A charged cork ball of mass m is suspended on a light string...

A charged cork ball of mass $m$ is suspended on a light string in the presence of a uniform electric field as shown in Figure $\mathrm{P} 23.55$ . When $\mathbf{E}=(A \hat{\mathbf{i}}+B \hat{\mathbf{j}}) \mathrm{N} / \mathrm{C},$ where $A$ and $B$ are positive numbers, the ball is in equilibrium at the angle $\theta .$ Find $(a)$ the charge on the ball and $(b)$ the tension in the string.

Key Concepts

Tension in a String

Tension is the force transmitted through a string or rope when it is pulled tight by forces acting from opposite ends. In equilibrium problems, the tension in the string balances both the weight of the object and any additional horizontal forces, making it key to solving the system of forces acting on the suspended mass.

Force Decomposition

In problems involving forces at angles, it is necessary to break down vector forces into their orthogonal components, typically along horizontal and vertical axes. This technique simplifies the analysis by allowing separate consideration of forces in each direction, facilitating the application of equilibrium conditions.

Gravitational Force

This is the force with which the Earth attracts any object with mass. It is given by the product of the object's mass and the acceleration due to gravity, acting in the downward direction. Recognizing the role of gravitational force enables the proper balancing of forces in problems involving suspended masses.

Electrostatic Force

This represents the force experienced by a charged object in an electric field. The force is given by the product of the charge and the electric field vector and acts in the direction of the field for positive charges, while it reverses for negative charges. Understanding this concept is crucial for analyzing forces acting on charged bodies in uniform fields.

Static Equilibrium

An object in static equilibrium has zero net force acting on it. This means that all forces, including gravitational, electrostatic, and tension forces, balance each other out. This concept is essential to set up equations based on the equilibrium condition to solve for unknown quantities.

Transcript

00:01 In the given problem, there is a charged cork ball which is suspended in a uniform electric field with the help of a light string.

00:17 Here this is the cork ball.

00:20 This is the light string and in the electric field, this string is making an angle of 37 degree with the vertical.

00:29 This is the direction of uniform electric field e.

00:39 As the charge ball is having a positive charge on it.

00:45 So this positive charge will experience a force in the direction of electric field.

00:54 This is the electrostatic force experienced by this cork ball.

00:59 And here this is the weight of the ball.

01:03 Acting vertically downward.

01:06 And then we can consider this t as a tension created developed in the light string.

01:14 So if we draw a vertical line here, this angle here will also be 37 degree.

01:23 Now electric field is given as in vector form e bar is equal to 3 .00 i cap plus 5 .00 j -cap into 10 -d ratio to the power 5 newton per cull.

01:45 The magnitude of this electric field will be the square root of square of 3 into 10 to the power 5 plus square of 5 into 10 to the power 5.

02:02 We can take this 10 to the power 5 square common.

02:05 So it will come out of the root leaving behind 9 plus 25.

02:14 So it will be square root of 34 into 10 dash to the power 5 newton per coulant.

02:23 This is the magnitude of this electric field.

02:28 Now we want to find the angle this electric field makes with the direction of x -axis, let it be theta.

02:37 So to find this angle theta, we use the expression 10 theta is equal to ey, the vertical component of electric field by ex, its horizontal component.

02:53 Here ey is 5 .00 and ex is 3 .00, both having 10 dashed par 5 as a multiple.

03:04 So it will come out to be 1 .667.

03:12 So the value of this angle 10 theta is 10 inverse 1 .667, which comes out to be 59 .04 degree.

03:27 So now we can resolve this tension t and the electrostatic force f into horizontal and vertical components.

03:39 So this t will be having a vertical component as which is given as t cost 37 degree and horizontal component t sine 37 degree...

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