How is Coriolis acceleration in polar coordinate, different from Coriolis acceleration due to observation in non-inertial frame of reference?

Question

In Kleppner and Kolenkow's book: An Introduction to Mechanics, on page 34 (pasted below) on the topic titled "Acceleration in Polar coordinates", it has been mentioned that:

"when $r$ and $\theta$ both change, then Coriolis acceleration acts which is "real" and is "In contrast" to the Coriolis force which acts in a rotating frame of reference. "

What I'm trying to understand is that:

1. If we are analyzing the situation using polar coordinate system, then if r and theta both are changing, then Coriolis acceleration that acts (which by the way is real according to Kleppner and Kolenkow), is this Coriolis acceleration different from the Coriolis acceleration that would come into play due to Coriolis force, if we analyze the same situation from a rotating frame of reference?

2. Can Coriolis acceleration (real one) and Coriolis acceleration (due to Coriolis force when seen from rotating frame of reference) act simultaneously at a moving body(whose r and theta both are changing with time), if we observe from a rotating frame of reference using polar coordinate system?

Answer 1

Note the subtle difference in the wording.

Kleppner refers to a Coriolis acceleration which in an inertial frame is produced by a real force.
However, in a non-inertial rotating coordinate frame the Coriolis acceleration term with its sign reversed multiplied by the mass of the object is put on the force side of the equation $\vec F = m\,\vec a$ and called the (fictitious) Coriolis force.

Perhaps a simpler example will help.
Constant speed circular motion has an acceleration of $\vec a = -r\,\dot\theta ^2\hat r$ and so in an inertial frame one would write $\vec F_{\text{force causing centripetal acceleration }} = m\,\vec a = -m\,r\,\dot\theta ^2\hat r$.
Now in the rotating frame you observe the object to be stationary and yet it has a force, $F_{\text{force causing centripetal acceleration }}$, acting on it.
To be able to use Newton's second law a fictitious force, $+m\,r\,\dot\theta ^2\hat r$, is introduced and so now $F_{\text{force causing centripetal acceleration }} + m\,r\,\dot\theta ^2\hat r = 0$ and note there is no mention of any acceleration on the right hand side of the equation.

Answer 2

In physics there is unfortunately no general convention as to how the term 'coriolis acceleration' is used.

It's important to be aware that some authors may use the term 'coriolis acceleration' as saying the same thing as 'coriolis force'.

Kleppner and Kolenkov have opted to use 'coriolis acceleration' in a meaning that is distinct from 'coriolis force'.

I have my doubts about the description they give. I think it is ambiguous. It says: "present whenever $r$ and $\Theta$ both change with time."

That description is not particularly restrictive. For instance: let the object be in inertial motion with respect to the inertial coordinate system. Then in polar coordinates (inertial polar coordinate system) both $r$ and $\Theta$ are changing with time.

Hooke's law

Interestingly, there is in fact a case where an object is circumnavigating a point of attraction, resulting in an acceleration with respect to the rotating coordinate system that is in accordance with the expression $2 \Omega v$ (with $v$ for the velocity relative to the rotating coordinate system, and $\Omega$ for the rotation rate of the rotating coordinate system.)

In the animation:
An object is subject to a centripetal force that increases linear with distance to the point of attraction. That force profile is referred to as Hooke's law.

The left hand side shows the motion with respect to the inertial coordinate system. The right hand side shows the motion as seen from a point of view that is co-rotating with the rotating platform

On the rim of the rotating platform 4 sections are indicated, to make it clear that the left hand side shows a platform rotating at a constant angular velocity.

As stated earlier: the centripetal force increases linear with radial distance.

The following parametric equation describes the motion of the object.

$x = a \cos(\Omega t)$
$y = b \sin(\Omega t)$

$a$ half the length of the major axis
$b$ half the length of the minor axis
$\Omega$ 360 degrees divided by the duration of one revolution

So the motion is a superpositon of two harmonic oscillations, perpendicular to each other.

In the animation the radial velocity and the angular velocity of the object are both changing, in a particular correlated way.

The parametric equation can be rearranged as follows:

$$ x = \frac{a+b}{2} \cos(\Omega t) + \frac{a-b}{2} \cos(\Omega t) $$ $$ y = \frac{a+b}{2} \sin(\Omega t) - \frac{a-b}{2} \sin(\Omega t) $$

After transformation of the motion to the rotating coordinate system the motion relative to the co-rotating coordinate system is as follows:

$$ x = \frac{a-b}{2} \cos(2 \Omega t) $$ $$ y = - \frac{a-b}{2} \sin(2 \Omega t) $$

We can think of the motion with respect to the rotating coordinate system as a symmetry that arises from the linear character of Hooke's law: if the centripetal force is exactly according to Hooke's law then the motion with respect to the rotating coordinate system is perfectly circular.

The next step is to obtain an expression for the acceleration of the moving object relative to the co-rotating coordinate system.

Let $a_c$ be the acceleration relative to the co-rotating coordinate system.
The magnitude of the acceleration towards the center of the epi-circle is given by the standard expression for the required centripetal acceleration for sustained circular motion:

$$ a_c = \omega^2r_{epi} \tag{1} $$

(I use the lowercase $\omega$ here for the motion along the epi-circle to distinguish it from the uppercase $\Omega$ that I used for the angular velocity of the system. $r_{epi}$ stands for 'radius of the epi-circle'.)

This expression gives the magnitude. To sustain uniform circular motion the acceleration must be perpendicular to the instantaneous velocity. That gives the direction of $a_c$: perpendicular.

The form of (1) is not practical. The first step to change the form is to convert the radial distance $r_{epi}$ to velocity with respect to the rotating coordinate system. We have the general relation $v=\omega r$ that is valid when the motion is circular motion. Substituting a factor $\omega r$ with $v$:

$$ a_c = \omega v \tag{2} $$

Next next we use the fact that the angular velocity $\omega$ with respect to the center of the epi-circle is twice the angular velocity of the rotating coordinate system with respect to the inertial coordinate system: $\omega = 2 \Omega$

$$ a_c = 2 \Omega v \tag{3} $$

This demonstrates:
When an object is circumnavigating a point of attraction, with the centripetal force according to Hooke's law, then the acceleration with respect to the rotating coordinate system is given by the expression $ a_c = 2 \Omega v$

As a reminder: in every classical mechanics textbook it is stated: the features that characterize 'coriolis effect' are the following:

• the direction of acceleration with respect to the rotating coordinate system: perpendicular to the velocity with respect to the rotating coordinate system.
• the magnitude of acceleration with respect to the rotating coordinate system: $2 \Omega v$

The motion presented in the animation satisfies the above criteria.

Whenever rotation is involved a centripetal force according to Hooke's law tends to arise naturally. Example: the underlying dynamics of a liquid mirror telescope. As the dish filled with Mercury is spinning the cross section of the surface assumes the shape of a parabola. For a rotating liquid the equilibrium shape is the shape such that at every distance to the center of rotation the required centripetal force is provided. So inevitably you get a centripetal force according to Hooke's law.

We have that the circumnavigating motion depicted in the animation is due to a real force, in this case a centripetal force that increases linear with radial distance.

The period of the circumnavigating motion is determined by the magnitude of the centripetal force.

The choice of rotating coordinate system is obvious: use a rotating coordinate system with an angular velocity that matches the period of revolution of the circumnavigating motion.

That is how the rotation rate $\Omega$ ends up in the expression for the magnitude of the object's acceleration with respect to the rotating coordinate system: the only sensible choice of rotating coordinate system is the one that matches the period of the circumnavigating motion.