Exploring Types of Acceleration in Rotational Movement of Rigid Bodies

In summary, the body can have rotational acceleration through the centre of mass or rotational acceleration through a centre outside the centre of mass. The equations for these are a=r*alpha and a=r*alpha, respectively.
  • #1
physea
211
3
Hello,

I am confused about the types of acceleration in rotational movement of rigid bodies.

I am quite clear about the various types of movement of rigid bodies. The body can have translational movement where acceleration is dV/dt. But what are the other types of acceleration that the body may have?

I think we can categorise the other types of movements into rotational through the centre of mass and rotational through a centre outside the centre of mass.

Can you tell me please the equations that describe these? It is not clear and I see confusing things on the web, for example that a=ra.

Thanks!
 
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  • #2
physea said:
I am quite clear about the various types of movement of rigid bodies. The body can have translational movement where acceleration is dV/dt. But what are the other types of acceleration that the body may have?

I think we can categorise the other types of movements into rotational through the centre of mass and rotational through a centre outside the centre of mass.
Any rigid motion can be characterized as a translation plus a rotation. You can choose any point you like to describe the rotation. Depending on the point you choose, the corresponding translation may be different.

For instance, a rolling wheel can be described as a translation of the axle and a rotation about the axle. Or it can be described (momentarily at least) as a rotation about the instantaneous point of contact with the road.
 
  • #3
a=ra is the equation for rotational acceleration. It is usually written as a=r*alpha
 
  • #4
osilmag said:
a=ra is the equation for rotational acceleration. It is usually written as a=r*alpha

Can anyone explain what is a and what alpha?
 
  • #5
osilmag said:
a=ra is the equation for rotational acceleration. It is usually written as a=r*alpha
acceleration in circular movement is ##a=\omega^2\cdot r##
  • a is acceleration.
  • ##\omega## is angular velocity.
  • r is radius of trajectory.
 
  • #6
olgerm said:
acceleration in circular movement is ##a=\omega^2\cdot r##
  • a is acceleration.
  • ##\omega## is angular velocity.
  • r is radius of trajectory.

Ok, but you say it's omega squared, while @https://www.physicsforums.com/members/osilmag.640068/ said it's simply omega!
 
  • #7
physea said:
Ok, but you say it's omega squared, while @https://www.physicsforums.com/members/osilmag.640068/ said it's simply omega!
Omega squared is the correct one.
 
  • #8
sophiecentaur said:
Omega squared is the correct one.

Are you sure?
It's x = rθ
So derivative of x = r times derivative of θ
So second derivative of x = r times second derivative of θ which is linear acceleration = r times angular acceleration

How can angular velocity (ω) be the square of angular acceleration?
 
  • #9
physea said:
Are you sure?
If there is a misunderstanding here, then why not just look it up? Afaiac acceleration under circular motion is ω2r. Can you find a source that says otherwise.
This stuff is not really a matter of discussion. It's all written down and the definitions are accepted.
 
  • #10
sophiecentaur said:
If there is a misunderstanding here, then why not just look it up? Afaiac acceleration under circular motion is ω2r. Can you find a source that says otherwise.
This stuff is not really a matter of discussion. It's all written down and the definitions are accepted.

I don't want to just look it up, if I wanted to do that I would do it and wouldn't come here, but I suppose this is not the intention always as this forum would have no meaning.

I want to know how these are derived and related together.

So, I know that x=rθ.
From that, don't we derive that V=rω and γ=rα ? Aren't these correct so far?

How can α=rω^2 ?

By the way:
α = angular acceleration
γ = linear acceleration
 
  • #11
physea said:
How can α=rω^2 ?
Where did you get that from?
 
  • #12
A.T. said:
Where did you get that from?

olgerm said:
acceleration in circular movement is ##a=\omega^2\cdot r##
  • a is acceleration.
  • ##\omega## is angular velocity.
  • r is radius of trajectory.
 
  • #13
olgerm said:
a is acceleration.
physea said:
α = angular acceleration
 
  • #14
A.T. said:
...

Yeah, α = angular acceleration. I wrote the same.
α=rω^2
 
  • #15
physea said:
I wrote the same.
If you think you wrote the same as olgerm, then you need to use a bigger font, or better glasses.
 
  • #16
A.T. said:
If you think you wrote the same as olgerm, then you need to use a bigger font, or better glasses.

OK so you mean that γ=rω^2 then.
And we know that γ=rα
So α=ω^2. Is this true? The angular acceleration is the square of the angular velocity?
 
  • #17
physea said:
And we know that γ=rα
Where did you get that from?
 
  • #18
A.T. said:
Where did you get that from?
We know that x=rθ
Then x'=rθ'
So x''=rθ'' which is γ=rα.
 
  • #19
@physea What source are you using for your opinions and statements? I have a feeling that you are trying to self-drive through this topic and that you are trying to use Q and A to learn the stuff. This is not a good way (as you are demonstrating with many of your posts). You seem to be mixing up ideas and symbols, which may be why you arrive at things like "α=ω^2.", which is a nonsense statement where an acceleration is equated to a velocity squared. You would, I'm sure, never do that for linear motion.
You need a half decent mechanics book.
 
  • #21
physea said:
So, I know that x=rθ
What's x, what's r, what's theta and what's the situation?

The horizontal acceleration of the axle on a car moving down the highway where theta is the accumulated angle turned by the wheel is different from the centripetal acceleration of a bug on a tire that is rotating in place.
 
Last edited:
  • #22
jbriggs444 said:
What's x, what's r, what's theta and what's the situation?

The horizontal acceleration of the axle on a car moving down the highway where theta is the accumulated angle turned by the wheel is different from the centripetal acceleration of a bug on a tire that is rotating in place.

