Need help - about Coulomb's Law and SHM

In summary, the conversation discusses the application of Coulomb's Law and simple harmonic motion in a scenario involving two fixed positive charges and a negatively charged particle placed between them. The net force acting on the particle is calculated and it is shown that it follows a simple harmonic motion with a period that can be calculated using the given information. The individual discussing the topic also mentions the need to redo the calculation in order to show that the acceleration of the particle is equal to the negative of its displacement multiplied by the square of the angular frequency.
  • #1
elvisphy
1
0
need help -- about Coulomb's Law and SHM

Two positive charges +Q are held fixed a distance d apart.
A particle of negative chage -q and mass m is placed midway between them, then is given a small displacement perpendicular to the line joining them and released. Show that the particle describes simple harmonic motion and find the period.
----------------------------
i've found that the net force acting on the -q is
F = - (qQd) / [4(pai)(z)(y^2 + (0.5d)^2)]^(3/2) = ma
where z is the permittivity, and y is the displacement of -q

how i can show that a = - w^2 y?
 
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  • #2
Well, you need to redo the calculation of the total net force.

Daniel.
 
  • #3


Hello,

Thank you for reaching out for help with Coulomb's Law and Simple Harmonic Motion (SHM). I am happy to assist you in understanding these concepts.

First, let's review Coulomb's Law. It states that the force between two charged particles is directly proportional to the product of their charges (q and Q) and inversely proportional to the square of the distance between them (d). This is represented by the equation F = kqQ/d^2, where k is the proportionality constant.

Now, let's look at the situation described in your question. We have two fixed positive charges, +Q, held d distance apart, and a negative charge, -q, placed midway between them. When the -q charge is given a small displacement, it will experience a force due to the electric field created by the two fixed charges, according to Coulomb's Law.

Using the equation for Coulomb's Law, we can calculate the net force on the -q charge:

F = kqQ/d^2

= (1/4πε0)(qQ/d^2) (where ε0 is the permittivity of free space)

= (qQd)/(4πε0d^3)

= (qQd)/(4πε0)(d^2)^3

= (qQd)/(4πε0)(y^2 + (0.5d)^2)^3 (since d = 2y)

= ma (according to Newton's Second Law, F = ma)

Therefore, we can see that the acceleration (a) of the -q charge is directly proportional to its displacement (y) and is in the opposite direction. This is the defining characteristic of SHM.

To show that a = -ω^2y, we can rearrange the equation for acceleration:

a = (qQd)/(4πε0)(y^2 + (0.5d)^2)^3

= (qQd)/(4πε0)(y^2 + (0.5d)^2)^2 * (y^2 + (0.5d)^2)^1

= (qQd)/(4πε0)(y^2 + (0.5d)^2)^2 * (y^2 + (0.5d)^2)^1/2

= (qQd)/(4πε0)(
 

Related to Need help - about Coulomb's Law and SHM

1. What is Coulomb's Law?

Coulomb's Law is a fundamental principle in physics that describes the electrostatic interaction between two charged particles. It states that the force between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.

2. How is Coulomb's Law mathematically expressed?

Coulomb's Law can be mathematically expressed as F = k(q1q2)/r^2, where F is the force between the two charges, q1 and q2 are the magnitudes of the charges, r is the distance between them, and k is a constant known as the Coulomb's constant.

3. What is the relationship between Coulomb's Law and SHM?

Coulomb's Law and Simple Harmonic Motion (SHM) are related through the concept of the electric force being a restoring force. Just like how a spring exerts a force to bring an object back to its equilibrium position in SHM, Coulomb's Law describes how the electric force between two charged particles brings them back to their equilibrium positions.

4. How does distance affect the force in Coulomb's Law?

The force between two charges in Coulomb's Law is inversely proportional to the square of the distance between them. This means that as the distance between the charges increases, the force between them decreases. This relationship is known as the inverse square law.

5. What are some real-life applications of Coulomb's Law?

Coulomb's Law has many real-life applications, such as in the design and operation of electronic devices, understanding the behavior of atoms and molecules, and in the study of electromagnetic fields and radiation. It is also used in industries such as telecommunications, energy production, and medical technology.

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