Question

In some circumstances, it is useful to look at the linear velocity of a point on the blade. The linear velocity of a point in uniform circular motion is measured in meters per second and is just like the linear velocity in kinematics, except that its direction continuously changes. Imagine taking a part of the circle of the motion and straightening it out to determine the velocity. One application of linear velocity in circular motion is the case in which the lift provided by a section of the blade a distance r from the center of rotation is directly proportional to the linear speed of that part of the blade through the air.
What is the equation that relates the angular velocity omega to the magnitude of the linear velocity v?

Answer 1

Answer:

$v=wr$

Explanation:

Tangent and Angular Velocities

In the uniform circular motion, an object describes the same angles in the same times. If $\theta$ is the angle formed by the trajectory of the object in a time t, then its angular velocity is

$\displaystyle w=\frac{\theta}{t}$

if $\theta$ is expressed in radians and t in seconds the units of w is rad/s. If the circular motion is uniform, the object forms an angle $2\theta$ in 2t, or $3\theta$ in 3t, etc. Thus the angular velocity is constant.

The magnitude of the tangent or linear velocity is computed as the ratio between the arc length and the time taken to travel that distance:

$\displaystyle v=\frac{\theta r}{t}$

Replacing the formula for w, we have

$\boxed{ v=wr}$