A 2.75-gram air inflated balloon is given an excess negative charge, q1 = -3.00 × 10-8 C, by rubbing it with a blanket. It is found that a charged rod can be held above the balloon at a distance of d = 5.80 cm to make the balloon float. In order for this to occur, what polarity of charge must the rod possess? How much charge, q2, does the rod have? Assume the balloon and rod to be point charges. The Coulomb force constant is 1/(4π ε0) = 8.99 × 109 N·m2/C2 and the acceleration due to gravity g = 9.81 m/s2.

Question

A 2.75-gram air inflated balloon is given an excess negative charge, q1 = -3.00 × 10-8 C, by rubbing it with a blanket. It is found that a charged rod can be held above the balloon at a distance of d = 5.80 cm to make the balloon float. In order for this to occur, what polarity of charge must the rod possess? How much charge, q2, does the rod have? Assume the balloon and rod to be point charges. The Coulomb force constant is 1/(4π ε0) = 8.99 × 109 N·m2/C2 and the acceleration due to gravity g = 9.81 m/s2.

Answer 1

To solve the problem, we need to consider the forces acting on the balloon: the electrical force between the balloon and the rod and the gravitational force pulling the balloon downward.

First, let's determine the mass of the balloon. Since we know the mass is 2.75 grams, we can convert it to kilograms by dividing by 1000:

m = 2.75 g / 1000 = 0.00275 kg

Next, let's calculate the gravitational force acting on the balloon. The gravitational force is given by:

F_gravity = m * g

where g is the acceleration due to gravity (9.81 m/s^2).

F_gravity = 0.00275 kg * 9.81 m/s^2 = 0.0269275 N

Now, let's consider the electrical force between the balloon and the rod. The electrical force is given by Coulomb's law:

F_electric = k * |q1| * |q2| / d^2

where k is the Coulomb force constant (8.99 × 10^9 N·m^2/C^2), q1 is the charge on the balloon (-3.00 × 10^-8 C), q2 is the charge on the rod (which we need to find), and d is the distance between the balloon and the rod (0.058 m).

We want the electrical force to balance the gravitational force, so we have:

F_electric = F_gravity

Plugging in the values and rearranging the equation, we get:

k * |q1| * |q2| / d^2 = m * g

Now we can solve for q2. Rearranging the equation gives us:

|q2| = (m * g * d^2) / (k * |q1|)

|q2| = (0.00275 kg * 9.81 m/s^2 * (0.058 m)^2) / (8.99 × 10^9 N·m^2/C^2 * 3.00 × 10^-8 C)

Calculating this expression will give us the magnitude of the charge |q2|. The negative sign of q2 will represent the polarity of the charge on the rod.

|q2| = 5.397 * 10^-8 C

Therefore, the rod must possess a positive charge (+5.397 × 10^-8 C), and the magnitude of the charge on the rod is 5.397 × 10^-8 C.