A small sphere of mass m=.0064kg and charge q1= +28.9 nC is attached to the end of a string and hangs vertically. A second charge of equal mass and charge q2= -58nC is located below the first charge by a distance d= 2.00cm.
a. Find the Tension in the string
b. If the string has a maximum tension of .180 N what is the smallest value d can have in cm before the string breaks?
My answers: a. .063 N b. 59cm
are wrong and i don't know why.
To solve this problem, we will analyze the forces acting on the small sphere and use Newton's second law of motion (F=ma) to find the tension in the string.
a. Find the Tension in the string:
The forces acting on the small sphere are the gravitational force (weight) and the electrostatic force due to the interaction with the second charge.
1. Gravitational Force:
The gravitational force is given by F_gravity = m*g, where m is the mass of the sphere and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_gravity = (0.0064 kg) * (9.8 m/s^2) = 0.06272 N
2. Electrostatic Force:
The electrostatic force between two charges is given by Coulomb's Law:
F_electrostatic = k * |q1*q2| / (d^2)
where k is the electrostatic constant (approximately 9.0 x 10^9 Nm^2/C^2), q1 is the charge of the first sphere, q2 is the charge of the second sphere, and d is the distance between them.
F_electrostatic = (9.0 x 10^9 Nm^2/C^2) * |28.9 x 10^(-9) C * (-58 x 10^(-9) C)| / (0.02 m)^2
F_electrostatic = 0.29 N (approx)
3. Tension in the string:
Since the small sphere is in equilibrium and not accelerating in the vertical direction, the tension in the string must balance the combined weight and electrostatic force.
Tension = F_gravity + |F_electrostatic|
Tension = 0.06272 N + 0.29 N
Tension = 0.35272 N (approx)
So, the tension in the string is approximately 0.35272 N.
b. If the string has a maximum tension of 0.180 N, then the smallest value d can have before the string breaks can be determined by rearranging the equation for electrostatic force:
F_electrostatic = k * |q1*q2| / (d^2)
d = sqrt(k * |q1*q2| / F_electrostatic)
Substituting the given values:
d = sqrt((9.0 x 10^9 Nm^2/C^2) * |28.9 x 10^(-9) C * (-58 x 10^(-9) C)| / (0.180 N)^2)
Calculating this expression gives d ≈ 0.030434 m or 3.0434 cm (approx).
Therefore, the smallest value d can have before the string breaks is approximately 3.0434 cm, not 59 cm as you mentioned.
To calculate the tension in the string, we need to consider the forces acting on the small sphere:
1. The gravitational force (F_gravity): Since the sphere is hanging vertically, the gravitational force acting on it is given by F_gravity = m * g, where m is the mass of the sphere and g is the acceleration due to gravity (9.8 m/s^2).
2. The electrostatic force (F_electric): The first charge (q1) exerts an electrostatic force on the second charge (q2) given by Coulomb's law: F_electric = k * |q1 * q2| / d^2, where k is the electrostatic constant (8.99 x 10^9 N * m^2 / C^2) and d is the distance between the charges.
3. The tension in the string (T): This is the force exerted by the string to keep the sphere in equilibrium, and it is equal to the sum of the gravitational force and the electrostatic force.
Now let's calculate the tension in the string (T):
a. Tension in the string (T) = F_gravity + F_electric
T = m * g + k * |q1 * q2| / d^2
Substituting the given values:
T = (0.0064 kg) * (9.8 m/s^2) + (8.99 x 10^9 N * m^2 / C^2) * |(28.9 x 10^-9 C) * (-58 x 10^-9 C)| / (0.02 m)^2
T ≈ 0.063 N
Therefore, the tension in the string is approximately 0.063 N.
Now let's move on to the second part of the question:
b. In this case, we need to find the smallest value that d can have before the tension in the string exceeds the maximum tension (0.180 N), causing the string to break.
Setting T = 0.180 N in the equation from part a:
0.180 N = (0.0064 kg) * (9.8 m/s^2) + (8.99 x 10^9 N * m^2 / C^2) * |(28.9 x 10^-9 C) * (-58 x 10^-9 C)| / d^2
Rearranging the equation to solve for d:
d^2 = (8.99 x 10^9 N * m^2 / C^2) * |(28.9 x 10^-9 C) * (-58 x 10^-9 C)| / (0.180 N - (0.0064 kg) * (9.8 m/s^2))
d^2 ≈ 5.9 x 10^-4 m^2
So, the smallest value of d in meters is approximately √(5.9 x 10^-4) ≈ 0.0243 m = 2.43 cm.
Therefore, the smallest value d can have before the string breaks is approximately 2.43 cm, not 59 cm.