Calculating Charge and Force Using Coulomb's Law and Trigonometry

In summary: Where did you get that?In summary, the conversation discusses a problem involving two small balls with a mass of 5.0 g each, attached to silk threads that are tied to the same point on the ceiling. When given the same charge Q, the balls hang at an angle of 5.0° to the vertical. The conversation includes equations such as Coulomb's Law and F = mg to determine the magnitude of Q. The correct solution is found to be 9.17 * 10^-7C. However, there is a discussion about torque and using it to solve the problem, rather than determining the tension in the string.
  • #1
Civil_Disobedient
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Homework Statement


Two small balls, each of mass 5.0 g, are attached to silk threads 50 cm long, which are in turn tied to the same point on the ceiling, as shown below. When the balls are given the same charge Q, the threads hang at 5.0° to the vertical, as shown below. What is the magnitude of Q? What are the signs of the two charges?

Homework Equations


Coulomb's Law: F = k*q^2 / r^2

The Attempt at a Solution


Here's the drawing of the problem:
F0BhDQH.png


So far I've found distance r by dividing each half into 2 triangles and using trig:
0.05 * sin5 = r/2
r = 0.0087m

To find Q, I rearranged Coulomb's Law like so:
F = k*q^2 / r^2
F*r^2 = k*q^2
q = sqrt(F*r^2 / k)

To find force, I used F = mg
F = 5g * 10 = 50N per ball, 100N total for two balls

From there, I plugged in what I got
q = sqrt(F*r^2 / k
q = sqrt(100*0.0087^2 / 9*10^9)
q = 9.17 * 10^-7C

So I ended up with 9.17 * 10^-7C, which I feel is wrong (I suspect I may have calculated force wrong) but I don't have an answer or solution key and sadly can't check mines or any other solutions. Any help is appreciated!
 

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  • #2
50 cm is not 0.05 m
5 g is not 50 N, and you don't add the two balls together.
Is F equal to mg? Consider that they are acting in orthogonal directions.
Take one ball, and consider the torque on it.
 
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  • #3
mjc123 said:
50 cm is not 0.05 m
5 g is not 50 N, and you don't add the two balls together.
Is F equal to mg? Consider that they are acting in orthogonal directions.
Take one ball, and consider the torque on it.

I figured out some of the incorrect unit conversions I made. 50 cm = 0.5 m, so r = 0.087.
5g = 0.005 kg --> 0.049 N
But I don't quite get the part about torque. I've tried torque = rFsinθ.
For q = sqrt(F*r^2 / k), shouldn't I use force of tension in the diagonal direction instead of torque?
 
  • #4
Civil_Disobedient said:
I've tried torque = rFsinθ.
I assume you are taking that from a generic equation where those three variables are defined suitably, but they are not the same as the three variables of the same names in the present problem.
@mjc123 means consider the torque balance on a ball+string system about the point where the string is attached to the ceiling. What forces have a torque about that point?
 
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  • #5
Okay, so that force would be the horizontal force of tension. This creates a right triangle with angle 5 degrees and a hypotenuse of 0.049N, which is the tension. Using trig, 0.049N * sin(5) = 0.0043N (horizontal tension). From there, I ended up with 6.01 * 10^-8C for charge.

Tm6Z0v6.png

I used the solutions to help a bit on this last part with tension and horizontal tension. My last question is how did you find the force of tension to be 0.049N? Was it just F = mg = 0.05 kg * 9.8 m/s^2?
 

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  • #6
Civil_Disobedient said:
so that force would be the horizontal force of tension.
No, the tension acts through the point of contact of the string with the ceiling, so has no torque about that point. Which forces on the ball do have a torque about it?
Civil_Disobedient said:
hypotenuse of 0.049N, which is the tension
Nobody said the tension was 0.049N. The point of mjc's method is to avoid having to determine the tension. If you are not comfortable with that method, just write out the usual two equations for horizontal and vertical balances of forces, using an unknown for the tension.
 
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  • #7
Civil_Disobedient said:
My last question is how did you find the force of tension to be 0.049N? Was it just F = mg = 0.05 kg * 9.8 m/s^2?

Nope, that's nothing at all. Nothing is 0.05 kg.
 

Related to Calculating Charge and Force Using Coulomb's Law and Trigonometry

What is Coulomb's Law?

Coulomb's Law is a fundamental law of physics that describes the electrostatic interaction between two charged particles. It states that the force between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.

How does Coulomb's Law relate to two balls?

In the context of two balls, Coulomb's Law can be used to calculate the force of attraction or repulsion between two charged balls. This can help determine the behavior of the balls when they are brought close to each other.

Can Coulomb's Law be applied to all types of charges?

Yes, Coulomb's Law can be applied to all types of charges, whether they are positive or negative. However, it is important to note that the force between two like charges (both positive or both negative) will be repulsive, while the force between two opposite charges (one positive and one negative) will be attractive.

How does the distance between two balls affect the force of attraction or repulsion?

The force between two balls will decrease as the distance between them increases, according to the inverse square law in Coulomb's Law. This means that the force will be stronger when the balls are closer together and weaker when they are further apart.

What are the units of measurement for the force calculated using Coulomb's Law?

The units of force in Coulomb's Law are given in Newtons (N), which is the standard unit of force in the International System of Units (SI). This unit is equivalent to kg*m/s^2.

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