23. Two identical balls each have an excess charge of 5.0 x 10-7 C, and they are attached to the ceiling by light string 0.80 m long as shown in the figure below. If the angle that the threads make with respect to each other is 300, determine the force experienced by each ball. Assume that the balls are small enough that they can be treated as particles.
To determine the force experienced by each ball, we can use Coulomb's law. However, we first need to calculate the distance between the charged balls.
Since the balls are attached to the ceiling by light strings 0.80 m long, we can consider them in the form of an isosceles triangle with the top angle of 300. The distance between the balls (d) can be calculated using the triangle's properties.
Let's start by finding the vertical distance (h) between the balls using trigonometry:
h = 0.80 m × sin(30°)
h = 0.80 m × 0.5
h = 0.40 m
Now, let's find the horizontal distance (x) between the balls using trigonometry:
x = 0.80 m × cos(30°)
x = 0.80 m × √3/2
x ≈ 0.693 m
The distance between the balls (d) can be calculated using the Pythagorean theorem:
d = sqrt(x² + h²)
d = sqrt(0.693² + 0.40²)
d ≈ 0.802 m
Now that we have the distance between the balls, we can use Coulomb's law to find the force experienced by each ball:
F = (k * q₁ * q₂) / d²
where:
F is the force experienced by each ball,
k is the electrostatic constant (8.99 x 10^9 N·m²/C²),
q₁ and q₂ are the excess charges on each ball (5.0 x 10^-7 C),
and d is the distance between the balls (0.802 m).
Plugging in the values:
F = (8.99 x 10^9 N·m²/C²) * (5.0 x 10^-7 C)² / (0.802 m)²
Calculating this expression:
F ≈ 3.12 x 10^-3 N
Therefore, each ball experiences a force of approximately 3.12 x 10^-3 N.