Two identical small insulating balls are suspended by separate 0.17 m threads that are attached to a common point on the ceiling. Each ball has a mass of 7.40 10-4 kg. Initially the balls are uncharged and hang straight down. They are then given identical positive charges and, as a result, spread apart with an angle of 26° between the threads.
What is the charge on each ball?
What is the tension in the threads?
13 degrees from vertical
T = tension
T cos 13 = m g
so T = m g/cos 13
solve for T
T sin 13 = horizontal force
so
T sin 13= k Q^2/(2*.17* sin 13)^2
To determine the charge on each ball, we can use Coulomb's law. Coulomb's law states that the force between two charged objects is given by the equation:
F = k * (q₁ * q₂) / r²,
where F is the force between the objects, k is the electrostatic constant (approximately 9 x 10^9 N m²/C²), q₁ and q₂ are the charges on the two objects, and r is the separation between the objects.
In this case, the two balls are hanging at an angle of 26°. Let's call the angle between each thread and the vertical axis θ. If we draw a diagram, we can see that the two threads form an isosceles triangle, with the angle between them equal to 180° - 2θ.
Using trigonometry, we can determine the separation between the balls, r, as follows:
r = 2 * L * sin(θ),
where L is the length of each thread. In this case, the length of each thread is 0.17 m.
So, r = 2 * 0.17 m * sin(26°) = 0.17 m * 2 * 0.438 = 0.147 m.
Now we can find the force between the balls using Coulomb's law. At equilibrium, the forces between the balls must balance out. Since the balls are identical, the forces exerted on each other by Coulomb's law are equal in magnitude but opposite in direction. Therefore, we have:
F = k * (q₁ * q₂) / r² = k * (q * q) / r²,
where q is the charge on each ball.
The angle between each thread and the vertical axis is 26°, which means that the angle between any thread and the horizontal axis is 90° - 26° = 64°.
Now we can consider the forces acting on each ball. We have the weight (mg) acting vertically downward and the tension in the thread pulling upward at an angle of 64°.
For each ball, in the vertical direction, we have:
T * cos(64°) + T * cos(64°) = 2T * cos(64°) upward,
mg = downward.
For equilibrium, these two forces must be equal in magnitude but opposite in direction. Therefore:
T * cos(64°) = mg.
Using the values provided, T * cos(64°) = (7.40 * 10^-4 kg) * (9.8 m/s²) = 7.252 * 10^-6 N.
The tension, T, in each thread is 7.252 * 10^-6 N.
Now we can solve for the charge on each ball using the equation:
F = k * (q * q) / r²,
and substituting the values for the force and separation we found above:
k * (q * q) / r² = T = 7.252 * 10^-6 N.
Solving this equation for q² gives us:
q² = (T * r²) / k,
q² = (7.252 * 10^-6 N) * (0.147 m)² / (9 * 10^9 N m²/C²),
q² = 1.031 * 10^-14 C²,
Taking the square root of both sides gives us:
q = ±1.015 * 10^-7 C,
Since the balls have identical positive charges, the charge on each ball is q = 1.015 * 10^-7 C.