A positively charged bead having a mass of 1.00 g falls from rest in a vacuum from a height of 5

A positively charged bead having a mass of 1.00 g falls from...

Key Concepts

Work-Energy Principle

The work-energy principle states that the work done by all forces acting on an object is equal to the change in its kinetic energy. In problems where multiple forces (such as gravitational and electric forces) act on a particle, this principle helps in determining the net work done over a displacement, thereby linking force, displacement, and changes in velocity. It is a key tool in analyzing how potential energies convert into kinetic energy when various forces are doing work.

Kinematics of Free Fall

Kinematics in the context of free fall involves describing motion under a constant gravitational acceleration. In the absence of air resistance, the equations of kinematics (such as v² = u² + 2as) allow one to determine the speed of an object falling from a certain height. This concept is used to compare the expected speed from gravity alone to the observed speed when additional forces, like those from an electric field, are present.

Vector Addition of Forces

In scenarios where more than one force is acting on a particle, the net force is determined by the vector sum of these forces. Understanding the vector addition of forces is essential to correctly identify the overall acceleration experienced by the particle, which in turn affects its motion. This concept allows for the distinction between cases where forces are aligned or opposed, as seen in problems combining gravitational and electric forces.

Uniform Electric Field

A uniform electric field is one that has the same magnitude and direction at every point in space. In such a field, a charged particle experiences a constant force, which can be calculated by multiplying the charge by the electric field strength. This concept is fundamental in understanding the behavior of charged particles as they move through regions where the electric field does not vary with position.

Electric Force on a Charged Particle

When a charged particle is placed in an electric field, it experiences an electric force given by F = qE, where q is the charge and E is the electric field. The direction of this force depends on the sign of the charge; for a positive charge, the force acts in the same direction as the field, while for a negative charge, it acts in the opposite direction. This principle is crucial when analyzing the particle’s motion under the influence of both gravitational and electric forces.

Transcript

00:01 So we have this charged beat that is the motion of the beat is only in the vertical direction, but better to say on the vertical axis.

00:15 So, for example, the beat is like this and it is charged, so and it is inside an electric field.

00:26 So we have two forces, one force is certainly the force of gravity, which is downwards.

00:31 And we have another force that is the force of electric field, but we don't really know yet the direction of this force, and that is for the part a we need to actually find direction of the force.

00:51 So we do know that the b is positively charged.

00:57 So when we have positively charged bodies or points, we know that positive charges move in the same level.

01:08 Direction as the direction of the electric field itself.

01:13 So what we need to find here is we need to use certain, we need to use some kinematic equations to find the most of the data we have from what is given.

01:30 We know that, for example, final velocity, but final velocity squared will be equal to the initial velocity squared plus two accelerations times and in this case the vertical displacement we know that this we know this displacement it is 5 meters we know that the initial velocity is zero and we know the final velocity so basically we have this zero so this is final velocity squared is equal to two accelerations times the displacement and we need to find acceleration now a deceleration is why we're searching for the acceleration if the acceleration is positive then we know from the second newton's law so that the total acceleration of the of the bead is we can actually say fb not f a so the total the total force let's say the force on the bead is the sum of two forces the of the gravitational force and the sum of the force due to electrostatic influence.

02:58 Now, these two forces together, so mg, and now we don't know is it plus or minus, but we have some plus or minus f, will contribute to the final acceleration.

03:15 If the final acceleration is positive and greater than the acceleration of the gravity, which is g, then the acceleration that comes due to electrostatic force is in the same direction as the acceleration of the gravities.

03:34 And in that case, we know that the force of the electric field and the force, the gravitational field, are the same.

03:43 They are in the same direction.

03:45 And that is why we're able to search for the sign and the magnitude of the acceleration.

03:50 And we're using this kinematic equation here to start from...

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