OMG, that's what I am trying to clarify with this thread but no-one answers properly.

So, again, I am asking, in rotational movement, what are the different types of acceleration and their formulas?

Will someone eventually post a 'comprehensive' reply instead of posting bits?
 
  • #23
What you mean by type of acceleration? If one pointmass is in circular movement then its coordinates are:
##
x=r\cdot sin(t\cdot\omega+\alpha)\\
y=r\cdot cos(t\cdot\omega+\alpha)
##
acceleration is:
##
a_x(t)=-r\cdot\omega^2\cdot sin(t\cdot \omega+\alpha)\\
a_y(t)=-r\cdot\omega^2\cdot cos(t\cdot \omega+\alpha)
##

If by rotational movement you that the body is spinning, then different parts of the body have different accelerations.
 
  • #24
olgerm said:
What you mean by type of acceleration? If one pointmass is in circular movement then its coordinates are:
##
x=r\cdot sin(t\cdot\omega+\alpha)\\
y=r\cdot cos(t\cdot\omega+\alpha)
##
acceleration is:
##
a_x(t)=-r\cdot\omega^2\cdot sin(t\cdot \omega+\alpha)\\
a_y(t)=-r\cdot\omega^2\cdot cos(t\cdot \omega+\alpha)
##

If by rotational movement you that the body is spinning, then different parts of the body have different accelerations.

You introduce terms that I don't understand.
If v=rω, what is α?

Also, from what I have gathered, there are two types of acceleration. The centripetal acceleration and the ... not sure how it is called, the acceleration that is tangent to the trace of the point/particle.

What are these accelerations equal to? Is there any other type of acceleration in circular/rotational motions?
 
Last edited:
  • #25
physea said:
You introduce terms that I don't understand.
If v=rω, what is α?
if ##x(0)=0## and ##y(0)=r##, then ##\alpha=0##.
It is hard to explain. If you understand what other variables mean and, what circular orbit is, then you should understand what ##\alpha## means.
 
  • #26
physea said:
The centripetal acceleration and the ... not sure how it is called, the acceleration that is tangent to the trace of the point/particle.
These equations assume constant angular speed therefore tangential acceleraition is 0.
 
  • #27
physea said:
So, again, I am asking, in rotational movement, what are the different types of acceleration and their formulas?

Will someone eventually post a 'comprehensive' reply instead of posting bits?
This is actually quite unreasonable. You are not asking one question or introducing a point for discussion. What you need is a whole Dynamics Course, which is not the brief of PF.
If it's too hard for you to find a suitable course then I suggest some on line course like the Kahn Academy.
 
  • #28
##a_{tangential}=r \cdot \frac{\partial \omega}{\partial t}##
 
  • #29
olgerm said:
##a_{tangential}=r \cdot \frac{\partial \omega}{\partial t}##
That's fine but he is demanding a "Comprehensive reply". There is no end to what that could entail.
 
  • #30
physea said:
OMG, that's what I am trying to clarify with this thread but no-one answers properly.

So, again, I am asking, in rotational movement, what are the different types of acceleration and their formulas?

Linear acceleration and angular acceleration.

Linear acceleration ##\vec{a}=\frac{d \vec{v}}{dt}## and angular acceleration ##\alpha=\frac{d \omega}{dt}##.

This article explains the meanings of the symbols in those equations, and provides an in-depth answer to your question: https://en.wikipedia.org/wiki/Rotation_around_a_fixed_axis
 
  • #32
So we have two accelerations, the tangential acceleration and the centripetal acceleration.
The centripetal is the one that makes the particle move in a circular route.
The tangential is the one that makes the particle move faster/slower on the circular route.
Right?

And the first one is v^2/r.
And the second is γ/r.
Right?

I don't understand why it should take tons of posts for a comprehensive and simple reply like this.
 
  • #33
physea said:
I don't understand why it should take tons of posts for a comprehensive and simple reply like this.
Thread is closed for Moderation...
 
  • #34
physea said:
Will someone eventually post a 'comprehensive' reply instead of posting bits?

physea said:
I don't understand why it should take tons of posts for a comprehensive and simple reply like this.

You have a misconception about how PF works.

A "comprehensive reply" is open-ended; you could be asking for a course in physics 101, and that's way outside the scope of PF. All we can do is point you in the right direction; you have to do the work.

Several posters in this thread have given you good links to sources of information. Please read them, and take some time to build your own understanding. Then, if you have specific questions about something you find, you can start a new thread asking that specific question.

This thread will remain closed.
 
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1. What is rotational acceleration?

Rotational acceleration is the rate at which the angular velocity of a rigid body changes over time. It is a measure of how quickly the object is rotating.

2. How is rotational acceleration different from linear acceleration?

Rotational acceleration refers to the change in angular velocity, while linear acceleration refers to the change in linear velocity. Rotational acceleration is also measured in radians per second squared, while linear acceleration is measured in meters per second squared.

3. What factors affect rotational acceleration?

The factors that affect rotational acceleration include the mass and distribution of mass of the object, the applied torque, and the moment of inertia, which is a measure of how resistant the object is to rotational motion.

4. How is rotational acceleration measured?

Rotational acceleration is typically measured using a device called an accelerometer, which measures the change in angular velocity over time. It can also be calculated using the formula α = Δω/Δt, where α is the rotational acceleration, Δω is the change in angular velocity, and Δt is the change in time.

5. Why is understanding rotational acceleration important?

Understanding rotational acceleration is important in many fields, including physics, engineering, and sports. It allows us to analyze and predict the behavior of rotating objects, such as the motion of planets, the movement of vehicles, and the performance of athletes. It also helps us design and improve technologies that rely on rotational motion, such as engines, turbines, and amusement park rides.

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