A small 6.8 g plastic ball is suspended by a
37.3 cm long string in a uniform electric field
of 4790 N/C, as shown.
The acceleration of gravity is 9.8 m/s2 , and
the Coulomb constant is 8.99×109 N · m2/C2..
If the ball is in equilibrium when the string
makes a 11.2 degree angle with the vertical as indicated, what is the net charge on the ball?
Answer in units of μC.
Write two force balance equations: vertical and horizontal. The string tension force T will appear in each equation but can be eliminated since there are two unknowns and two equations.
T sin 11.2 = Coulomb force
T cos 11.2 = gravity force, M g
You will need to know the direction of the E field. I assumed it was horizontal; you should have said what it was since no figure was shown.
To find the net charge on the ball, we can use the concept of electrostatic force and gravitational force balancing each other in equilibrium.
Step 1: Find the force of gravity acting on the ball.
The force of gravity (Fg) can be calculated using the mass (m) of the ball and the acceleration due to gravity (g): Fg = m * g.
Given:
mass of the ball (m) = 6.8 g = 6.8 * 10^(-3) kg
acceleration due to gravity (g) = 9.8 m/s^2
Fg = (6.8 * 10^(-3)) kg * 9.8 m/s^2
Fg ≈ 6.664 * 10^(-2) N
Step 2: Find the electrostatic force acting on the ball.
The electrostatic force (Fe) can be calculated using the equation: Fe = q * E.
Given:
length of the string (L) = 37.3 cm = 0.373 m
electric field strength (E) = 4790 N/C
Since the string makes an angle of 11.2 degrees with the vertical, only the vertical component of the electrostatic force will balance the force of gravity.
Fe = q * E * cos(theta), where theta is the angle between the electric field and the vertical.
theta = 11.2 degrees = 11.2 * (pi/180) radians (converting degrees to radians)
Fe = q * E * cos(11.2 * (pi/180))
Step 3: Equate the force of gravity and the vertical component of the electrostatic force to find the charge (q).
In equilibrium, the forces must balance each other.
Fe = Fg
q * E * cos(11.2 * (pi/180)) = Fg
q = Fg / (E * cos(11.2 * (pi/180)))
q ≈ (6.664 * 10^(-2) N) / (4790 N/C * cos(11.2 * (pi/180)))
Calculating this expression will give you the net charge on the ball.