A small sphere of mass m = 6.40 g and charge q1 = 30.3 nC is attached to the end of a string and hangs vertically as in the figure. A second charge of equal mass and charge q2 = −58.0 nC is located below the first charge a distance d = 2.00 cm below the first charge as in the figure.
(a) Find the tension in the string.
N
(b) If the string can withstand a maximum tension of 0.180 N, what is the smallest value d can have before the string breaks?
cm
Which string? Assuming the string holding between the masses.
Tension= mg-kq1q2/.02^2
To solve this problem, we need to analyze the forces acting on the small sphere hanging from the string.
First, let's consider the forces acting on the sphere:
1. Gravitational Force (Fg):
The sphere has a mass of 6.40 g, so we can convert it to kg:
m = 6.40 g = 6.40 × 10^(-3) kg
The gravitational force can be calculated using:
Fg = m * g
where g is the acceleration due to gravity, approximately 9.8 m/s².
2. Electric Force (Fe):
The electric force between two charges can be calculated using Coulomb's law:
Fe = k * (|q1| * |q2|) / r²
where k is the electrostatic constant (8.99 × 10^9 N m²/C²), q1 and q2 are the charges of the two spheres, and r is the distance between them.
Considering the sphere is at equilibrium (not accelerating), the tension in the string is equal in magnitude but opposite in direction to the net force acting on the sphere.
(a) Find the tension in the string:
Since the sphere is in equilibrium, the net force acting on it is zero.
Net Force = Fg + Fe = 0
Setting Fg = -Fe (as they are opposite in direction), we have:
m * g = k * (|q1| * |q2|) / r²
Now, solving for the tension (T) in the string:
T = m * g + k * (|q1| * |q2|) / r²
(b) If the string can withstand a maximum tension of 0.180 N, we can set up an equation to find the smallest value d can have before the string breaks:
Tension = m * g + k * (|q1| * |q2|) / d²
Substituting the given values and solving for d:
0.180 N = 6.40 × 10^(-3) kg * 9.8 m/s² + (8.99 × 10^9 N m²/C²) * (30.3 × 10^(-9) C) * (58.0 × 10^(-9) C) / d²
Now, rearrange the equation to solve for d:
d = √((8.99 × 10^9 N m²/C²) * (30.3 × 10^(-9) C) * (58.0 × 10^(-9) C) / (0.180 N - 6.40 × 10^(-3) kg * 9.8 m/s²))
Note: Make sure to convert all units to be consistent before performing the calculation.
By substituting the values and solving this equation, you can find the smallest value of d in centimeters before the string breaks.