Pendulum and rotation

neufneufneuf

It's very strange, I don't find where I'm wrong in this scene. When I test a circle on an ellipse, nothing happens, but an ellipse on a circle this give more and more energy. I try to change all parameters but nothing change the result. Have you test ? It seems the energy added arrive when the ellipse change its direction of angular velocity.


With Algodoo, even with a linear movement, the energy is not fixed.

dlouth15

dlouth15 wrote: »

I think the reason you are going wrong is that you are skipping important steps in the derivation of the formulas.

Anyway it is taking to long to explain so here it is

[latex]$$F_{x}=\frac{GM_{1}M_{2}}{\pi}\int_{r=0}^{1}\int_{\theta=0}^{2\pi}\frac{\left(d-r\cos\theta\right)}{\left(\left(d-r\cos\theta\right)^{2}+r^{2}\sin^{2}\theta\right)^{3/2}}r\mathrm{\, d\theta\,\mathrm{d}}r$$[/latex].

This is then used to calculate the position of the centre of gravity.

Here's a simple program that computes the COG. It doesn't do the integral based on polar coordinates above but simply fills a circle of unit radius with points, calculates the force in the x-direction for each of the points. The resultant force is then used to calculate the cog. The program dumps out a distance (in units of radii of the disk) and then the position of the cog from the centre of the disk (also in units of radii). The results can then be plotted.

#include <iostream>
#include <math.h>
using namespace std;

double cog(double d){
	double TotForce_x = 0.0d;
	long count = 0;
	for (double x = -1.0d; x <= 1.0d; x += 0.001d){
		double maxY = sqrt(1.0d - x*x);
		for (double y = -maxY; y <= maxY; y+=0.001d){
			count ++;
			TotForce_x += (d-x)/pow(y*y+(d-x)*(d-x), (double) 3/ (double) 2);
		}
	}
	double AvgForce_x = TotForce_x / (double) count;
	double distFromAttractor = 1 / sqrt(AvgForce_x);
	double cog = d - distFromAttractor;
	return cog;
}

int main() {
	for (double d = 1; d <= 5.0d; d+= 0.01d ){
		cout << d << '\t' << cog(d) << endl;
	}
	return 0;
}

Oh and happy New Year!

neufneufneuf

Happy New Year too Dlouth15 ! thanks for the program.

1/ You said the force from wall is applied to center of mass not to center of gravity. It's logical if the energy is conserved. I would to verify this point. How can I do ? I look at each point and see if force from wall is perpendicular for each movement of each point, is it good like that ?


2/ If I think with all points of a circle, each point move like a cycloid (if the wheel turn before and no friction): http://en.wikipedia.org/wiki/Cycloid. So, for me if I'm looking for the works of the force from wall, a part works in positive and a part in negative, it is true ? it's possible to sum the work from force and movement of each point, it must be 0. But the force from wall increase with distance. I know it's possible to think with mathematical equation but if I think like that the sum of works of force from wall is not zero. I don't know how to draw the system for "see" good forces.

dlouth15

neufneufneuf wrote: »

Happy New Year too Dlouth15 ! thanks for the program.

1/ You said the force from wall is applied to center of mass not to center of gravity. It's logical if the energy is conserved. I would to verify this point. How can I do ? I look at each point and see if force from wall is perpendicular for each movement of each point, is it good like that?

The force from wall is called the normal force and is always perpendicular to the wall at the point of contact. Depending on the geometry of the object it may or may not go through the centre of mass. In the case of a disk of uniform density it will go through the centre of mass due to its shape.

These are basically rules that have been built up over the years as a way of calculating the motion of objects. For example, I drew a diagram a while back on this thread that allows you, if you know the force due to gravity, to calculate the acceleration of the object down the slope of the outer wall and also the amount of force the object is exerting on the wall itself.

2/ If I think with all points of a circle, each point move like a cycloid (if the wheel turn before and no friction): http://en.wikipedia.org/wiki/Cycloid. So, for me if I'm looking for the works of the force from wall, a part works in positive and a part in negative, it is true ? it's possible to sum the work from force and movement of each point, it must be 0. But the force from wall increase with distance. I know it's possible to think with mathematical equation but if I think like that the sum of works of force from wall is not zero. I don't know how to draw the system for "see" good forces.

You see the diagram that I have included in this post? You need to draw something similar for the example you have given. The only difference is that you have a flat rather than curved surface. When you have done this (and done it properly) you will have the answer to your question.

neufneufneuf

When you have done this (and done it properly) you will have the answer to your question.

I'm ok, if I build a diagram with the center of mass, all is fine, the force from wall is always perpendiculary to the object.

I would like to think only with points, not the center of mass, the result must be the same. A circle is composed with a infinite number of point. I can draw all trajectory for each point. At a time t (look at image), all trajectories have a slope with an angle different but it's only a part of circle, another part (in the same time) is far away and slopes are negative. In a homogeneous field all is fine, but here the normal force increase (or decrease). For me, the normal force works, so what's wrong with my diagram ?

dlouth15

neufneufneuf wrote: »

I'm ok, if I build a diagram with the center of mass, all is fine, the force from wall is always perpendiculary to the object.

I would like to think only with points, not the center of mass, the result must be the same. A circle is composed with a infinite number of point. I can draw all trajectory for each point. At a time t (look at image), all trajectories have a slope with an angle different but it's only a part of circle, another part (in the same time) is far away and slopes are negative. In a homogeneous field all is fine, but here the normal force increase (or decrease). For me, the normal force works, so what's wrong with my diagram ?

Draw the diagram as I have suggested. All will become clear.

neufneufneuf

Yes, with diagram like you drawn it's ok.

I come back to center of gravity. If an object attrack another object we take the cg not the center of mass. So, here the trajectory of the cg is like the image, a part works and another part seems to be perpendicular to the trajectory, this part don't works ? I don't understant.

dlouth15

neufneufneuf wrote: »

I come back to center of gravity. If an object attrack another object we take the cg not the center of mass. So, here the trajectory of the cg is like the image, a part works and another part seems to be perpendicular to the trajectory, this part don't works ? I don't understant.

No you can take the centre of mass because the centre of gravity is in a straight line between the centre of mass and the point of attraction.

Work out the force using [latex]$$F=\frac{GM_{1}M_{2}}{\pi}\int_{r=0}^{1}\int_{\theta=0}^{2\pi}\frac{\left(d-r\cos\theta\right)}{\left(\left(d-r\cos\theta\right)^{2}+r^{2}\sin^{2}\theta\right)^{3/2}}r\mathrm{\, d\theta\,\mathrm{d}}r$$[/latex] for the disk and apply that force to the centre of mass.

This is why the diagram I suggested you draw is valid.

neufneufneuf

ok, thanks !