Two volleyballs, each of mass 0.300 kg, are charged by an electrostatic generator. each is attached to an identical string and suspended from the same point, as shown in Fiq.2. They repel each other and hang with separation 50.0 cm. The length of the string from the point of support to the center of a ball is 250 cm. Determine the charge on each ball.
Fiqure 2
is trinagle with equal sides 2.5 m and base is 0.5m
F = k q^2/(.5)^2 = m g sin theta
where theta is angle from vertical
tan theta = theta in radians = sin theta for small angle = .25 m (half of distance between)/2.5 = 0.1
so
q^2 = .5^2 (.3)(9.81)(0.1)/(9*10^9)
To determine the charge on each ball, we can make use of Coulomb's Law. Coulomb's Law states that the electrostatic force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Let's derive the equation for the force between the two volleyball balls:
1. The first step is to find the distance between the centers of the balls. We subtract half of the base (0.5m) from the length of the string (2.5m) to get the vertical distance between the centers: sqrt(2.5^2 - 0.5^2) = sqrt(6.75) ≈ 2.598m.
2. The force between the two charged balls is given by Coulomb's Law:
F = k * (q1 * q2) / (r^2),
where F is the electrostatic force, q1 and q2 are the charges on the balls, r is the distance between their centers, and k is Coulomb's constant (8.99 x 10^9 N m^2/C^2).
3. We know that the two balls repel each other, so the electrostatic force acts in the opposite direction to the gravitational force. In this case, the vertical component of the tension force in the string counteracts the weight of the ball. Therefore, we can say the electrostatic force equals the weight of the ball.
Now, let's calculate the charge on each ball:
4. Weight of a ball is given by:
W = m * g,
where W is the weight, m is the mass (0.300 kg), and g is the acceleration due to gravity (9.8 m/s^2).
Therefore, each ball has a weight of: W = 0.300 kg * 9.8 m/s^2 = 2.94 N.
5. The electrostatic force equals the weight of the ball. Therefore, we can write:
F = k * (q1 * q2) / (r^2) = 2.94 N.
6. Now, substitute the values into the equation:
2.94 N = (8.99 x 10^9 N m^2/C^2) * (q1 * q2) / (2.598 m)^2.
7. Since the balls have the same charge (q1 = q2 = q), the equation becomes:
2.94 N = (8.99 x 10^9 N m^2/C^2) * (q^2) / (2.598 m)^2.
8. Rearrange the equation to solve for q^2:
q^2 = (2.94 N * (2.598 m)^2) / (8.99 x 10^9 N m^2/C^2).
9. Solve for q:
q = sqrt((2.94 N * (2.598 m)^2) / (8.99 x 10^9 N m^2/C^2)).
10. Calculate the numerical value for q using a calculator to get the charge on each ball.