An electron is a subatomic particle (m=9.11 ×10^-31 kg) that is subject to electric forces. An e

An electron is a subatomic particle (m=9.11 ×10^-31 kg) that...

An electron is a subatomic particle $\left(m=9.11 \times 10^{-31} \mathrm{kg}\right)$ that is subject to electric forces. An electron moving in the $+x$ direction accelerates from an initial velocity of $+5.40 \times 10^{5} \mathrm{m} / \mathrm{s}$ to a final velocity of $+2.10 \times 10^{6} \mathrm{m} / \mathrm{s}$ while traveling a distance of 0.038 $\mathrm{m}$ . The electron's acceleration is due to two electric forces parallel to the $x$ axis: $\overrightarrow{\mathbf{F}}_{\mathbf{1}}=+7.50 \times 10^{-17} \mathrm{N},$ and $\overrightarrow{\mathbf{F}}_{2}$ which points in the $-x$ direction. Find the magnitudes of $(a)$ the net force acting on the electron and $\quad(b)$ the electric force $\vec{F}_{2}$ .

Key Concepts

Newton's Second Law

This principle states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F_net = m * a). It is fundamental for determining how the sum of forces affects the motion of a particle, such as an electron, and enables the calculation of acceleration when the net force is known, or vice versa.

Kinematics Equations for Constant Acceleration

Kinematics provides mathematical relationships that describe the motion of objects under constant acceleration. These equations relate displacement, initial velocity, final velocity, acceleration, and time. They are used to calculate the acceleration when the initial and final velocities and the displacement are known, which is crucial in solving problems involving changes in motion.

Vector Addition of Forces

Forces are vector quantities, meaning they have both magnitude and direction. The net force acting on an object is the vector sum of all individual forces. Understanding how to add vectors allows one to account for forces acting in different directions, which is essential when analyzing multiple simultaneous forces, such as when one electric force is opposed by another.

Transcript

00:01 We begin this question by calculating what is the net force that acts on the electron.

00:06 And how can we do that? well, remember that from newton's second law, the net force is given by the mass times the acceleration.

00:15 We know the mass of that electron.

00:17 The mass of that electron is equal to 9 .11 times 10 to minus 31 kilograms.

00:24 Now, we have to determine what is the acceleration.

00:27 And how can we do that? well, take a look at the data.

00:31 So, the electron goes from 5 .4 times 10 to the 5th meters per second to 2 .1 times 10 to the 6 meters per second.

00:40 Then the variation in the velocity is 2 .1 times 10 to the 6 minus 5 .4 times 10 the 5th.

00:51 And this happens while the electron was traveling a distance of 0 .038 meters.

00:57 So all we know is delta v and delta distance.

01:02 How can you know what is the acceleration? we can use the orichelli's equation, which tells us that the final velocity squared is equal to the initial velocity squared plus two times the acceleration times the displacement.

01:17 Then we have the following.

01:19 The final velocity 2 .1 times 10 .6 squared is equal to the initial velocity 5 .4.

01:28 Times 10 to the fifth squared plus two times the acceleration times 0 .038.

01:36 Then we have to solve this equation for the acceleration.

01:40 In order to do that, we have first to do some things.

01:44 So we send this term to the other side to get the following.

01:49 So we get 2 .1 times 10 to the 6 squared minus 5 .4 times 10 to the 6 squared, 10 to the 5th squared is equal to 2 times the acceleration times 0 .038...

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