Two pith balls below each have a mass of 5.0 g and equal charge. One pith ball is suspended by an insulating thread. The other is brought to x = 4.0 cm from the suspended ball. The suspended ball is now hanging with the thread forming an angle of 30.0° with the vertical. The ball is in equilibrium with FE, Fg, and FT. Calculate each of the following.
Fg = 0.049
FE = 0.028
see other post.
To calculate the force exerted by the electric field (FE) and the force of gravity (Fg), we first need to understand the equilibrium condition of the suspended pith ball.
In equilibrium, the net force acting on the pith ball must be zero. This means that the sum of all forces in the horizontal and vertical direction must balance out.
Let's break down the forces acting on the pith ball:
1. Force due to electric field (FE): This force is exerted on the suspended pith ball by the other pith ball, given that they have equal and opposite charges. The direction of this force is along the line joining the two pith balls.
2. Force of gravity (Fg): This force is exerted on the suspended pith ball due to its mass. It acts vertically downwards, opposing the force exerted by the electric field.
3. Tension force (FT): This force is exerted by the thread holding the pith ball in place. It acts along the direction of the string.
To find the magnitude of the electric field (E) generated by the other pith ball, we need to use Coulomb's Law:
FE = k * (q1 * q2) / r^2
Where:
k is Coulomb's constant (8.99 x 10^9 Nm²/C²)
q1 and q2 are the charges of the two pith balls (-q and q, since they have equal and opposite charges)
r is the distance between the centers of the pith balls (given as 4.0 cm = 0.04 m)
Substituting values into the equation:
FE = (8.99 x 10^9 Nm²/C²) * ((5.0 x 10^-3 kg) * (5.0 x 10^-3 kg)) / (0.04 m)^2
FE = 8.99 x 10^9 Nm²/C² * 2.5 x 10^-5 kg² / 0.0016 m²
FE = 8.99 x 10^9 N * (2.5 x 10^-5 kg / 0.0016 m²) * (1 C² / 1 Nm²)
FE ≈ 8.99 x 10^9 * 0.015625
FE ≈ 0.140 N (rounded to three significant figures)
Now, since the suspended pith ball is in equilibrium, the tension force in the string must balance the horizontal and vertical components of the gravitational force.
Using trigonometry, we can determine the vertical component of the gravitational force:
Fy = Fg * sin θ
Where:
Fy is the vertical component of the gravitational force
Fg is the force of gravity (given as 0.049 N)
θ is the angle formed by the thread with the vertical (given as 30.0°)
Substituting values into the equation:
Fy = 0.049 N * sin(30.0°)
Fy = 0.025 N (rounded to three significant figures)
Since the tension FT must balance Fy, we can write:
FT = Fy
Therefore, the tension force FT is also equal to 0.025 N.
So, to summarize:
FE = 0.140 N
Fg = 0.049 N
FT = 0.025 N