5.00x 10^-2 mass with charge +0.75 micro coulombs is hung by a thin insulated thread. Charge of -0.9 micro coulombs is held .15meters directly to the right so that the thread makes an angle with the vertical. What is the angle and the tension in the string?
28.9 & .56n
To find the angle and tension in the string, we can use Coulomb's Law to calculate the electrostatic force between the two charges. We can then equate this force with the tension in the string to solve for the unknowns.
Step 1: Find the electrostatic force:
Coulomb's Law states that the force between two charges is given by:
F = k * (|q1| * |q2|) / r^2
Where F is the force, k is the electrostatic constant (9 x 10^9 N.m^2/C^2), q1 and q2 are the charges, and r is the distance between the charges.
Given:
q1 = +0.75 μC = +0.75 x 10^-6 C
q2 = -0.9 μC = -0.9 x 10^-6 C
r = 0.15 m
Calculating the force:
F = (9 x 10^9 N.m^2/C^2) * [(+0.75 x 10^-6 C) * (-0.9 x 10^-6 C)] / (0.15 m)^2
Step 2: Equate the force with the tension in the string:
In a system at equilibrium, the vertical component of the tension in the string must balance the weight of the mass. The horizontal component of the tension provides the centripetal force to keep the mass in its circular path.
Since the mass is in equilibrium, the vertical component of the tension is equal to the weight of the mass:
T * cos(θ) = m * g
Where T is the tension, θ is the angle with the vertical, m is the mass, and g is the acceleration due to gravity (9.8 m/s^2).
The horizontal component of the tension provides the centripetal force:
T * sin(θ) = F
Step 3: Solve the equations:
Substituting the values and rearranging the equations:
T * cos(θ) = m * g
T * sin(θ) = F
T * cos(θ) = (5.00 x 10^-2 kg) * (9.8 m/s^2) [Substituting the given mass and acceleration due to gravity]
T * sin(θ) = (result obtained from step 1)
To find the tension, divide the two equations:
T * cos(θ) / T * sin(θ) = [(5.00 x 10^-2 kg) * (9.8 m/s^2)] / (result obtained from step 1)
This will give you the tangent of the angle (θ). Taking the inverse tangent (arctan) of this value will give you the actual angle.
Finally, using the value of the angle, substitute it into one of the equations to solve for the tension in the string:
T * cos(θ) = (5.00 x 10^-2 kg) * (9.8 m/s^2)
This will give you the tension in the string.