A 1.00 g mass hangs by a string next to an identical stationary mass. When both masses are given identical positive charges, the hanging charge swings out 3.00 cm from the vertical so that its string makes an angle of 30.0 degrees with the vertical. Find the magnitude of the charge.

Horizontal force= kqq/.03^2
vertical force=mg

tan30= horizontalforce/verticalforce

solve this for q

To find the magnitude of the charge, we need to solve for q using the given information.

First, let's analyze the forces acting on the hanging charged mass.

The force exerted horizontally on the hanging mass is given by Coulomb's Law:

horizontal force (F_h) = (k * q^2) / r^2

Given that the distance (r) is 0.03 m (3.00 cm) and the angle made with the vertical is 30.0 degrees, we can find the vertical force (F_v) as:

F_v = m * g

where m is the mass (1.00 g = 0.001 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Now, let's calculate the vertical force:

F_v = (0.001 kg) * (9.8 m/s^2) = 0.0098 N

Next, we can use the given angle to determine the ratio of the horizontal force to the vertical force:

tan(30°) = F_h / F_v

Solving for F_h:

F_h = F_v * tan(30°)
= (0.0098 N) * tan(30°)
= 0.0098 N * 0.577
≈ 0.00566 N

Now, we can solve for q by rearranging the equation for the horizontal force:

F_h = (k * q^2) / r^2

Rearranging to solve for q:

q^2 = (F_h * r^2) / k

Substituting the known values:

q^2 = (0.00566 N * (0.03 m)^2) / (9 × 10^9 N m^2/C^2)

Simplifying:

q^2 = 0.00004851 C^2

Taking the square root of both sides:

q = √0.00004851 C
≈ 0.00697 C

Therefore, the magnitude of the charge is approximately 0.00697 C.

To find the magnitude of the charge, we can start by analyzing the forces acting on the hanging mass.

1. Start with the gravitational force:
The vertical force acting on the hanging mass is its weight, given by the equation:
Vertical force = mg,
where m is the mass (1.00 g), and g is the acceleration due to gravity (approximated as 9.8 m/s^2).

2. Consider the electrostatic force:
The horizontal force acting on the hanging mass is the electrostatic force, given by Coulomb's Law:
Fe = k * (q1 * q2) / r^2,
where Fe is the electrostatic force, k is Coulomb's constant, q1 and q2 are the charges, and r is the distance between the charges.

In this case, both masses have the same charge, so q1 = q2, and the electrostatic force becomes:
Fe = k * (q^2) / r^2.

3. Set up the equation for the forces:
The gravitational force and the electrostatic force combine to create a net force that causes the hanging mass to swing.

Since the hanging mass forms a right-angled triangle with the string, we can use trigonometry to relate the forces:
tanθ = (Fe / Fg),
where θ is the angle made by the string with the vertical, Fe is the horizontal electrostatic force, and Fg is the vertical gravitational force.

In this case, θ = 30.0 degrees, Fe = k * (q^2) / r^2, and Fg = mg.

4. Solve the equation for q:
Replace all variables in the equation with their respective values:
tan 30.0 = [(k * (q^2) / r^2)] / (mg).

Simplify the equation:
tan 30.0 = k * (q^2) / (m * g * r^2).

Rearrange the equation to solve for q:
(q^2) = (m * g * r^2 * tan 30.0) / k.

Finally, take the square root of both sides of the equation to find the magnitude of the charge, q:
q = √[(m * g * r^2 * tan 30.0) / k].

By substituting the given values for m (mass), g (acceleration due to gravity), r (distance), and using the known value of Coulomb's constant k, you can calculate the magnitude of the charge, q.