Exploring Types of Acceleration in Rotational Movement of Rigid Bodies

[physea]

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!

 

[jbriggs444]

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.

 

[osilmag]

a=ra is the equation for rotational acceleration. It is usually written as a=r*alpha

 

[physea]

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?

 

[olgerm]

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.

 

[physea]

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!

 

[sophiecentaur]

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.

 

[physea]

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?

 

[sophiecentaur]

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.

 

[physea]

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

 

[A.T.]

How can α=rω^2 ?

Where did you get that from?

 

[physea]

Where did you get that from?

acceleration in circular movement is $a=\omega^2\cdot r$

• a is acceleration.
• $\omega$ is angular velocity.
• r is radius of trajectory.

 

[A.T.]

a is acceleration.

α = angular acceleration

 

[physea]

...

Yeah, α = angular acceleration. I wrote the same.
α=rω^2

 

[A.T.]

I wrote the same.

If you think you wrote the same as olgerm, then you need to use a bigger font, or better glasses.

 

[physea]

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?

 

[A.T.]

And we know that γ=rα

Where did you get that from?

 

[physea]

Where did you get that from?

We know that x=rθ
Then x'=rθ'
So x''=rθ'' which is γ=rα.

 

[sophiecentaur]

@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.

 

[A.T.]

We know that x=rθ
Then x'=rθ'
So x''=rθ'' which is γ=rα.

This is a 2D problem. You need to use vectors, and distinguish between tangential and radial direction:
https://en.wikipedia.org/wiki/Circular_motion#Acceleration

 

[jbriggs444]

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.

 

[physea]

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?

 

[olgerm]

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.

 

[physea]

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?

 

[olgerm]

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.

 

[olgerm]

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.

 

[sophiecentaur]

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.

 

[olgerm]

$a_{tangential}=r \cdot \frac{\partial \omega}{\partial t}$

 

[sophiecentaur]

$a_{tangential}=r \cdot \frac{\partial \omega}{\partial t}$

That's fine but he is demanding a "Comprehensive reply". There is no end to what that could entail.

 

[Herman Trivilino]

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