What is the Tension Force and Charge of Two Separated Balls in Equilibrium?

[fawk3s]

Homework Statement

Two balls, each with a mass of 500 mg, are attached to the same point in the ceiling by strings with the length of 40 cm. The balls have been given equal and opposite charges, which caused them to separate in a way that the strings formed a 60-degree angle. Find the charges and the tension force in the string. Make a sketch.

The answers ought to be
q=~0,2*10^-6 C
F_t=~0,0044 N

Homework Equations

F_g=mg
F_C=kq₁q₂/d²

The Attempt at a Solution

http://desmond.imageshack.us/Himg513/scaled.php?server=513&filename=fdafdafa.png&res=medium

An equilateral triangle is formed with the strings, so the distance between the balls is also
d=40cm=0,4m

Gravitational force on a ball
F_g=mg=~5*10^-3 N (g=9,8 m/s²)

In order for equillibrium to be reached, the component vector of gravity, which is at a right angle with the string, must be equal to the component vector of Coulomb's force, which is also at a 90^o with the string.
That component vector of gravity is
F_g1=F_g*cos60^o=2,5*10^-3

So if F_C=kq₁q₂/d², then

F_g1=F_C*cos30^o, from which we get that
F_C=5*sqrt3*10^-3/3 N
and that
q₁=q₂=~0,2*10^-6 C

So far it seemed to me that everything went correctly, but I don't get the tension force to be 0,0044 N. Using sines and cosines on F_g and F_C, or Pythagoran theorem for that matter, I got

F_t=F_t1+F_t2
F_t1=cos30^o*F_g=5*sqrt3*10^-3/2
F_t2=cos60^o*F_C=2,5*sqrt3*10^-3/3
F_t=~0,0058 N

Now I've seen many mistakes in answers in this textbook, but I am not so sure it's the textbook which has a fault in it this time. So please tell me, did I go wrong and where, or is the answer in the textbook once again wrong?

(I didnt mark up all the angles, but they should be pretty easy to figure out. If you don't understand some of my calculations or think they are wrong, be sure to shout out or ask!)

Thanks in advance

 

[ehild]

You made calculation errors. m=500 mg=0.5 kg. g=9.8. Force of gravity is 4.9 N.
Do not round off the numerical results too early.

ehild

 

[fawk3s]

500 milligrams is equal to 0.0005 kg. So I can't have made a mistake there.
The reason I rounded them up so early (I'm usually pretty precise) is because the answer given in the back is always calculated that way. I was just trying to get a close answer.

Thanks in advance

 

[fawk3s]

I would really appreciate if someone would solve the problem independently to see what answers they get. If they match with the book's, I must have made a theoretical error somewhere (I don't think its a calculation error since I've checked the numbers twice). If they match with my answers, it must be the book what contains the error.
I hope it's not too much to ask.

What confuses me though, is that the charge I get matches with the book's answer.

Thanks in advance

 

[ehild]

500 milligrams is equal to 0.0005 kg. So I can't have made a mistake there.


The reason I rounded them up so early (I'm usually pretty precise) is because the answer given in the back is always calculated that way. I was just trying to get a close answer.

Thanks in advance

Sorry, I made the mistake!Your calculation is correct (althought I do not like that early rounding)

It is an other way for the calculation using the right triangle made of the forces of gravity, Coulomb force and tension.


ehild

 

[fawk3s]

It is an other way for the calculation using the right triangle made of the forces of gravity, Coulomb force and tension.


ehild

What you mean is I've made a mistake on calculating the magnitude of the tension? Could you specify please? Because I seem to miss it.

fawk3s

 

[ehild]

What you mean is I've made a mistake on calculating the magnitude of the tension? Could you specify please? Because I seem to miss it.

fawk3s

No, your calculation is right. But there is an easier way to get it:

T=mg/cos(30°), (and tan(30°)=Fc/mg; )

ehild

 

[fawk3s]

Oh, alrighty. But does this mean you got the same answer, and I have to assume the textbook is wrong? Sorry, just have to ask again for confirmation.

 

[ehild]

I have got the same answer:Tension=4.9x10^-3/cos(30°)=5.7x10^-3 N.

ehild

 

[fawk3s]

Thank you for your help ehild, I really appreciate it.