two identically charged sphere of mass 3 gm each are suspended from a point by threads of length 13 cm. they are separated 10 cm due to repulsion. find the tension of the thread and charge on the sphere?
To find the tension in the thread and the charge on the sphere, we can use Coulomb's law and equilibrium conditions.
Step 1: Determine the gravitational force on each sphere.
Since both spheres have the same mass of 3 gm, and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the gravitational force using the formula:
Gravitational Force = mass * acceleration due to gravity
Gravitational Force = (3 gm) * (9.8 m/s^2)
Gravitational Force = 29.4 gm·m/s^2
Since the unit of force in the SI system is Newton (N), we need to convert grams·m/s^2 to Newtons (N).
1 gm·m/s^2 = 0.001 N
So, Gravitational Force = 0.001 * 29.4 N
Gravitational Force = 0.0294 N (approximately)
Step 2: Analyze the forces acting on one of the spheres.
The forces acting on one of the spheres are:
- Gravitational force pulling it downward.
- Electrical repulsive force between the two spheres.
Step 3: Find the tension in the thread.
The tension in the thread is a result of the combination of the gravitational force and the electrical repulsive force. Since both of these forces act along the same line of action, the net force in the vertical direction must be zero for the spheres to be in equilibrium. Therefore, the tension in the thread is equal to the gravitational force.
Tension in the thread = 0.0294 N
Step 4: Calculate the electrical repulsive force.
We can use Coulomb's law to find the electrical repulsive force.
Coulomb's law states that the magnitude of the electrical force between two point charges is given by:
Electrical force = (k * |q1 * q2|) / r^2
where k is Coulomb's constant (9 * 10^9 N·m^2/C^2), q1 and q2 are the charges on the spheres, and r is the distance between them.
Given:
k = 9 * 10^9 N·m^2/C^2
r = 10 cm = 0.1 m
Since the spheres have the same charge (let's call it q), the electrical repulsive force is equal in magnitude and opposite in direction on each sphere.
So, the electrical repulsive force acting on each sphere is:
Electrical force = (k * |q * q|) / r^2
The electrical force must balance the gravitational force for the spheres to be in equilibrium.
Gravitational Force = Electrical Force
0.0294 N = (9 * 10^9 N·m^2/C^2 * |q * q|) / (0.1 m)^2
Simplifying the equation:
(q^2) = (0.0294 N * (0.1 m)^2) / (9 * 10^9 N·m^2/C^2)
(q^2) = 0.00003267 C^2
Taking the square root of both sides:
q = √(0.00003267 C^2)
q = 0.00571 C (approximately)
Step 5: Determine the charge on the sphere and the tension in the thread.
The charge on each sphere is 0.00571 C, and the tension in the thread is 0.0294 N.