Coulomb’s law & Electrons: Solve for L

Question

Two small metallic spheres, each of mass m = 0.200 g, are suspended pendulums by light strings of length (L) as shown in figure below. The spheres are given the same electric charge of 7.2 nC, and they come to equilibrium when each string is at an angle of θ = 5.00 with the vertical. How long are the strings?

Given:

1. m=0.200m --> .0002 kg
2. g=9.81
3. q=7.2×10⁻⁹
4. θ=5.00∘
5. k=8.988×10⁹

I have tried and cannot figure out how to solve this problem. I want to put my work below to show you where I am because I feel like I'm so close but I keep getting a weird answer that doesn't make sense. Here is my work:

Tcosθ=mg

Tsinθ=Fe​

Fe​=kq^2​/r^2

r=2Lsinθ

Fe ​= kq²/2Lsinθ)² ​= 4kq²/4L2sinθ​²

Tsinθ=kq²/4L2sinθ​²

(mg/cos​θ)sinθ = kq²/4L2sinθ​²

**and basically I isolated L and got this:

L=kq²/4mgsinθ²cosθ

I would appreciate it if anyone could help me solve it. I plugged in the numbers given and got .45 m and then my second attempt at it I got .299 m.

Answer 1

Hi, Paige!

Your setup is correct: the physics and trig are good, and your first 4 equations are correct. It looks like you've just got a couple of algebra errors (some of which may be a result of typing equations into this website :).

In the future, one way to spot this immediately is to do a dimensional analysis.

Your equation L=kq2/4mgsinθ2cosθ will give you (Nm²/C²)C²/(kg m/s²) = Nm²/N = m²

k q² m g

This tells you right away that your L should be L², and you can go look for where you dropped a square.

But back to the actual question...

Your Fe equation is Fe=kq²/(2Lsinθ)²

which simplifies to Fe=kq²/(4L²sin²θ) [which is not what you got]

substituting T=mg/cosθ was very nice. Now your Tsinθ equation becomes

(mg/cosθ)sinθ = kq²/(4L²sin²θ)

Now solve for L², take the square root, and you should get the right answer. Pro tip: put the units into your equation along with the numbers. So q=7.2×10⁻⁹ C, etc. The units should all cancel out, and you should end up with meters.

Answer 2

I followed your calculations, and it seems that when you substituted 2Lsin(theta) in for r in the expression for electrostatic force, you didn't square L in the denominator.

r^2=(2Lsin(theta))^2=4L^2sin^2(theta).

See if this changes the expression for L you get in the end.