Two spheres A and B have an equal mass and are...

Two spheres $A$ and $B$ have an equal mass and are electrostatically charged such that the repulsive force acting between them has a magnitude of $20 \mathrm{mN}$ and is directed along line $A B$. Determine the angle $\theta$, the tension in cords $A C$ and $B C,$ and the mass $m$ of each sphere.

Key Concepts

Coulomb's Law

Coulomb's Law describes the force between two point charges. It states that the magnitude of the electrostatic force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This concept is crucial when calculating the repulsive force acting between charged objects, such as spheres in this problem.

Static Equilibrium

The concept of static equilibrium involves analyzing situations where the net force acting on an object is zero. In such cases, all forces, including gravitational, tension, and electrostatic forces, must balance each other. This equilibrium condition is essential for determining unknown quantities like mass, tension in cords, and angles in systems where forces are acting in different directions.

Vector Decomposition

Vector decomposition is the process of breaking a force into its horizontal and vertical components. This technique is necessary for analyzing the forces acting on the spheres, as the tension in the cords must be resolved into components that counteract both the gravitational force and the electrostatic repulsive force. Understanding vector decomposition helps clarify how different forces interact to maintain equilibrium.

Tension in Cords

Tension in cords is the force transmitted through a string, rope, or cable when it is pulled tight by forces acting from opposite ends. In this context, analyzing the tension in the cords is linked to the static equilibrium conditions and vector decomposition, as the tension must balance the resultant forces on the spheres. This analysis allows the determination of both the magnitude of the tension and the angle at which the cords are positioned.

Transcript

00:01 So in this problem we're dealing with static electric field and the relationship between electric force caused by electric fuel with another forces like forces of gravity and tension in this problem.

00:15 So here we have two ball suspended on two light strings that have their length and we know the angle theta between the string and the normal.

00:32 And they are in a somewhat static situation.

00:36 So there are a few forces, actually two forces.

00:39 The force of electric field which will act differently.

00:42 So it will propel one ball and attract another ball.

00:47 Of course the ball themselves have interaction because they're oppositely charged and there is the gravitational field which is pulling them down.

00:57 Of course in the static situation all of these forces sum up and cause the force of the the tension inside the string.

01:05 So to find this electric field we can actually start from, we can actually use only one ball and the situation considering one ball.

01:16 So we will use the positive chart ball and we will say that the total force which is equal to electric field times the q, the charge of the positive ball is equal to the forces which are attracting, this is both.

01:34 So a basic electric field is attracted in this way, since it's, that is how it is directed.

01:42 It's directed like this.

01:44 This is electric field.

01:46 This force must be plus with two forces.

01:49 So f2, which is the force of attraction between negative and positive charge, and the component of the tension force, which is in that direction and this tension force is coming from the mg or from the nass.

02:03 So basically we need to find, we need to say that this eq equals t tangent sinus, theta plus f1.

02:16 Now, of course, the tension force itself can be obtained if we say that, we can say that the tension t cosine theta is equal to mg.

02:27 So mg is equal to this and you can find a tangent force if we divide.

02:36 So let's do that first...

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