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

To solve this problem, we can use Coulomb's Law, which states that the force between two charges is equal to the electrostatic constant (k) multiplied by the product of their charges and divided by the square of the distance between them.

Given:
Mass of each volleyball (m) = 0.300 kg
Distance between the volleyballs (r) = 50.0 cm = 0.5 m
Length of the string (L) = 250 cm = 2.5 m

We know that the gravitational force acting on each volleyball is balanced by the electrostatic force of repulsion.

Step 1: Calculate the gravitational force (F_gravity) acting on each volleyball.
We can use the formula:
F_gravity = m * g
where g is the acceleration due to gravity (approximately 9.8 m/s²).

F_gravity = 0.300 kg * 9.8 m/s² = 2.94 N

Step 2: Calculate the electrostatic force (F_electrostatic) between the volleyballs.
Using Coulomb's Law:
F_electrostatic = k * (q₁ * q₂) / r²
where k is the electrostatic constant (approximately 9.0 × 10⁹ N m²/C²) and q₁, q₂ are the charges on the volleyball.

Since the force of repulsion is equal to the gravitational force, we can equate the two formulas.
F_electrostatic = F_gravity

k * (q₁ * q₂) / r² = m * g

Substituting the given values and solving for q₁*q₂, we have:
(9.0 × 10⁹ N m²/C²) * (q₁ * q₂) / (0.5 m)² = 2.94 N

Step 3: Solve for q₁*q₂.

(q₁ * q₂) = (2.94 N * (0.5 m)²) / (9.0 × 10⁹ N m²/C²)

(q₁ * q₂) = 0.0735 C²

Step 4: Calculate the charges on each volleyball (q₁, q₂).

Since the volleyballs have identical charges, we can assume q₁ = q₂ = q.

q * q = 0.0735 C²

Taking the square root of both sides, we get:

q = √0.0735 C² = 0.271 C

Therefore, the charge on each volleyball is approximately 0.271 C.

To determine the charge on each ball, we can make use of Coulomb's Law and the principle of equilibrium in a system.

1. First, let's assume the charge on each ball is q.

2. Since the two balls repel each other, the electrostatic force between them can be expressed using Coulomb's Law:

F = k * (q1 * q2) / r^2,

where F is the electrostatic force, k is Coulomb's constant (9 x 10^9 N m^2/C^2), q1 and q2 are the charges on the balls, and r is the separation between the balls.

3. In equilibrium, the tension in the strings cancels out the electrostatic force. Considering the forces acting on one of the balls, we have:

T * cosθ = mg --- Equation (1),

where T is the tension in the string, θ is the angle the string makes with the vertical, and m is the mass of each ball.

4. The electrostatic force between the balls can also be decomposed into components along the strings. The vertical component of the electrostatic force is equal to the tension in the string:

T = F * sinθ --- Equation (2).

5. Solving Equations (1) and (2) simultaneously will allow us to determine the charge on each ball.

T * cosθ = mg,
T = F * sinθ.

6. Substituting the expressions for T and F into the equations above, we get:

(F * sinθ) * cosθ = mg,
(k * (q^2) / r^2) * (sinθ * cosθ) = mg.

7. We can simplify this further by noting that sinθ * cosθ = sin(2θ) / 2:

(k * (q^2) / r^2) * (sin(2θ) / 2) = mg.

8. Rearranging the equation, we can solve for q:

q^2 = (2 * m * g * r^2) / (k * sin(2θ)).

9. Now we can plug in the given values into the equation to find the charge on each ball. The mass of each ball is 0.300 kg, the separation between the balls is 50.0 cm (or 0.5 m), the gravitational acceleration is 9.8 m/s^2, and the length of the string is 250 cm (or 2.5 m).

q^2 = (2 * 0.300 kg * 9.8 m/s^2 * (0.5 m)^2) / (9 x 10^9 N m^2/C^2 * sin(2θ)).

10. We need the value of θ to proceed. In the given information, the length of the string from the point of support to the center of a ball is 250 cm, and the base of the triangle is 0.5 m. Since this forms a right-angled triangle with equal sides, we can use trigonometry to find θ:

sinθ = opposite / hypotenuse,
sinθ = 0.5 m / 2.5 m.

11. Finally, we can substitute the value of sinθ into the equation:

q^2 = (2 * 0.300 kg * 9.8 m/s^2 * (0.5 m)^2) / (9 x 10^9 N m^2/C^2 * sin(2θ)).

12. Taking the square root of q^2 will give us the charge on each ball:

q = sqrt((2 * 0.300 kg * 9.8 m/s^2 * (0.5 m)^2) / (9 x 10^9 N m^2/C^2 * sin(2θ))).

By calculating the final expression, you will find the charge on each ball.