Physical Variables of Circular Motion

Question

To me, the definitions of variables involved in circular motion are rather confusing (perhaps due to the lack of understanding on my part), hence the question.

As I understand it, there are two overarching kinds of motion involved in circular motion:

1. Tangential motion – motion that is tangent to the circle (and therefore often called 'linear').
2. Radial motion – motion that is directed towards the center of the circle.

Based on these two categories, I grouped up the variables involved:

• Tangential Motion

  • Displacement $s$ (m)
  • Linear velocity $v$ (m/s)
  • Linear acceleration $a$ (m/s$^2$) this only exists in non-uniform acceleration.

• Radial Motion

  • Angular displacement $\theta$ (rad)
  • Angular velocity $\omega$ (rad/s)
  • Angular acceleration $\alpha$ (rad/s$^2$)

Here is the list of problems I have encountered that need solving:

1. Where does centripetal acceleration and centripetal force fit into the picture?
2. How does the direction of angular velocity ($\omega$) suddenly change from being perpendicular to the plane of rotation to becoming along the plane of rotation?

3. What is the difference between angular acceleration and centripetal acceleration?

   • Is it because angular acceleration $\alpha = \dfrac{d \omega}{dt}$ whereas centripetal acceleration $a = \dfrac{v^2}{r}$?

   • If so, then there wouldn't be a difference between tangential (i.e., linear) acceleration and centripetal acceleration right?

Okay, I think that's about it. Really appreciate it guys!

Answer 1

Question 1

First of all, note that linear acceleration actually always exists in circular motion, and that it always points towards the center of the circle. That is, as a particle tries to move in a straight line, it keeps on getting pulled towards the center of the circle. Try imagining a particle moving in a straight line getting pulled towards a point to the side of its path to get an intuition of this.

Another way to think about this is to compare it with orbits. Try throwing a baseball really, really fast. It falls back to the ground, right? Try throwing it even faster. It falls back to the ground, but after a longer period of time. Notice that, because the Earth is a sphere, the ground actually curves downwards. What if you threw the ball so fast, so far, that by the time gravity finally pulls it "down", the ground had already curved downwards so much that it "misses" the ground? The ball then basically is constantly falling downwards but constantly missing the ground, because the ground actually curves downwards faster than the ball can land.

In circular motion, the particle is always feeling acceleration towards the center of the circle. If this acceleration stopped, the particle would just keep on going in a straight line.

This "type" of acceleration comes up a lot in Physics. Acceleration happens everywhere, but acceleration towards a central point is a special case of acceleration. We like to give names to things, and in this case, we call it "centripetal", or "center-seeking". Makes sense, right?

To rephrase -- linear acceleration is any time the velocity of an object changes. This can happen in uniform acceleration (i mean, it's in the name...uniform acceleration is acceleration), or non uniform acceleration...any time a velocity changes, we call that change an acceleration.

Sometimes we like to study a special case of acceleration, where the acceleration always points towards a center point. We give a label to this special case of study, for convenience -- we call it "centripetal".

There isn't anything necessarily magical about this definition. It's just like us saying

"hey...it's interesting if we study acceleration always towards the center."

"hey so you what happens when there is acceleration always towards the center?"

"hey so did you hear about that acceleration always towards the center people are thinking about"

"you know if we imagine an acceleration always towards the center, you get interesting things. like circular motion."

"someone needs to think of a better name for that because it's annoying always saying 'acceleration always towards a center point' every time we talk about it."

"oh i know how about 'centripetal' acceleration idk"

"oh ok that works whatever."

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Question 2

Okay so here's the thing about angular coordinates --

Every "concept" in linear coordinates has an "analog" in angular coordinates.

This shouldn't be that much of a surprise; coordinate systems are an invention of man, and the world does not care about it. There should be meaningful laws of physics no matter what coordinate systems you chose ... some just might be simpler than others.

What is "displacement"? If we think of a single dimension, it's simply the x coordinate.

What is "angular displacement"? If we think of a single dimension, it's simply the angle.

Now, what is "velocity"? It's simply the rate of change of the displacement ... the change in x over time.

What is the analogue of velocity in angular coordinates, then? It would be rate of change of the angle...the change in the angle over time.

