Revolution under the influence of a central force

[src2206]

Hello friends and fellow Physics enthusiasts

At the beginning let me confess that I am more of a sleeping member though I do follow this forum quite closely. Today, I am posting to seek some help from fellow Physics educators. I hope this is the right section of the forum to post, if not, I would like to request the moderator to be kind enough to move it to an appropriate section.

A few days back, I had an argument with a person on a social media platform regarding the effect of the absence of diurnal motion (rotation of Earth on its own axis). According to him, there would be a change in "Day-Night" even if only the Earth's revolution around the Sun is present. In support, he also showed a diagram which I am enclosing with this post. I disagreed, arguing that it is not possible that a point on the periphery of Earth will change its position relative to the Sun if there is no rotation of Earth. He then argued that if any other external star system/star is taken as a reference frame then it is possible.

I would like to request you all to help me out in the following two points:

1. Is he right? Am I making any mistake in understanding? Is the diagram attached right or wrong?

2. I am unable to formulate a pointed and logical reply in an organized manner to counter his argument. Provided he is wrong, I would like to request you to help me on this.

Under the influence of a central force and in absence of any torque, how can the position of a point on the periphery of the object change with respect to the central point or any other reference frame?

I hope I have clarified my dilemma and question clearly to the members. If you require any other input, please feel free to ask. I am unable to post the conversation, as it was in my mother tongue "Bengali" and I am sure most of the members here would have no use for that.

Thank you all, waiting eagerly for your reply.

 

[phinds]

View attachment 234236
1. Is he right?

yes

Am I making any mistake in understanding?

you have to be since you're wrong.

Think about this. You stand at a point on the Earth facing the sun. As the Earth rotates around the sun you continue to point towards exactly the same distant star. When The Earth gets to the other side of the sun 6 months later, you are facing directly away from the sun.

 

[src2206]

Let's put it in another perspective. If I tie a pebble with a string and rotate it around my finger, will the position of the knot change?
Please understand, I am not trying to argue, but I am trying to clear the misunderstanding.

 

[Tom.G]

Try putting your hands in front of you and point both index fingers up, finger nails facing away from each other.
Consider one fingertip to be the Sun and the other as the Earth.
You are the finger nail on the Earth.
Now without rotating either hand, cross your arms at the wrists.
Are you now facing the Sun?

 

[src2206]

Try putting your hands in front of you and point both index fingers up, finger nails facing away from each other.
Consider one fingertip to be the Sun and the other as the Earth.
You are the finger nail on the Earth.
Now without rotating either hand, cross your arms at the wrists.
Are you now facing the Sun?

Thanks so much, Tom. I totally agree. But why the example of rotating something by a string is not conforming with this idea?

 

[Tom.G]

But why the example of rotating something by a string is not conforming with this idea?

Probably because in keeping the string taut, you are rotating the pebble thereby keep the knot always pointing to your finger.

 

[src2206]

Nope, there is no rotation of the pebble about its own axis

 

[Tom.G]

If there was no rotation it would behave as your finger nail did. Can you draw a sketch or something to better describe the situation with the string and pebble?

 

[PeroK]

Hello friends and fellow Physics enthusiasts

At the beginning let me confess that I am more of a sleeping member though I do follow this forum quite closely. Today, I am posting to seek some help from fellow Physics educators. I hope this is the right section of the forum to post, if not, I would like to request the moderator to be kind enough to move it to an appropriate section.

A few days back, I had an argument with a person on a social media platform regarding the effect of the absence of diurnal motion (rotation of Earth on its own axis). According to him, there would be a change in "Day-Night" even if only the Earth's revolution around the Sun is present. In support, he also showed a diagram which I am enclosing with this post. I disagreed, arguing that it is not possible that a point on the periphery of Earth will change its position relative to the Sun if there is no rotation of Earth. He then argued that if any other external star system/star is taken as a reference frame then it is possible.

View attachment 234236

I would like to request you all to help me out in the following two points:

1. Is he right? Am I making any mistake in understanding? Is the diagram attached right or wrong?

2. I am unable to formulate a pointed and logical reply in an organized manner to counter his argument. Provided he is wrong, I would like to request you to help me on this.

Under the influence of a central force and in absence of any torque, how can the position of a point on the periphery of the object change with respect to the central point or any other reference frame?

I hope I have clarified my dilemma and question clearly to the members. If you require any other input, please feel free to ask. I am unable to post the conversation, as it was in my mother tongue "Bengali" and I am sure most of the members here would have no use for that.

Thank you all, waiting eagerly for your reply.

The attached diagram shows what would happen if the Earth were not rotating on its axis.

