two pith balls each of mass 2 mg are suspended by very light thread of 20 cm long from a common pt. each ball has the same charge so that the threads form angles of 30° with the vertical. what is the charge on each ball?
To determine the charge on each ball, we can use Coulomb's Law and a few trigonometric relationships.
Given:
- Mass of each pith ball = 2 mg
- Length of each thread = 20 cm
- Angle formed between the thread and the vertical = 30°
First, we need to convert the mass of each pith ball from milligrams (mg) to kilograms (kg).
1 mg = 1 x 10^(-6) kg
So, mass of each pith ball = 2 x 10^(-6) kg
Next, let's calculate the tension in the thread for one of the pith balls.
The tension can be resolved into two components: one parallel to the vertical and one perpendicular to the vertical.
The tension component parallel to the vertical is given by:
T_parallel = T × sin(θ)
Where:
T_parallel = Tension component parallel to the vertical
T = Tension in the thread
θ = Angle formed between the thread and the vertical (30°)
The tension component parallel to the vertical is required to balance the force of gravity acting on the pith ball:
T_parallel = m × g
Where:
m = Mass of the pith ball
g = Acceleration due to gravity (9.8 m/s^2)
Now, let's calculate the tension in the thread:
T = T_parallel / sin(θ)
Substituting the known values:
T = (m × g) / sin(θ)
T = (2 x 10^(-6) kg × 9.8 m/s^2) / sin(30°)
Next, let's calculate the electrostatic force acting between the two pith balls.
Since the threads form an isosceles triangle, the angle between the threads is 60° (180° - 2 × 30°).
The electrostatic force can be expressed as:
F = (k × q^2) / r^2
Where:
F = Electrostatic force
k = Coulomb's constant (8.9875 x 10^9 Nm^2/C^2)
q = Charge on each pith ball
r = Distance between the pith balls
The distance between the pith balls is given by:
r = 2 × (20 cm) = 40 cm = 0.4 m
Now, we can calculate the charge on each pith ball:
F = (k × q^2) / r^2
Simplifying and rearranging the equation:
q^2 = (F × r^2) / k
Taking the square root of both sides:
q = √((F × r^2) / k)
Substituting the known values:
q = √((((m × g) / sin(θ)) × r^2) / k)
Calculating q:
q = √((((2 x 10^(-6) kg × 9.8 m/s^2) / sin(30°)) × (0.4 m)^2) / (8.9875 x 10^9 Nm^2/C^2))
Simplifying the expression will give us the charge on each pith ball.
To find the charge on each ball, we can use Coulomb's law and the principle of equilibrium for the forces acting on the pith balls. Here's how you can solve this problem step by step:
1. Draw a diagram: Sketch a diagram representing the situation. Draw two pith balls suspended by light threads from a common point, making angles of 30 degrees with the vertical.
2. Identify the forces: There are three forces acting on each pith ball: the gravitational force (mg), the tension in the thread (T), and the electrostatic force (Fe). The gravitational force acts vertically downward, the tension in the thread acts toward the center of the circle, and the electrostatic force acts radially outward.
3. Set up equations: Apply the principle of equilibrium to the forces acting on each pith ball. The vertical components of the tension in the thread must balance the weight of the pith ball, and the horizontal components must balance each other. Also, since the pith balls have the same charge, the electrostatic forces on them must be equal in magnitude.
4. Solve for tension: To find the tension in the thread, consider the vertical components of the tension. Since the threads make angles of 30 degrees with the vertical, the vertical component of the tension (Tsin30°) must balance the weight (mg) of the pith balls.
Tsin30° = mg
Solve for T.
5. Solve for electrostatic force: To find the electrostatic force, consider the horizontal components of the tension. The horizontal components of the tension in the threads must balance each other.
Tcos30° = Fe
Solve for Fe.
6. Calculate the charge: Now, use Coulomb's law to relate the electrostatic force to the charge on the pith balls.
Fe = k * (q^2) / r^2
Plug in the values for Fe, k (Coulomb's constant), and the distance between the pith balls (r) which is the length of the thread.
Solve for q, the charge on each pith ball.
7. Substitute values and calculate: Substitute the known values, such as the mass (2 mg), the length of the thread (20 cm = 0.2 m), and the known constants (such as the value of k). Then solve the equation to find the charge on each ball.
8. Finalize your answer: Once you have calculated the charge on each pith ball, state your answer. Be sure to include the appropriate units for charge (e.g., coulombs, C).
By following these steps, you should be able to calculate the charge on each pith ball in this given scenario.