What is "acceleration"? As we have discussed before, it's a measure of the change in velocity -- specifically, the change in velocity over time.

In the same sense, "angular acceleration" is the change in angular velocity over time.

Now, what is the "direction" of "rate of change of the angle"? There are a couple of answers to this.

1. Positive, or negative -- the angle could either be getting bigger and bigger, or lower and lower.
2. Clockwise or anti-clockwise -- the particle could either be moving clockwise or counter-clockwise.

So angular velocity has two "directions": clockwise, or counter-clockwise.

Now, look at a particle in circular motion and draw me a "tangent arrow" on the circle that signifies clockwise. At one point in time, the arrow might point upwards. But later on in that path, that arrow is pointing downwards. The arrow changed...but the angular velocity did not.

This is because this tangent arrow is actually the velocity of the point. It is a line in x,y coordinates. But angular velocity isn't "actually" that line. It has nothing to do with lines in space. It doesn't make any sense to think of it as a line in space. Becuase it's not a line in space.

Another way to look at this is that angular velocity and linear velocity -- clockwise/counter-clockwise and xyz -- have different "dimensions"...and putting them on the same scale is like measuring distance in kilograms, or measuring time in meters.

Obviously the arrow in xyz space does not actually represent angular velocity. Because angular velocity stayed the same, but the line changed. It is clear that the line isn't actually a good indication of the nature of angular velocity.

So what is?

Simply "counter-clockwise" or "clockwise" (or positive or negative) is the best way you can describe it, if you can establish which way you are looking at it from.

In physics however, we love arrows, so we made up some rules that we can describe counter-clockwise and clock-wise with arrows, like this:

"How do we invent a convenient way of keeping track of angular momentum...using arrows? we love arrows."

"How about we make the arrow come from the center of the circle."

"That's kind of confusing...which direction would it point at?"

"It should point perpendicular to the plane of the circle! That way nobody will actually confuse it for a real physical quantity."

"Oh yes that's right because it's just an imaginary thing we made up for convenient notation."

"Let's make it...if it's clockwise, it points down, into the page...and if it's counter-clockwise, it points up, out of the page."

"Why?"

"No real physical reason......I just like it that way."

"Ok. That's kind of cool because now we can 'add' angular velocities, the same way we can 'add' vectors. Like if we had a length 3 vector going up and we added it with a lengh 1 vector going down, we'd get a length 2 vector going up.

"Similarly if we had a counter-clockwise angular velocity of 3 and we add it with a clockwise angular velocity of 1...we get a counter-clockwise angular velocity of 2. Perfect!"

"Yes it appears that this imaginary physically meaningless notation actually makes it really convenient to add angular velocities! And it makes a lot of other things convenient too!"

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Question 3

It is my hope that at by this point you know enough about the nature of angular acceleration (which is merely the change in angular velocity...and like angular velocity is either positive or negative, with no "physical" linear direction) and the "definition" of centripetal acceleration to answer your own questions.

Centripetal acceleration is a "linear", or "normal" acceleration, with x y z coordinates and everything. It has a direction. Its direction is always towards the center of the circle. Why? That's just its definition -- "centripetal" is just a name we give to acceleration towards the center of the circle.

Asking "why" is like asking why change in velocity is called acceleration. "Acceleration" is just a word we invented because we don't want to say "change in velocity" all the time.

Now I noticed you said "tangential acceleration". Be careful here. Tangential acceleration is not necessarily linear acceleration. In our case, our linear acceleration is always towards the center of the circle, and this is the case for circles of constant angular velocity. Because of this, tangential acceleration is actually zero.

Tangential acceleration starts showing up when the circular motion begins speeding up...which is something I'll leave for another tale.

Answer 2

Probably you may know the three Newtonian axiom as the foundation of classical mechanics of which the first one states, that "an object either is at rest or moves at a constant velocity, unless acted upon by a force". This implies that the velocity, as a vectorial quantity whose direction describes the direction of movement and whose absolut value is the speed at which a mass is moving, is constant unless you have a force that change either the direction or the speed. A reference system in which this first axiom is valid is refered to as an inertial system.