If point P were attached to the Sun, then the Earth would have to rotate once per year. For example, if this were an experiment with the central force being provided by a rope between a central point and an object, then the object would have to rotate once per orbital revolution.

If you analyse this carefully, then you'll see that in this case you do not have a perfect central force. The force is applied to a single point on the surface, not to the centre of mass. The rope, therefore, also provides a torque in order to rotate the object.

 

[Dale]

Is he right? Am I making any mistake in understanding? Is the diagram attached right or wrong?

Your friend is correct.

The question can be addressed using a laser ring gyroscope attached to the Earth to measure the rotational velocity of the earth. If the rotational velocity were exactly $2\pi/86400 \: rad/s = 7.2722 \: 10^{-5}\: s^{-1}$ then if it were changed to 0 there would be no day night cycle. The actual measured value is $7.2921150 \: s^{-1}$. Not coincidentally these two values differ by about one part in 365.

 

[A.T.]

But why the example of rotating something by a string is not conforming with this idea?

The force of the string is usually not acting on the center of mass of the object, so it can create torques, which force the object to spin at the same same rate it revolves around the center.

 

[A.T.]

I disagreed, arguing that it is not possible that a point on the periphery of Earth will change its position relative to the Sun if there is no rotation of Earth.

The red part is your misunderstanding. It should be "relative to inertial coordinate axes". In the image the body axes of the planet stay fixed relative to the axes of the inertial image axes, so the planet is not rotating.

 

[sophiecentaur]

I'm not sure that I understand what the OP is trying to say (or his on-line friend). Rather than trying to unpick all that, I think it's best to state the accepted situation and then see if that first post fits - and it doesn't seem to.
Despite the fashion to discuss these things in terms of various different frames of reference, I think the frame of the Fixed Stars is by far the easiest one to discuss and most familiar with most people. In a period of a year, the Sun doesn't move significantly so let's call the Sun stationary and non rotating. To a first approximation, the Sun has no influence on our rotation. (Tidal locking with the Moon etc. is not relevant to this first stab at things)
If the Earth were in a circular orbit round the Sun but not rotating relative to the fixed stars, then one point on the Earth would be facing in the direction of one particular star all the time. We would go round the Sun once a year and our 'day' would last exactly one year too.
In fact, we do rotate - daily. We have 365 1/4 Noons (Sun overhead) each year and the fixed star we were using will be overhead one more time than the Sun, during our solar year and the sidereal day (time between our star being overhead) is about 4 minutes shorter. There's enough difference in the time scales of the Solar System and the Galaxy to allow this approximation to work well enough for starters.
There could be endless Q and A about this topic but, unless the OP actually asks the 'right' questions, he will not get the 'right' answers. I suggest the OP Googles the term Sidereal Day and gets familiar with what it all implies. This topic is not altogether intuitive and there are a number of YouTube videos that could help - take your pick.
PS The title of the thread seems to involve two concepts that are not related.

 

[src2206]

Thank you all for your replies and insights. I have been extremely busy during the last few days, so could not reply in a timely manner, my apologies for that. I would surely go through the points raised here in detail and come back. Though one thing I must disagree, that my title consists of two different concepts.

 

[sophiecentaur]

Though one thing I must disagree, that my title consists of two different concepts.

Translation and rotation are two different concepts - as are Momentum and Angular Momentum. I would say that common scientific usage of the word "rotation" (and especially in the context of astronomical motion) implies angular momentum around the axis of the object itself. If the term is not used exclusively for that, it would need to be qualified every time, to distinguish it from "orbit" or "trajectory" under an external force.
Can you find a significant number of examples where an orbit is described as rotation? (You may well find one or two but that goes for many examples of inappropriate use of words.)

 

[Mister T]

If I tie a pebble with a string and rotate it around my finger, will the position of the knot change?

The pebble will rotate once per revolution. If that pebble were a planet the length of the day and the length of the year would be the same.

There's an old riddle that goes something like this. You have two coins, one has a diameter of one inch and the other has a diameter of three inches. Both lie flat on a table top, touching. The smaller coin is rotated in such a way that it doesn't slip against the larger coin as it rolls around it. When the coin first returns to its original location how many times will it have rotated?

 

[A.T.]

If I tie a pebble with a string and rotate it around my finger, will the position of the knot change?

The pebble will rotate once per revolution.

Yes

If that pebble were a planet the length of the day and the length of the year would be the same.

No, there would be no day-night-cycle in that situation (if the finger is the sun).

 

[Dale]

No, there would be no day-night-cycle in that situation (if the finger is the sun).

It would have no solar day, but it would have a sidereal day equal to its year.

