Athin rod rotates at a constant angular speed. consider the tangential speed of each point on the rod for the case when the axis of rotation is perpendicular to the rod at its center and at one end. explain for each case whether there are any points on the rod that have the same tangential speeds.

    v = R w    

With this expression we see that for each point at different radius the tangential velocity is different

Explanation:

They indicate that the angular velocity is constant, that is

            w = dθ / dt

Where θ is the radius swept angle and t the time taken.

The tangential velocity is linear or

           v = dx / dt

Where x is the distance traveled in time (t)

In the definition of radians

          θ = s / R

Where s is the arc traveled and R the radius vector from the pivot point, if the angle is small the arc (s) and the length (x) are almost equal

         θ = x / R

We substitute in the speed equation

         v = d (θ R) / dt

The radius is a constant for each point

         v = R dθ / dt

         v = R w

With this expression we see that for each point at different radius the tangential velocity is different