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.
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:
F = (k * q1 * q2) / r^2
Where:
F is the force between the objects,
k is the electrostatic constant (9 x 10^9 N m^2 / C^2),
q1 and q2 are the charges on the objects, and
r is the distance between the objects.
In this case, we can assume that the force of gravity on the balls is negligible compared to the electrostatic force between them.
First, let's find the mass per unit length of the string. We have two balls of mass 0.300 kg each, so the total mass is 0.600 kg. The total length of the string is 250 cm, which is 2.5 m. Therefore, the mass per unit length of the string is:
m/L = (0.600 kg) / (2.5 m) = 0.240 kg/m
Next, let's find the tension in the string. The tension in the string at the midpoint is equal to the weight of the balls, which is the force due to gravity. The formula for the weight is:
Weight = mass * gravitational acceleration
Since we know the mass of the balls and the gravitational acceleration, we can find the weight:
Weight = (0.300 kg) * (9.8 m/s^2) = 2.94 N
Since the balls hang with separation 50.0 cm, half of the weight (1.47 N) is acting on each ball. Therefore, the tension in the string is 1.47 N.
Now, we can use the tension in the string to find the force of repulsion between the balls. The tension in the string is equal to the force of repulsion, so:
F = 1.47 N
We can rearrange Coulomb's law to solve for the charges on each ball:
q1 * q2 = (F * r^2) / k
Plugging in the known values:
q1 * q2 = (1.47 N * (0.5 m)^2) / (9 x 10^9 N m^2 / C^2)
Simplifying:
q1 * q2 = 0.3685 x 10^-9 C^2
Since the balls have the same charge magnitude, we can assume q1 = q2 = q:
q * q = 0.3685 x 10^-9 C^2
Taking the square root of both sides:
q = sqrt(0.3685 x 10^-9) C
Calculating this value, we get:
q ≈ 6.07 x 10^-5 C
Therefore, the charge on each ball is approximately 6.07 x 10^-5 C.