 

[A.T.]

It would have no solar day, but it would have a sidereal day equal to its year.

Right, and in the diagram in the OP it would be the other way around.

 

[src2206]

My dear friends and fellow Physics enthusiasts

At the very beginning, I offer my unconditional apologies for such a delayed response. Unfortunately, I was too tied up to come back here and post a coherent reply to this discussion.

What I am looking for is a mathematical way to represent this mathematically in the equation of motion, the way we can mathematically show the Coriolis force while analyzing rotational motion. I understand that there is going to be a change in the location of a point on the periphery of Earth relative to the Sun. I am trying to figure out a way to mathematically show/express the "origin" of this "pseudo-rotation".

I hope I have made my point clear. Thank you again for all your support.

 

[phinds]

I am trying to figure out a way to mathematically show/express the "origin" of this "pseudo-rotation".

Uh ... what is it that you think is "pseudo" about this very real rotation (relative to the sun)?

 

[src2206]

Uh ... what is it that you think is "pseudo" about this very real rotation (relative to the sun)?

Thanks so much. I actually used the word "pseudo" to signify that the rotation appears with reference to the sun, and there is no real rotation on its axis. My choice of words may be wrong, but I hope I have conveyed the meaning.

Anyways, is it possible to have the derivation of an equation in which this motion can be mathematically represented?

Thanks again.

 

[A.T.]

What I am looking for is a mathematical way to represent this mathematically in the equation of motion, the way we can mathematically show the Coriolis force while analyzing rotational motion.

The Coriolis force appears if you use a rotating frame of reference, not if you analyze rotational motion. If you analyze rotational motion from an inertial frame, there is no Coriolis force.

I understand that there is going to be a change in the location of a point on the periphery of Earth relative to the Sun.

If the planet doesn't rotate relative to the inertial frame, then any point of it moves on a circle in the inertial frame. All these circles have the same radius, but different centers.

If the planet rotates with the same rate and direction as it orbits, then any point of it moves on a circle in the inertial frame. All these circles have the same center, but some different radii.

 

[Dale]

Anyways, is it possible to have the derivation of an equation in which this motion can be mathematically represented?

I am not sure exactly what motion you want an equation for. Can you be specific? What are the variables that you are hoping to relate through this equation.

 

[src2206]

Hello Dale and A.T.

Thank you for your reply.

I am trying to figure out the equation of motion of Earth in case there is no rotation, and to figure out whether that equation would include a term to show the "rotation" of a point on the periphery of Earth relative to Sun. My ultimate aim is to draw a clear comparison between this "revolution" of Earth and rotation of a stone tied by a string.

 

[A.T.]

I am trying to figure out the equation of motion of Earth in case there is no rotation, and to figure out whether that equation would include a term to show the "rotation" of a point on the periphery of Earth relative to Sun.

If there is no rotation, why would there be a term showing the rotation?

Maybe you have to define the reference frame properly. I assumed "relative to the Sun" means "in the inertial rest frame of the Sun". Are you actually talking about the rotating rest frame of Sun's and Earth's centers?

 

[Dale]

It is still a little unclear what variables you want in this formula, but here is a simple one that might be what you are looking for:
$$N_{sidereal}=1+N_{solar}$$Where N is the number of (sidereal or solar) days per year.

In the case of the pebble on the string, $N_{solar}=0$ so $N_{sidereal}=1$

 

[A.T.]

It is still a little unclear what variables you want in this formula, but here is a simple one that might be what you are looking for:
$$N_{sidereal}=1+N_{solar}$$Where N is the number of (sidereal or solar) days per year.

In the case of the pebble on the string, $N_{solar}=0$ so $N_{sidereal}=1$

What if $N_{sidereal}=0$ ? Maybe its better to write this as:
$$|N_{sidereal}-1|=N_{solar}$$

 

[Dale]

What if $N_{sidereal}=0$ ? Maybe its better to write this as:
$$|N_{sidereal}-1|=N_{solar}$$

If $N_{sidereal}=0$ then $N_{solar}=-1$ meaning that the sun rises in the west and sets in the east once per year. The formula assumes prograde rotation, so retrograde rotation is indicated by negative numbers.

 

[A.T.]

If $N_{sidereal}=0$ then $N_{solar}=-1$ meaning that the sun rises in the west and sets in the east once per year.

Yeah, I got that. Just though that the concept of negative days might be confusing.

 

[Dale]

Yeah, I got that. Just though that the concept of negative days might be confusing.

You are right. There is a separate formula for retrograde rotation, which gives positive days. But I would rather use one formula with negative numbers than two formulas with positive numbers.