Angular motion versus Linear motion

[shehri]

We have linear & angular velocities related by:

v=r.omega.
&
a=r.alpha

On my textbook it's said" Above two equations show that on a rotating body,points that're at different distances from the axis do not have the same speed or acceleration,but all points on a rigid body rotating about a fixed axis do have same angular displcement,angular speed & angular acceleration at any instant".Plz. explain this statement.Thanks.

 

[tim_lou]

when omega is constant, v varies as r varies. which means, if a object is rotating with a constant angular velocity, the speed of the different parts of the body at varies distance (r) from the axis of rotating will have varying speed according to $v=r\omega$

similarly, for angular acceleration.

 

[gabee]

Right--linear velocity measures distance traveled in a unit of time; angular velocity measures an angle rotated through in a unit of time.

The angular velocity is measured in radians/second, which means if a wheel rotates once every second its angular velocity is $$2\pi$$ rad/s. However, the linear velocity of points farther away from the axis of rotation is faster.

The center of the wheel (where r = 0) has linear velocity 0 rad/s (because it doesn't go anywhere, just spins around). A point 1 m from the center of the wheel travels a certain distance given by $$2\pi r$$ every second. In this case, $$2\pi (1) = 2\pi$$, so this point must travel $$2\pi$$ meters in one rotation. A point 2 m from the center will travel faster (linearly) since it has to cover a greater distance in the same time. In this case, $$2\pi (2) = 4\pi$$ so this point on the same wheel with the same angular velocity is traveling $$4\pi$$ meters every rotation.

Angular velocity is just a handy way of measuring quantities so our calculations don't get very complicated trying to convert linear coordinates to polar and so on.

Let us know if there's anything else that needs clarification!

 

[shehri]

Thanks for clarification.This forum is doing a wonderful job to help science students clear their concepts;especially students from developing countries like me avail ourseves a lot.Thanks once again.

 

[shehri]

But according to v=rw=>w=v/r, angular velocity w also inversly proportional to r.Then how can angular velocity be same at various points of rigid body if radious 'r' varies?Thanks.

 

[Tomsk]

It is the linear velocity that varies so that the ratio v/r is constant. So points on a rigid body further from the centre move faster (as r goes up, v goes up), so that the angular velocity is constant.

 

[shehri]

Thank u friend.Now I'm clear about it.