Recent content by anuttarasammyak

The data is not written suggests that they do not matter, a big hint from your teacher, I suppose, :smile: We may check our reasoning by our experience of jet coaster ride or ging a down smooth slope on bycycle with/without braking.

Another solution F is on AC. ##\angle ACD = \pi/4## ##\angle AOD=\pi/2## ##\triangle AEO \equiv \triangle OGD## ##EO=GD## Thus ##r_1^2+r_2^2=R^2## where ##r_1## and ##r_2## are radii of two smaller circles.

As an example, for b=R/2, this vector is $$(\sqrt{3},0,1)^T=2* ( \frac{\sqrt{3}}{2}, 0 , \frac{1}{2} )^T$$ I would like to understand how this (x,0,z) type vector of magnitude ##\frac{R}{b}## tells us about the final angle position of the ball.

x-component of this vector increases from 0 to infinity as the path latitude increases from 0 to 90 degree. Right?

NOT ELEGANT solution The eqation of the circles $$x^2+y^2=R^2$$ $$(x-x_1)^2+y^2=R^2-x_1^2$$ $$(x-x_2)^2+y^2=R^2-x_2^2$$ By subtraction of the two smaller circles equatins, we see they touch at ##x=x_1+x_2,y=0##. Thus $$x_1^2=R^2-x_2^2$$ $$x_1^2+x_2^2=R^2$$ Area of the sum of small circles...

@wrobel thanks to your result of the rotation angle of the globe, which is around North-South axis, $$2\pi(1-\sqrt{1-\alpha})$$ where $$0<\alpha=(\frac{b}{R})^2\ (1-\frac{1}{(1+\frac{mR^2}{J})^2})<1$$ , I noticed that my conjecture in post #2 holds for ##b=R##, latitude 0 contour, but it does...

An example of infinite collection of infinite sets is ##\{p^n\}## where p is prime number. We can do similar on n and so on to get infinite sequence of infinity. It is .. as it is.

Let consuming time to infinitesimal part of pass ##dl## be ##dt## with speed ##v##. $$dt=\frac{dl}{v}$$ v depends on where on the path the ball is, ##v=v(l)##. Integrating it along the path L, consuming time to go through the path L is $$T =\int_L \frac{dl}{v}$$ Conservation of energy...

What do you mean by slicing numbers? Could you give us some examples?

v<c holds for Inertial Frame of Reference (IFR) s. Two buckets and the spinning platet Earth are rotation systems which are not IFR. There things far away from the observer can and must have v >c . Moving actually or inactually should not matter in the sence of relativity theory.

Though I am lazy and awkward in pursuing numerical calculation, I am curious to know it supports my conjecture post #2 or not.

We on planet Earth observe that the stars have rotation orbit of 24 hours period. ##\omega R > c ## for R > 4 * 10^12 m　～　30 a.u. Is it OK in your point ?

The insect moving around the latitude circle has angular momentum both in the z-direction and in the direction normal to the latitude circle. The globe has the same amount of angular momentum with the opposite sign. Therefore, the globe rotates not only about the z-axis but also about the axis...

We draw latitude lines and longitude lines on the globe and set it in a recess for the free rotation with the north pole top initially. We let a beetle walk around on a latitude x line and make it complete one rotation. After the rotation the north pole or NS axis tilts with angle...

As for movements with regard to trajectory circle, the cases : a. a bettle spins at the circle center b. long snake forms a ring shape and crawls around the circle c. four beetles with 90 degree angles separation, like North, South, East, and West, crawl around the circle. d. two beetles with...