Angular velocity and rotational motion

[MIA6]

For the rotational motion, on my book, it says "Since w (angular velocity) is the same for all points of a rotating object, a (alpha) also will be the same for all points.w and a are properties of the rotating object as a whole." The thing that I don't get is that why velocity is the same at all points no matter if there is an acceleration? Its speed is supposed to increase at each point? Hope you can explain, thank you.

 

[Dale]

Remember that velocity is a vector. A vector has a magnitude and a direction. Speed is the magnitude of the velocity vector. So a velocity can change direction without changing speed. This is still a change in velocity and therefore an acceleration.

Uniform circular motion is a special case of continuous acceleration without a change in speed. Only the direction changes.

 

[Laszlo Lemhenyi]

No, velocity is not the same for all points !
It differs from point to point, if not in value then in its direction.
Yet, the angular velocity associated with any point of the rotating
object is the same, direction - and value-wise.
Acceleration has nothing to do with this property.

 

[MIA6]

Remember that velocity is a vector. A vector has a magnitude and a direction. Speed is the magnitude of the velocity vector. So a velocity can change direction without changing speed. This is still a change in velocity and therefore an acceleration.

Uniform circular motion is a special case of continuous acceleration without a change in speed. Only the direction changes.

You just said that there is a change in velocity no matter its speed or direction. But why it stated that w (angular velocity) is the same for all points of a rotating object? It implies no change?

 

[Astronuc]

ln motion, on my book, it says "Since w (angular velocity) is the same for all points of a rotating object, a (alpha) also will be the same for all points.w and a are properties of the rotating object as a whole."

I believe that the inference is that the object is 'solid' or more precisely 'rigid', i.e. the points in the object are fixed within the object, in which case if all objects have the same angular velocity, then all points would have the same angular acceleration, i.e. the rate of change of angular velocity would the same for all points in the object.

Do not confuse angular velocity with linear velocity, since the linear velocity is constantly changes (direction) and the magnitude depends on r, the distance from axis of rotation.

The same angular velocity means the radius to a point sweeps the same angle (angular displacement) during a given time period (same length of time).

 

[MIA6]

I believe that the inference is that the object is 'solid' or more precisely 'rigid', i.e. the points in the object are fixed within the object, in which case if all objects have the same angular velocity, then all points would have the same angular acceleration, i.e. the rate of change of angular velocity would the same for all points in the object.

Do not confuse angular velocity with linear velocity, since the linear velocity is constantly changes (direction) and the magnitude depends on r, the distance from axis of rotation.

The same angular velocity means the radius to a point sweeps the same angle (angular displacement) during a given time period (same length of time).

but wouldn't the angular velocity change any direction or increase speed when there is an acceleration since I was thinking about the same concept as linear velocity.

 

[Astronuc]

but wouldn't the angular velocity change any direction or increase speed when there is an acceleration since I was thinking about the same concept as linear velocity.

Yes, an angular acceleration implies a change in angular velocity, but that is not what the statement is addressing.

In a rotating object, the only way that every point can have the same angular velocity \omega is for the object to be rigid, and that rigidity also means that at a given instant, each point has the same angular acceleration \alpha if a torque is applied to the object. That would not be the case for a fluid (liquid or gas).

 

[MIA6]

Yes, an angular acceleration implies a change in angular velocity, but that is not what the statement is addressing.

In a rotating object, the only way that every point can have the same angular velocity itex]\omega[/itex] is for the object to be rigid, and that rigidity also means that at a given instant, each point has the same angular acceleration \alpha if a torque is applied to the object. That would not be the case for a fluid (liquid or gas).

So if there is an acceleration, then the angular velocity would change, let say, from initial velocity 20m/s to 30m/s (maybe we can think of this as instantaneous velocity), then in every instant, it would be 30m/s? but wouldn't the speed keep increasing?from 20 -->25-->30-->35?

 

[Astronuc]

Well angular acceleration is measured in rad/s², and angular velocity in rad/s, where rad means radians. One revolution represents an angular displacement of 2\pi radians. Radians are dimensionless, i.e. they don't have length.

The magnitude of linear (tangential) velocity of a point in a rotating object is given by v=\omegar, where r is the distance of said point from the axis of rotation.

So the linear (tangential) velocity varies with r.

See if this reference helps - http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html

http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html

Note the equation \omega\,=\,\omega_0\,+\,\alpha{t}, where \omega\,=\,\omega(t)

 

[MIA6]

Well angular acceleration is measured in rad/s², and angular velocity in rad/s, where rad means radians. One revolution represents an angular displacement of 2\pi radians. Radians are dimensionless, i.e. they don't have length.

The magnitude of linear (tangential) velocity of a point in a rotating object is given by v=\omegar, where r is the distance of said point from the axis of rotation.

So the linear (tangential) velocity varies with r.

See if this reference helps - http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html

http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html

Note the equation \omega\,=\,\omega_0\,+\,\alpha{t}, where \omega\,=\,\omega(t)

RIght, I forgot it should be radian/second. According to the formula, average alpha=delta w/ delta t, can we know that at every point, or every instant, the average velocity is increasing or remains at a new velocity at every point? let's say, if the original is w=24 rad/s, then the new w is 30rad/s, and will it remain at every point or it will accelerate so will have bigger and bigger speed at each point.

 

[Astronuc]

If every point has angular velocity 24 rad/s, then some time later, every point has angular velocity of 30 rad/s, then every point accelerated at the same angular acceleration, since every point had the same change in angular velocity over the same period of time.

 

[MIA6]

If every point has angular velocity 24 rad/s, then some time later, every point has angular velocity of 30 rad/s, then every point accelerated at the same angular acceleration, since every point had the same change in angular velocity over the same period of time.

Oh, so what you said was, suppose the angular acceleration is 5 rad/s^2, then every point or every instant will increase the same amount 5 rad/s^2 at one second, and then the next second, it will still increase 5 rad/s^2, so that their velocity at every instant is the same, which is unlike translational motion that each point has different/increasing velocity when it undergoes acceleration?

 

[Astronuc]

Oh, so what you said was, suppose the angular acceleration is 5 rad/s^2, then every point or every instant will increase the same amount 5 rad/s^2 at one second, and then the next second, it will still increase 5 rad/s^2, so that their velocity at every instant is the same, which is unlike translational motion that each point has different/increasing velocity when it undergoes acceleration?

Close.

If angular acceleration is 5 rad/s^2, then in one second, the angular velocity increases by 5 rad/s (i.e. \Delta\omega\,=\,\alpha*\Delta{t} => 5 rad/s = 5 rad/s² * 1 s), and if the angular acceleration remains constant, then after another second the angular velocity increases by 5 rad/s.

Acceleration implies a change in velocity with time, whether it's linear or angular.

The point of the comment "Since w (angular velocity) is the same for all points of a rotating object, a (alpha) also will be the same for all points.w and a are properties of the rotating object as a whole." is that if all points of a rotating body rotate simultaneously at the same angular velocity, then the body is rigid, and this then implies that all points in a rigid body will experience that same angular acceleration simultaneously. The key is RIGID body.