Two identical small insulating balls are suspended by separate 0.25-m threads that are attached

Two identical small insulating balls are suspended by...

Two identical small insulating balls are suspended by separate $0.25-\mathrm{m}$ threads that are attached to a common point on the ceiling. Each ball has a mass of $8.0 \times 10^{-4} \mathrm{kg} .$ Initially the balls are uncharged and hang straight down. They are then given identical positive charges and, as a result, spread apart with an angle of $36^{\circ}$ between the threads. Determine (a) the charge on each ball and (b) the tension in the threads.

Two identical small insulating balls are suspended by separate $0.25-\mathrm{m}$ threads that are attached to a common point on the ceiling. Each ball has a mass of $8.0 \times 10^{-4} \mathrm{kg} .$ Initially the balls are uncharged and hang straight down. They are then given identical positive charges and, as a result, spread apart with an angle of $36^{\circ}$ between the threads. Determine
(a) the charge on each ball and (b) the tension in the threads.

Key Concepts

Coulomb's Law

Coulomb's law quantifies the force between two point charges. The law states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This concept is foundational for analyzing interactions between charged objects in electrostatics.

Equilibrium of Forces

In static equilibrium problems, the net force acting on an object must be zero. This requires that all forces, including gravitational, tension, and electrostatic forces, balance out in each direction. Understanding equilibrium allows one to set up equations that resolve these forces into their components and solve for unknown quantities.

Trigonometric Decomposition of Forces

When forces act at an angle, they must be resolved into perpendicular components using trigonometric functions such as sine and cosine. This technique is crucial for dissecting forces acting along directions that do not align with the standard horizontal and vertical axes, enabling the accurate application of equilibrium conditions.

Transcript

0:00 Hi.

00:02 In the given problem, length of the threads which are joined at the same point from the upper end of the threads and their lower ends are joined with two balls.

00:25 The length of these thread has been given as 0 .25 meter.

00:33 Suppose they are joined at point a.

00:35 Here this ball, this is b and here this is c.

00:41 The distance between them is b, c, which is for the time being not known to us.

00:51 If you draw a vertical lines which separates the two balls at the midpoint between them, let this point to be d.

01:01 The angle between the two threads when a charge q is given to both the balls.

01:11 So suppose the charge given to the balls is q, the identical charge, due to which, using coulin's law, the walls will repel each other with a force, electrostatic force of repulsion f .e.

01:36 Mass of the ball is given as 8 .0 into 10 dash per minus 4 kilogram.

01:46 So here this is the weight of the ball m g this is the tension t in the threat due to the due to this identical charge over both the balls the balls will repel each other then they will make an angle between them which is given as 36 degree here.

02:09 So the angle made by one of the threads with the vertical means theta is half of this 36 degree means this is 18 degree hence if this angle is theta this angle made by the vertical component of the tension with the thread will also be theta.

02:34 So this vertical component will be t cost theta and horizontal component will be p sine theta here in this triangle a b d which is a right angled triangle in right angled triangle triangle a, b, if you want to find this bd, here this bd by ab means perpendicular by hypotenuse is actually sine theta.

03:19 So this bd comes out to be a b into sine theta or we can say this is l, sine theta.

03:29 Here this l is 0 .25 and this is sign 18 degree.

03:39 So the distance between the two balls becomes twice of bd means this is two times of 0 .25 into sign 18 degree meters or we can say this is 0 .5 into sign 18 degree meter.

04:00 Now as this ball is in equilibrium...

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