg) * (3 x 10

^{-10}m)

^{3}/ ((8.99 x 10

^{9}N⋅m

^{2}/ C) * (1.602 x 10

^{-19}C)

^{2}))

= π * √( 1.031103 x 10

^{-54}kg⋅m

^{2 }/ 2.307 x 10

^{-28}N⋅m

^{2})

= 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 1x10

^{1}.