Two positive point charges +q are on the y axis at y = +a and y = a. A bead of m
ID: 2004044 • Letter: T
Question
Two positive point charges +q are on the y axis at y = +a and y = a. A bead of mass m and charge q slides without friction along a taut thread that runs along the x axis. Let x be the position of the bead.
(a) Show that for x << a, the bead experiences a linear restoring force (a force that is proportional to x and directed toward the equilibrium position at x = 0) and therefore undergoes simple harmonic motion. (Do this on paper. Your instructor may ask you to turn in this work.) DON'T WORRY ABOUT THIS PART!
(b) Find the period of the motion. (Start with the period of a simple harmonic oscillator. Use the following as necessary: m, a, k, and q.)
T = ______________
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need by 10pm EST. THANKS!!!
Explanation / Answer
You say “dont worry about this part” for part A, but you have to do part A to get part B.
The bead will experience a force of
F = k q2 / r2 from each of the two charges on the y axis, where r2 = a2 + x2.
Since x << a, we can assume that r is approximately equal to a.
The component of the forces in the y direction will cancel; the components in the x direction will add. So the total force is given by
Ftot = 2 Fx = 2 ( k q2 / a2 ) cosq = 2 ( k q2 / a2 ) (x / a) = A x
where A = 2 k q2 / a3 and since k = 1/4peo A = q2 / 2peoa3
Technically, we can say that Ftot = - Ax since the force is in the positive direction when x is negative and in the neg direction when x is positive. This is the result for part A.
Now for part B... we can use Newton’s second law: F = ma so -Ax = ma or
a + (A/m) x = 0
This is the same result for the simple spring & mass system. Using the same derivation, we get:
w2 = A/m or w = (A/m)1/2
We also know that T = 2p / w so the final result is
T = 2p (m/A)1/2 = 2p (2peo m / q2)1/2 =
= 2p (2peo m)1/2 / q
NOTE: for all the above equations, my "p" is for pi and my "eo " is epsilon naught. I did my answer in MS Word and tried to copy it over, but the greek letters didnt copy properly.