1. To answer your first question, the $\textbf{centripetal force}$ is the froce that makes a body moving along a curved path. The direction of this force $\boldsymbol{F}_{c}$ is always orthogonal to the velocity $\boldsymbol{v}$. The magnitude of this force is given by $F_{c}=\frac{mv^{2}}{r}=ma_{c}$, where $a_{c}$ is the $\textbf{centripetal acceleration}$. "The direction of the force is toward the center of the circle in which the object is moving, or the osculating circle, the circle that best fits the local path of the object, if the path is not circular."

2. "The rotation itself is represented by the angular velocity vector $\boldsymbol{\Omega}$, which is normal to the plane of the orbit (using the right-hand rule) and has magnitude given by: $\left|\boldsymbol{\Omega}\right|=\frac{d\theta}{dt}=\omega$ with θ the angular position at time $t$." For more information see http://en.wikipedia.org/wiki/Centripetal_force.

For your particular question the direction of angular velocity is by definition perpendicular to the plane of motion and lies never in the plane. The velocity $\boldsymbol{v}=r\times\boldsymbol{\omega}$ by which a moving particle along the axis, defined by the direction of $\omega$, is in contrast to $\omega$ dependent on the radius of the curved motion. In contrast to linear motion where $\boldsymbol{p}=m\cdot\boldsymbol{v}$ ensures that momentum and velocity are always parallel to each other, for a rotation the corresponding relation $\boldsymbol{L}=\Theta\cdot\boldsymbol{\omega}$ (between angular momentum $\boldsymbol{L}$ and angular momentum $\boldsymbol{\omega}$) is more complicated, since $\Theta$ is in general not a scalar but a tensor quantity(matrix), and so angular momentum and angular velocity are in general not parallel to each other.

1. You already gave the definition of angular acceleration, as the time rate of change of the angular velocity. $\boldsymbol{\alpha}$ is related to the acceleration $\boldsymbol{a}$ of the moving body by $\boldsymbol{\alpha}=\frac{\boldsymbol{a}_{T}}{R}$, where $\boldsymbol{a}_{T}$ denotes the tangential component of centripetal acceleration or simply acceleration along the curved path and $R$ is the distance to the center around which the body is moving.

Answer 3

Attempt to answer. But my knowledge of this stuff is old, and I am not sure I use the concepts that are now being taught (though it seems ok here). This is only classical mechanics. No or very little math. I try to use math only when I do not understand (and you can find the formulae everywhere).

Question 1:

Left to its own devices, a body in motion will just keep moving in a straight line. Newton's first law. (velocity is a vector, so constant velocity means motion in a straight line).

Centripetal acceleration is the acceleration resulting from the centripetal force (second law) that bends the trajectory into a circular motion rather than leave it straight (1st law). The centripetal force can come through any kind of physical phenomenon: gravity from a central body, pull of a string, normal force from the side of a rotating satellite (so-called artifical gravity), push from a rocket engine, etc.

Question 2:

I do not understand the question. Angular velocity is represented by a vector on the axis of rotation, with a length proportional to the speed and a direction dependent of the orientation of the rotation. hence it is always orthogonal to the plane of rotation. But, afaik, nothing moves in the direction of that vector. (except for black holes ... only a joke, ignore)

Question 3:

Angular acceleration is completely different from centripetal acceleration, though one may affect the other.

Centripetal acceleration is fixed for a fixed value of angular velocity at a given distance. If you have angular acceleration, it means that you start rotating faster or slower. In that case, if you are to stay on the same circular track, you have to change the centripetal acceleration (you are now talking of third derivative). Else you will move to a different circle with a different radius.

In the context of a circular motion, tangential acceleration and angular acceleration are essentially equivalent, up to a multiplicative factor which is the radius. Remember the length of an arc is $l=\alpha r$, sorry for the math, and you can compute the relation between the second derivatives of $l$ and $\alpha$. And as I said above centripetal acceleration and angular acceleration are quite distinct, though have interaction. Hence the same is true for centripetal acceleration and tangential acceleration.
(I hope the change in terminology will satisfy everyone. It is indeed more appropriate, and particularly from a pedagogical point on view. -- see the comments)

Answer 4

Centrifugal force is fictitious force, it is inertial force which tends to put an object out of rotational motion. There are two types of centrifugal force, one is due to motion of an object and directed toward perpendicular to central attracting force. Other is due to motion of frame and directed radially outward to normal of the surface. As there is no radial inward force, so its direction is radially outward to the circle of motion. It has many practical utility from dryer of washing machine to separate chemicals in laboratory. To clear misconception about centripetal and centrifugal force, let discuss about the case of an object moving in circle.

