Two identical balls each have a charge of -7.5x10^-6C. The balls hang from identical strings that are 8�‹ from the vertical because of the repulsive force between the charged balls. The balls are separated by a distance of 10cm. What is the tension in one of the strings? What is the mass of one of the balls?
To find the tension in one of the strings, we can use the concept of electrostatic force and equilibrium.
Step 1: Find the electrostatic force between the charged balls.
The electrostatic force between two charges can be calculated using Coulomb's law:
F = k * (q1 * q2) / r^2
Where:
F is the electrostatic force
k is the electrostatic constant (9 * 10^9 Nm^2/C^2)
q1 and q2 are the charges on the balls (-7.5 * 10^-6C for both)
r is the distance between the balls (10cm = 0.1m)
Plugging in the values:
F = (9 * 10^9 Nm^2/C^2) * (-7.5 * 10^-6C) * (-7.5 * 10^-6C) / (0.1m)^2
Simplifying:
F = (9 * 10^9 Nm^2/C^2) * (56.25 * 10^-12C^2) / 0.01m^2
F = 506.25 N
Step 2: Find the vertical component of the tension in the string.
The tension in the string can be decomposed into two components: the vertical component (T_v) and horizontal component (T_h). Since the balls are in equilibrium, the vertical component of the tension balances the weight of one ball.
T_v = mg
Where:
T_v is the vertical component of the tension
m is the mass of one ball (to be determined)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Step 3: Equate the vertical component of tension and the electrostatic force.
Since the electrostatic force is repulsive, it is balanced by the vertical component of the tension in one string.
T_v = F
Substituting the values:
mg = 506.25 N
Step 4: Solve for the tension in one of the strings.
Dividing both sides by g:
T_v / g = m
506.25 N / 9.8 m/s^2 = m
51.633 kg = m
Therefore, the tension in one of the strings is 506.25 N, and the mass of one of the balls is 51.633 kg.
To find the tension in one of the strings, we can use the concept of electrostatic force:
Step 1: Calculate the electrostatic force between the two charged balls.
The formula to calculate the electrostatic force between two charged objects is given by Coulomb's Law:
F = k * (q1 * q2) / r^2
Where:
F is the electrostatic force
k is the electrostatic constant, approximately equal to 9 * 10^9 Nm^2/C^2
q1 and q2 are the charges of the balls (-7.5 x 10^-6C each)
r is the distance between the balls (10cm = 0.1m)
Calculating the electrostatic force:
F = (9 * 10^9 Nm^2/C^2) * ((-7.5 x 10^-6C)^2) / (0.1m)^2
= 6.75 x 10^-3 N
Step 2: Determine the angle made by the string with the vertical.
The angle can be found using trigonometry. We have a right triangle, where the opposite side is the distance given (8�‹) and the hypotenuse is the tension in the string.
Using the sine function:
sin(theta) = opposite / hypotenuse
sin(theta) = 8 / T
Step 3: Calculate the tension in one of the strings.
Rearranging the equation:
T = 8 / sin(theta)
T = 8 / sin(8�‹)
Step 4: Calculate the tension in the string.
T = 8 / sin(8�‹)
T ≈ 53.99 N
Therefore, the tension in one of the strings is approximately 53.99 N.
Now let's calculate the mass of one of the balls:
Step 5: Use Newton's second law to find the mass.
The tension in the string is equal to the weight of the ball:
T = mg
Where:
T is the tension in the string (53.99 N)
m is the mass of one of the balls
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Calculating the mass:
m = T / g
m = 53.99 N / 9.8 m/s^2
m ≈ 5.51 kg
Therefore, the mass of one of the balls is approximately 5.51 kg.