A small, 2.00 -g plastic ball is suspended by a 20.0-cm- long string in a uniform electric field

A small, 2.00 -g plastic ball is suspended by a 20.0-cm-...

A small, 2.00 -g plastic ball is suspended by a $20.0-\mathrm{cm}-$ long string in a uniform electric field as shown in Figure $P 19.67 .$ If the ball is in equilibrium when the string makes a $15.0^{\circ}$ angle with the vertical, what is the net charge on the ball?

Key Concepts

Gravitational Force

Gravitational force is the attractive force exerted by the Earth on an object, calculated as the product of the object's mass and the acceleration due to gravity. This force is a fundamental element in mechanical equilibrium problems, providing the vertical component that must be balanced by other forces.

Uniform Electric Field and Electric Force

In a uniform electric field, a charged object experiences an electric force defined by the product of its net charge and the field strength. This concept is crucial for understanding how electric fields influence charged bodies, similar to how gravitational fields act on masses.

Static Equilibrium

Static equilibrium is the condition in which an object remains at rest because the net force acting on it is zero. In such problems, the sum of all forces in every direction must cancel, which allows analysts to set up balance equations that equate opposing forces.

Decomposition of Forces

Decomposition of forces involves breaking a force into its perpendicular components, typically along horizontal and vertical axes. This technique simplifies the analysis of forces acting at angles by letting one treat each directional component independently to set up equations for force balance.

Transcript

00:01 In this question, we're told that a 2 gram plastic ball is suspended by a 20 centimeter long string in a uniform electric field.

00:09 And we're directed to figure 19 .67.

00:13 If the ball is an equilibrium when the string makes a 15 degree angle with the vertical, we're asked to calculate the net charge.

00:20 So this is really a force balance question with a little bit of electricity thrown in there.

00:28 So we're going to start off with a free body diagram.

00:32 So of course we've got the tension force directed up at an angle.

00:38 We've got the gravitational force pointing straight down.

00:42 And the electric force is pointing this way.

00:48 How do i know the electric force is pointing this way? i need something to balance out the tension force in the x direction here.

00:54 So it has to be going to the right.

00:57 And just as a side note, that automatically tells us that the charge on the ball is positive because the force and the electric field are going in the same direction.

01:07 Let's go ahead and draw the angle on here, just so we have it for reference.

01:11 So the angle with the vertical is 15 degrees.

01:15 So just be careful about that because, you know, a lot of times they're given with respect to the horizontal.

01:21 And the coordinate system we're going to use here is just the standard one.

01:24 So y is going to be vertical.

01:25 X, positive x will be towards the right.

01:29 So in order to go ahead and find what the charge is, we're definitely going to need to know what that electric force is.

01:37 And to find that, we need to know the x direction of the tension, but we don't know the tension.

01:43 So what i'm going to do is i'm going to analyze the y direction first so we can get a hold of what the tension in the string is.

01:52 So the ball is in equilibrium.

01:55 So we know that the net force have.

01:57 To be zero in the y direction...

Please give Ace some feedback