When the beautiful symmetry of Helen of geometry meets the inner beauty of Inertia

February 2015
Jaroslav Kopernicky
Jan Onderco


This paper builds on the fundamental understanding that inertia is a manifestation of mass. Inertia is an inner attribute, an intrinsic property of mass that causes many physical phenomena that are being observed by scientists. One of them is the centripetal and centrifugal forces of a constant uniform circular motion.
This paper focuses on explaining how inertia is a reason for breaking of symmetry in cycloidal motion.
This paper explains the effective mass increase, the time dilation, and the length contraction of an object moving at a constant velocity as effects caused by inertia.
The absolute frame of reference and the inertia are the fundamental building blocks of the world as we know it.

The circular motion

Circular motion
Figure 1: The Circle

Nomenclature summary

$\theta$ – The angle of the rotation of the circle in radians
$\omega$ $\space=\frac{d\theta}{dt}$ ; The angular velocity of the circle
$d\theta=dt$ ; $t$ corresponds to the angle $\theta$ therefore $\omega=1$ in our plots
$v$ – The velocity of the moving reference frame
$r$ – The radius of the circle
$A$ – The axis
$B$ – The body or a steel ball with mass $m$
$a_{cp}$ – The centripetal acceleration
$a_{ct}$ – The centrifugal acceleration

It all starts with an axis. The axis has to be backed up by a mass and its inertia. That’s the starting point of any circular motion analysis. If there would be no axis with its mass and its inertia then there is no circular motion.

Having said that let us assume there is an axis $A$ and a steel ball $B$ with its mass $m$ attached to the axis with an elastic spring. When the radius is $r$ then centripetal acceleration is $a_{cp} = r \omega^2$. The centripetal force is $m a_{cp}$ which is the force exerted by the spring on the ball. But we cannot forget Newton’s third law: “When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.”

Therefore there is a real centrifugal force $m a_{ct}$ of the same magnitude as the $m a_{cp}$. The centrifugal force is real because it can do work i.e. stretching the spring. A force that can do work cannot be called fictitious force. A force that does work is the acting force of the first part of Newton’s third law. Here is one example of the said case (Fig 2).

Circular motion
Figure 2: The Cycloid

Two bodies $B1$, $B2$ connected by a string rotating and translating without any physical axis. Assuming constant velocity $v$ and $\omega$ body $B2$ does work on body $B1$ but because $B1$ and $B2$ are far apart and not in contact then the only way to do work is by pulling. If the centrifugal force $m a_{ct}$ is not a real force and it does not ‘exist’ in this case then the only way that $B2$ can do work is by pushing the string with the centripetal force $m a_{cp}$. That’s just not going to happen.
Therefore we are left with a conclusion that the acting centrifugal force $m a_{ct}$ is as real as it gets. The centripetal and the centrifugal forces are inseparable twins.

The Cycloid motion

Now let us combine a constant circular motion (Figure 1) with a constant linear translatory motion. We’ll do analysis from the inertial reference frame that is not moving with the axis $A$.

When the axis $A$ moves $dx$ it takes time $dt$ for the ball $B$ to move due to its inertia and because we are using the elastic spring that will stretch then the radius will increase by $dr$. This will continue until we reach the top of the cycloid. The velocity will be $2v+dv$ at this point in time. Finding a proper $k$ for the elasticity where the radius stays longer after a few cycles should not be a problem.
This is the isochronic pendulum:
The Isochronic Pendulum
Figure 3: The Isochronic Pendulum

The ball $B$ starts at $R$, then is released and as it is going down the radius stretches; after a few cycles the radius stays longer with a proper rate of spring constant when compared to stationary hanging ball on the spring. The body $B’$ moves on a prolate cycloid now (Figure 4).

The Isochronic Pendulum
Figure 4: The Prolate Cycloid

When a properly calibrated elastic spring/rod/rubber band/… – ball system for a constant rotation without translatory motion moves at a constant velocity then the $dr$ change is an indicator of a constant linear motion of the system.

It should be noted that the above mentioned scenario is just one of the many. An oscillation would be the most common end-result.

The conclusions

The longer radius $r’$ gives us bigger $a_{cp}$ acceleration – bigger force; the effective mass increases.
The longer radius $r$ gives us longer period $T$; the time dilation.
This is my favorite, if there are two balls $B1$, $B2$ connected by an elastic spring and rotating in a cycloid motion then these balls are not in a potential energy well. They want to fall into it by tilting and going away from the cycloid motion and coming to a circular motion as a propeller. This is the length contraction. The balls and the spring in a gyro device will give us linear motion indication in the space when three devices starting in three right angled planes will end up with their axis being parallel to the direction of the motion.
The centrifugal force is as real as it gets! That’s the right hand rule in itself, $90^o$ to the velocity vector!