A small plastic ball of mass m = 3.00 g is suspended by a string of length L = 24.5 cm in a uniform electric field, as shown in the figure below. If the ball is in equilibrium when the string makes a θ = 16.0° angle with the vertical as indicated, what is the net charge on the ball?

To find the net charge on the ball, we can use the formula for the electric force on a charged object in an electric field.

The electric force on a charged object is given by the equation:

F = qE

Where F is the force, q is the charge of the object, and E is the electric field strength.

In this case, the electric force on the ball is balanced by the force of gravity pulling the ball downward, causing it to be in equilibrium.

The force of gravity is given by the equation:

F_gravity = mg

Where m is the mass of the ball and g is the acceleration due to gravity.

Since the ball is in equilibrium, the electric force and the force of gravity must be equal in magnitude:

F = F_gravity

qE = mg

To find the electric field strength, we can use the information given in the problem. The string makes an angle of 16.0° with the vertical, which means that the electric force is acting in the opposite direction of the force of gravity.

The component of the electric force in the vertical direction is given by:

F_electric_vertical = F * sin(θ)

Where θ is the angle the string makes with the vertical.

Since the ball is in equilibrium, the magnitude of the electric force in the vertical direction is equal to the magnitude of the force of gravity:

F_electric_vertical = mg

qE * sin(θ) = mg

We can now solve this equation for the net charge on the ball, q:

q = (mg) / (E * sin(θ))

Inserting the given values:

m = 3.00 g = 0.003 kg (Convert grams to kilograms)

θ = 16.0°

L = 24.5 cm = 0.245 m (Convert centimeters to meters)

g = 9.8 m/s² (Acceleration due to gravity)

Now we need to find the electric field strength, E.

Since the ball is in equilibrium, the electric force must also be balanced by the tension force in the string.

The tension force in the string is given by:

T = mg / cos(θ)

The vertical component of the tension force is equal to the component of the electric force in the vertical direction:

T * cos(θ) = qE

Solving this equation for the electric field strength, E:

E = T * cos(θ) / q

T = mg / cos(θ) = (0.003 kg * 9.8 m/s²) / cos(16.0°)

Now we can substitute this value into the equation for the net charge:

q = (mg) / (E * sin(θ)) = (0.003 kg * 9.8 m/s²) / ((0.003 kg * 9.8 m/s²) / cos(16.0°) * sin(16.0°))

Now we can calculate the value of q using this equation.

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