Solving Parallel Plate Capacitor Pendulum Oscillations

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1. Homework Statement

PLEASE DO NOT DELETE THIS POST, MODS, IT MAY LOOK LIKE THE SAME QUESTION AS BEFORE, BUT IT IS NOT, IT IS A TOTALLY DIFFERENT PART TO THE QUESTION.

consider a parallel=plate capacitor with square plates of side L and distance d (<<L) between them, charged with charges +Q and -Q. The plates of the capacitor are horizontal, with the lowest lying on the x-y plane, and the orientation is such that their sides are parallel to the x and y axis, respectively.

a simple pendulum of length d/2 and mass m, hangs vertically from the centre of the top plate, that can oscillate in the x-z plane.

recall that the differential equation for a mechanical simple pendulum in the gravitational field is ml *theta(double primed)=-mg*theta, where theta is the angular displacement from the vertical. Considering the electrical force only, and neglecting gravity, show that the period of small oscillations of the pendulum around its vertical axis is

T=2pi*sqrt[((L^2)*d*m*epsilon0)/(2Qq)]

The Attempt at a Solution

the notes say
omega0=sqrt[(Ze^2)/(4 pi *spsion 0*m(subscript e) *r^3)]

I tried to understand how this related to the question but don't see an obvious connection

 

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the mass has charge q, i neglected to say

 

[Doc Al]

PLEASE DO NOT DELETE THIS POST, MODS, IT MAY LOOK LIKE THE SAME QUESTION AS BEFORE, BUT IT IS NOT, IT IS A TOTALLY DIFFERENT PART TO THE QUESTION.

If it's a multiple part question, all parts belong in the same thread.

Where's your work?

Hint: Can you derive the period of a pendulum under gravity alone? It's the same problem, only now the force is due to the electric field, not gravity.

 

[ehild]

the notes say
omega0=sqrt[(Ze^2)/(4 pi *spsion 0*m(subscript e) *r^3)]

I tried to understand how this related to the question but don't see an obvious connection

No wonder, as it has no connections with your problem. It can be the angular velocity of an electron around a nucleus with Z protons.

Try to solve the problem using your knowledge and logic instead of trying to find a formula with omega at the left-hand side and wondering what to plug in for z and r, when you have q, Q and L.

ehild

 

[rwisz]

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I would like to start by acknowledging the complexity of this problem and the various factors that need to be considered. The solution involves combining principles from both classical mechanics and electromagnetism, and it requires a thorough understanding of both fields.

To solve this problem, we need to consider the forces acting on the pendulum. In this case, we have the electrical force due to the charged plates of the capacitor and the gravitational force, which we can neglect as per the problem statement. The electrical force follows the inverse-square law, similar to the gravitational force, and it can be calculated using Coulomb's law.

Next, we need to consider the motion of the pendulum in the x-z plane. This can be described using the differential equation mentioned in the problem statement, which is similar to the equation for a simple pendulum in a gravitational field. However, in this case, we need to use the electrical force instead of the gravitational force.

To find the period of small oscillations, we can use the equation for the natural frequency of a simple harmonic oscillator, which is given by ω0=√(k/m), where k is the spring constant and m is the mass. In this case, we can replace k with the force acting on the pendulum and m with the mass of the pendulum. This will give us the natural frequency of the system.

Now, we can use the relationship between the natural frequency and the period of oscillations, which is T=2π/ω0, to find the period of small oscillations of the pendulum. By substituting the force calculated from Coulomb's law and the mass of the pendulum into the equation for the natural frequency, we can derive the final equation for the period of oscillations, which is T=2π√((L^2*d*m*ε0)/(2*Q*q)).

In conclusion, solving this problem requires a combination of principles from classical mechanics and electromagnetism, and it demonstrates the interconnectedness of different fields of science. By understanding the underlying principles and using mathematical equations, we can arrive at a solution that accurately describes the behavior of the parallel plate capacitor pendulum.