If an object is in rotational motion, then it is subjected to two forces, centripetal toward centre and centrifugal toward tangential to circle. The resultant of two perpendicular force is circle if one of them is toward a fixed point. If centifugal force increases, by increasing tangential speed, then curve of path incline to tangential component and result in larger radius untill tangential speed becomes equal to previous one with bigger radius. In rotational dynamics, tangential change in speed due to external force result in torque applied. The applied torque changes the angular momentum, as fan attached with shaft moves in direction of shaft, fan rotates with proportional speed.

We know that acceleration of a particle in terms of radial and tangential components is,

$\mathbf a=(a_r−\omega^2r)\hat r+(2v\omega+\alpha r)\hat t\tag1$

To understand above expresson better, first take a case of stable orbit which is free from external torque, $\alpha r$ denotes torque and $a_r$ denotes external force but this is not contributing to torque as in radial direction. Now for a particle to remain in circular orbit, there must be inward radial acceleration or force equals to tangential acceleration. Let $a_r$ be acceleration required to keep a particle in orbit, then from above expression,

$−(ar−ω^2r)=2ω^2r⟹a_r=−ω^2r\tag2$

where $a_r$ is acceleration due to central force i.e. gravitational or electrostatic, second term in radial direction of $(1)$ is due to centripetal force, and first term in tangential direction is due to centrifugal force sometimes refer as coriolis force which becomes $2ω^2r$ if orbit is circular or stable. So centripetal force is not central force though both are same in magnitude and direction.

What happen if orbit is not stable, $α≠0$ or if there is no central force, $a_r=0$. If $α≠0$ then there is torque or perpendicular force to radial direction is applied and angular momentum is not conserved. So either there is change in orbital speed if particle is not free to move, part of a rigid body or change in direction of orbit, helical rising. The change in orbit’s radius is dependent on force in radial direction. So if force in radial direction become zero, then radial velocity of centrifugal force push it radially outward and angular velocity tries to keep in circular direction and a partical eventually comes out of orbit or circular motion spirally.

Addendum

Now what happens when a particle or a body is at rest and frame is moving with constant angular speed. From $(1)$, $\mathbf a=0$ then $a_r=0$ and $α=0$. Now putting $a_{cr}$ for any radial acceleration and $a_{ct}$ for any tangential acceleration due to rotation of frame, and let radial and tangential part separetely to zero. Thus,

$(a_{cr}−ω^2r)\hat r=0⟹\mathbf a_{cr}=ω^2r\hat r$, and

$(a_{ct}+2vω)\hat t=0⟹\mathbf a_{ct}=−2vω\hat t=−2ω^2r\hat t$

where last term in tangential direction is because frame is rigid and rotating with constant speed, so $v=ωr$. The term in radial direction is called as centrifugal acceleration and directed outward, while term in tangential direction is called as coriolis acceleration and directed opposite to rotation of the frame. If an acceleration applied to a body at rest is $\mathbf a$, then acceleration on the body in rotating or non-inertial frame is,

$\mathbf a_c=\mathbf a+\mathbf a_{cr}+\mathbf a_{ct}=\mathbf a+ω^2r\hat r−2ω^2r\hat t\tag3$

If there is no external force on a body in uniformly rotating frame then from $(3)$, the body forced to move outwardly and opposite to rotation of frame having spiral path. Generally coriolis acceleration’s term is not expressed as above. In case of a central force acting in radial direction inwardly then from $(3)$,

$\mathbf a_c=(−Ω^2R+ω^2r)\hat r−2ω^2r\hat t⇒g_c\hat r=(−g+ω^2r)\hat r$

where $Ω$ is angular speed of body in central force’s orbit, $R$ is radius of orbit due to central force, and $g_c$ is acceleration due to gravity on a body in rotating frame with speed $ω$ and $g$ is acceleration due to gravity in inertial frame.