Foucault pendulum - the physics and maths involved

[Thomas2054]

I have been looking at this paper (http://www.phys.unsw.edu.au/~jw/pendulumdetails.html) on the details of Foucault's pendulum. I am interested in understanding the details, but am having some trouble. My purpose is to study this as an example of how the analysis is done. I'd like to ask some questions and then plod along on my own, until the next question.

Let me start with this question. In the paragraph above equation (2) Wolfe has the equation: r" = r"p − r"o. I see where he then derives equation (2). However, it is equation (3) with which I am having a problem.

r" = r"p − r"o, which can be expanded to
mr" = mr"p − mr"o. Equation (1) is substituted for mr"p and equation (2), multiplied by m, is substituted for mr"o.

Why in equation (3) does the last term not have a minus sign?

Thanks.

Thomas

 

[Doc Al]

Why in equation (3) does the last term not have a minus sign?

I haven't looked it over too closely, but I suspect that the RHS of (2) is missing a minus sign.

 

[charng]

The Foucault pendulum is a fascinating example of the interplay between physics and mathematics. It demonstrates the principles of conservation of energy and angular momentum, as well as the effects of the Earth's rotation on a swinging object. Let's take a closer look at the equations involved in the analysis of this pendulum.

First, we have the equation r" = r"p − r"o, which represents the acceleration of the pendulum bob (r") as the sum of the acceleration due to the pendulum's motion (r"p) and the acceleration due to the Earth's rotation (r"o). This equation can be expanded to mr" = mr"p − mr"o, where m represents the mass of the pendulum bob.

In equation (3), we are solving for the acceleration of the pendulum bob (r") using the equations for r"p and r"o. The first term, mr"p, is derived from equation (1), which represents the acceleration of a simple pendulum. The second term, mr"o, is derived from equation (2), which represents the acceleration of an object due to the Earth's rotation. However, in this case, we are considering the acceleration of the pendulum bob due to the Earth's rotation, rather than the acceleration of the Earth's rotation itself. Therefore, the minus sign is not included in this term.

I hope this helps clarify the equations involved in the analysis of the Foucault pendulum. It is a complex system, but by breaking it down into individual components, we can better understand the physics and math behind it. Good luck in your studies!