I understand what is shown in this diagram of uniform circular motion (from Wikipedia):

Centripetal acceleration acts towards the center of the circle, and the velocity is perpendicular to it.

But I've also read that the direction of angular velocity is upwards for an object rotating counterclockwise, and downwards for an object rotating clockwise. So for example, if I was to stand on the right side of a car and see its wheels turning clockwise, the angular velocity would be directed into the axle. This doesn't make sense to me. Are the linear and angular velocity in different directions, or what?

 

Yes, linear and angular velocity are indeed oriented in different directions. This arises from the nature of vector algebra in 3 dimensions. In particular the cross product of the position vector with the linear velocity vector. As Yoda might say:

"Study vector algebra diligently you should young Jedi"

 

Yes, linear and angular velocity are indeed oriented in different directions. This arises from the nature of vector algebra in 3 dimensions. In particular the cross product of the position vector with the linear velocity vector. As Yoda might say:

"Study vector algebra diligently you should young Jedi"

OK, so does that mean that the linear and angular acceleration are also oriented in different directions?

 

Yes they are.

 

Yes they are.

OK, thanks for the clarification.

I also read that for an object moving with uniform circular motion, the angular velocity is constant but the angular acceleration is not. How can this be true? Since acceleration is a change in velocity, wouldn't a constant velocity necessitate a constant acceleration of zero?

 

Acceleration is composed of two components. Speed and direction.
If any of these variables change, we call that change, acceleration.

Any path other than a straight line, is acceleration.

 

Acceleration is composed of two components. Velocity and direction.
If any of these variables change, we call that change, acceleration.

Any path other than a straight line, is acceleration.

Yes, I know that acceleration is a vector and has both a magnitude and a direction. But since α = Δω/Δt, I don't understand how the angular acceleration can change if neither the magnitude nor the direction of the angular velocity is changing.

 

Any change in direction is acceleration.......whether that change is constant or not.

 

Any change in direction is acceleration.......whether that change is constant or not.

Yes, but:

the direction of angular velocity is upwards for an object rotating counterclockwise, and downwards for an object rotating clockwise. So for example, if I was to stand on the right side of a car and see its wheels turning clockwise, the angular velocity would be directed into the axle.

If I understand correctly, the linear velocity/acceleration experiences a change in direction, but the angular velocity doesn't experience any change in direction at all. For this reason, I would expect the angular acceleration to not have any change in direction either and thus equal zero, which is why I'm confused by what I read which stated otherwise.

 

When we take the derivative of a scalar quantity it is only the change in magnitude that is important. With a vector quantity when we take the derivative we get terms for the change in magnitude and terms for the change in directions. When we take the second derivative there are more terms for changing magnitudes and changing directions. I'd have to dig out an old textbook but there are surprising things that happen in vector calculus.

If the magnitude of the angular velocity is constant and the direction is constant then the angular acceleration will be zero. If you hold a spinning gyroscope sitting on a rotating chair you will be stable. Change the orientation of the axis of rotation and you will spin. The forces can actually be surprisingly large.

 

When we take the derivative of a scalar quantity it is only the change in magnitude that is important. With a vector quantity when we take the derivative we get terms for the change in magnitude and terms for the change in directions. When we take the second derivative there are more terms for changing magnitudes and changing directions. I'd have to dig out an old textbook but there are surprising things that happen in vector calculus.

I haven't learned calculus yet, so I'm not exactly sure what you mean. Are you saying that it is possible for there to be a non-zero acceleration when neither the magnitude nor the direction of the velocity are changing?

 

Here's a good series of slides on the subject
http://hyperphysics.phy-astr.gsu.edu/hbase/rotv.html
Your intuition on the basics is correct.
Now let the rotating wheel and shaft together rotate about another axis other than the axis of the shaft.

 

I haven't learned calculus yet, so I'm not exactly sure what you mean. Are you saying that it is possible for there to be a non-zero acceleration when neither the magnitude nor the direction of the velocity are changing?

No I'm not saying that at all. I am saying that the problems get complicated quickly because there are counter intuitive things going on with angular and rotational motion.

 

OK, thanks for the clarification.

I also read that for an object moving with uniform circular motion, the angular velocity is constant but the angular acceleration is not. How can this be true? Since acceleration is a change in velocity, wouldn't a constant velocity necessitate a constant acceleration of zero?

Where did you read that?

If angular velocity, which has both magnitude and direction, is constant, then how can the time derivative of it be anything other than zero?

I think what you read is that the angular velocity is constant but that the acceleration (as apposed to angular acceleration) is neither zero nor it is even constant.

Just as there is a difference between velocity and angular velocity, so too is there a difference between acceleration and angular acceleration.

 

Where did you read that?

If angular velocity, which has both magnitude and direction, is constant, then how can the time derivative of it be anything other than zero?

I think what you read is that the angular velocity is constant but that the acceleration (as apposed to angular acceleration) is neither zero nor it is even constant.

Just as there is a difference between velocity and angular velocity, so too is there a difference between acceleration and angular acceleration.

Thank you, you exactly addressed my confusion. My solutions guide, which said that the correct answer to a multiple choice question dealing with uniform circular motion is that angular velocity is constant and angular acceleration is not, must contain a mistake.

 

Hello,

I think that most of the confusion here was just about how we measure speed.
While we can also measure velocity from some external point, velocity is basically an intrinsic property while direction is based on some external reference point. In other words, velocity is really measured along the path of the object itself so the speed can in fact stay constant while the direction changes.

You can imagine an ant walking across the ground searching for food. As it curves right and left and back again, if it takes one small step every tenth of a second the speed along that curve is constant although the direction changes frequently. It takes acceleration to change the direction but none to change the speed.
Alternately, if it walks in a straight line taking small steps that vary in duration, it takes acceleration to force a change in speed and none to change direction (because it is not changing direction now).
So you see either a change in speed or a change in direction (or both at the same time) requires acceleration but just because there is some acceleration does not mean that the speed itself must be changing because it might be only the direction.

It is a little different if we want to measure the speed of an object approaching the x and y axis for example. In this case we would have to account for the change in x and the change in y too. But in most simple cases we measure the speed as the velocity of the object referenced to it's own path of travel.

Just to be complete, we can also measure direction intrinsically. For example, the ant turns 'right', then the ant turns 'left', then the ant turns 'left' again, then the ant turns 'right' again. These statements 'right' and 'left' are intrinsic properties that depend only on the objects current position along the path.

Another simple example:
You are driving a car at 60 mph along a highway that curves right and left. Your speedometer reads "60 mph" all the while yet your path is curving back and forth. If you make a sharp turn along a sharp curve in the road, you feel some force on your body that tries to push you in the opposite direction. That force comes from an acceleration even though the speedometer reading did not change.