V=(omega)(radius). One side of the eqn has a direction, the other side is just a magnitude...?

Question

PLEASE READ FIRST!!!!

So i noticed this equation as the magnitude of the linear velocity of a rotating object is equal to the angular velocity (radians/s) times the radius.

What i don't understand is that the magnitude of the velocity is strictly a positive value. However, omega can have a negative value (as something travels backwards in the clockwise direction).

Does this mean V is not actually |V|, does v have a direction in this equation? I was told if it were to have a direction, as linear velocity, its magnitude would be different based on where its measured.


If this V is a constant linear velocity, how come when measured at a point on the circle, it changes (mirroring a trig function)

Answer 1

I second Richard B's compliment to your sense of asking the right questions, and not just blithely plodding along, accepting a bunch of equations without attempting to understand them.



And unlike Richard, I fairly sailed through vector algebra and vector calculus, except for a significant stumble over the treatment of alternating differential forms. But I cleared that up a few years later, in a different course. Luckily, your problem here doesn't require anything greatly complex or hairy as those.



When "properly" written, that equation you quote is



v =ω xr



where each of those three quantities is a vector. If you haven't dealt with vector algebra, and cross- and dot-products of vectors in particular, all you need to know is that the magnitude of both sides of that equation gives:



v = ω r sinθ



where v, ω, and r, are magnitudes of the above vectors, and so, are always non-negative.

And that θ is the angle between the vectorsω andr, and so is always between 0º and 180º inclusive, so that the sine is always non-negative.

Now the vectorω is defined in a way that will seem strange at first, but bear with me. If you picture whatever is turning, and you position the fingers of your right hand together in a 'curl,' with the thumb pointing out to the side, then just place your curled fingers so that they point in the direction of the rotation. Your thumb then points perpendicular to the plane of that rotation, in the direction of the vectorω. For circular motion, that will be always perpendicular to the radius vector,r, whenr is taken with the center of rotation as its origin. And in that case, θ is always ½π, and sinθ=1.



And that's where

v = ω r

comes from.

Answer 2

For simple problems, v = ωr can be used as if v, ω and r are scalars., ie. just use the magnitudes.

So v is treated as a speed.



But at a higher level and for more complex problems, the full equation is:

v = ωXr

where v, ω and r are treated as vectors, and 'X' denotes what is called the 'vector product' (or sometimes called the 'cross product'). This is more difficult to deal with mathematically but allows for the direction problems you describe. E.g see link.



Note, r is not a constant, it is the position vector from the centre to the rotating object and constantly changes direction. That gives v a value which is constantly changing direction.



By the way, v is not 'a constant linear velocity', it is the instantaneous velocity.

Answer 3

good thought



Omega = radial velocity is a vector with direction at right angle to Radius vector which can change with time

the radian is a pure number

radian = distance on curved path / Radius: circumference of one full circle = 2 Pi R so the radian angle of a full circle is 2PiR/R 2Pi a unit less, irrational transcendental number that can only be approximated



in V velocity (vector no matter what system is used) = omega/Sec * R vector

therefore V in Spherical or cylindrical coordinates is a vector that changes direction with time but not scalar value



note that in rectangular coordinates x,y.z it is constantly changing (x,y,z) but the vector (,x,y,z) keeps the same scaler value



VECTOR ANALYSES AND CALCULUS IS DIFFERENT FROM SCALER

it takes years of math study to really understand the difference. I ddi not get the point for a long time. Physics teachers may assume you have taken vector math already



Scalers are only one possible class of numbers.

for example in multiplication a * b = c all scaler numbers

but vectors A dot B = scaler and A Cross B = C vector and B cross A = MINUS C vector



and vector differential calculus is even worse

Answer 4

Many quantities that are treated as scalars can also have a direction i.e positive and negative.



Money in the bank, current flow, water flow, even mass.



So if you want |V| then you have to interpret it accordingly.



But there are reasons for different uses of the formula.



Two arms from a common axle ( like a clock ) are rotating.

One has a velocity V2 the other V1



We wish to state the result in such a way as Velocity of 2 as seen by 1 is V2 - V1



and if both were only |V2| and |V1| respectively then it is NOT true that |V2 -V1| = |V2|-|V1|

in all conditions.

so the preservation of sign is important.



Indeed almost all scalars are actually better considered as 1 dimensional vectors.

Answer 5

What you must remember is v=omega x radius (it is a cross product).

Now omega has a direction. It can be found by using the right hand thumb rule, or corkscrew rule, whatever you wan to call it. And this radius is the radius vector. It point from to the object in motion. Hence both the right hand side and the left hand side are vector quantities.