A cylinder rotating in Cartesian coordinate system

[Earthland]

Homework Statement

In Cartesian coordinaate system, we describe the rotation of a cylinder. The axis of the cylinder has the same direction as the basis vector e3. Angular velocity is described by vector w = 2e1 - 5e2 + 7e3 rad/s. I must find the velocity vector (v) of a point P that is described by position vector OP = 1e2 + 3e3.

The Attempt at a Solution

I know the answer is just wXOP = (2,-5,7)X(0,1,3)=(-22,-6,2) and it's supposed to be an easy one. However, I can't quite imagine what is actually going on. I guess these are pretty stupid questions, but:

1) I know how the angular velocity vector is given. If the cylinder rotated around its axis, the angular velocity vector should be something like (0,0,x). Fine, cylinder don't have to rotate around its axis, but if so, doesn't the direction of the axis change and if so, what's the point of telling that its direction is e3?

2) A point on a cylinder would be moving so how can we describe it with a constant position vector? Or is it just a point "in space" through which the cylinder rotates?

3) In case like this on the picture

I understand that angular velocity X position vector (r) would give right answer, since v must be perpendicular with both w and r. Well, that is just what wXr gives us, but in given task, w and position vector OP are not perpendicular to each other. They do define a plane to witch v can be perpendicular, but ... well I just don't get what it means.

If someone could draw me a picture, it would be most helpful.

 

[Simon Bridge]

I must find the velocity vector (v) of a point P that is described by position vector OP = 1e2 + 3e3.

Are e1, e2, and e3 supposed to be the Cartesian unit vectors?
Are you expected to find the instantaneous velocity of point P in the rotating reference frame?

To understand it, look at how the equivalent problem would be done for an arbitrary linear velocity and the definition of angular velocity.

 

[Earthland]

Are e1, e2, and e3 supposed to be the Cartesian unit vectors?

yes

Are you expected to find the instantaneous velocity of point P in the rotating reference frame?

I think it's called the linear velocity vector that I was told to find, but I'm not sure what it's called in english.

 

[Simon Bridge]

I think it's called the linear velocity vector that I was told to find, but I'm not sure what it's called in english.

OK - but which reference frame?

 

[TSny]

1) I know how the angular velocity vector is given. If the cylinder rotated around its axis, the angular velocity vector should be something like (0,0,x). Fine, cylinder don't have to rotate around its axis, but if so, doesn't the direction of the axis change and if so, what's the point of telling that its direction is e3?

Yes, the axis of the cylinder will change orientation. The question is asking for the velocity of point P at the instant the axis of the cylinder is in the e₃ direction.

2) A point on a cylinder would be moving so how can we describe it with a constant position vector? Or is it just a point "in space" through which the cylinder rotates?

The position of P is changing. But you just want to find its velocity at the instant when it is located at the given position.

I understand that angular velocity X position vector (r) would give right answer, since v must be perpendicular with both w and r. Well, that is just what wXr gives us, but in given task, w and position vector OP are not perpendicular to each other. They do define a plane to witch v can be perpendicular, but ... well I just don't get what it means.

If someone could draw me a picture, it would be most helpful.

I don't know if this picture will help. Each point P of the cylinder will have a velocity perpendicular to the plane formed by w and r. If w remains constant in magnitude and direction, then each point P will move in a circle around w. (I drew w vertical even though in your problem it tilts off in some direction.)

 

[Simon Bridge]

Each point P of the cylinder will have a velocity perpendicular to the plane formed by w and r. If w remains constant in magnitude and direction, then each point P will move in a circle around w. (I drew w vertical even though in your problem it tilts off in some direction.)

It's like the stars viewed from the surface of the Earth move in a circle about the axis of the Earth's rotation.

 

[Earthland]

I think I got it, thank you :)