When an object rotates on an fixed axle (axis) without any translational motion, we look at its rotational kinematics.  Consider the pulley in the diagram below. As the mass rises, the string feeds off the edge of the pulley. If the mass rises 20 cm, then 20 cm of string will move along the arc length (edge) of the circle (pulley)

 

Mathematically, the formula relates the angle, , through which the pulley rotates and the arc length, s.

 

For example, if the pulley makes one complete revolution, the arc length, s, will equal one circumference, , while  = 360º or

 

Substituting these values gives us the equality

 

      

 

Therefore this formula requires that the angle of rotation be measured in radians. Recall from trigonometry that

 

 

If a pulley is rotating at a constant rate, we say that it has a constant angular velocity omega, , which is measured in radians/sec. The equation analogous to s = vt when an object translates at a constant velocity is .

 

Let's examine a horizontal turntable rotating at 33¹/₃ revolutions per minute, or rpm. This is equivalent to

 

 

Let's investigate the turntable a little further. Suppose there are three figurines located at three different distances from the center, or axis of rotation, on the surface of the turntable.

 

Next we want to compare their angular velocities and linear velocities. Since they are on the same rotating platform, all three share the same omega, . That is, all three revolve through . However, they each travel through a linear distance based on their unique radius and subsequent circumference. Our new relationship is

 

.

 

Therefore, the figurine closed to the center has the least linear velocity (speed in m/sec) while the figurine closest to the edge has the greatest linear velocity. In the picture below, four students are on a rotating merry-go-round. The student at the bottom right has the greatest linear velocity while the two students curled forward on the left have equal but smaller linear velocities.

 

image courtesy of http://ligiatcolindres.blogspot.com/2013/05/torque.html

 

 

Now let's suppose that we have two pulleys which share a common belt that passes along their circumferences.

 

image courtesy of http://tech.texasdi.org/gearsandpulleys

 

One pulley is called the driving pulley (usually connected to a motor) while the other is called the driven pulley. The common belt assures that the perimeters of both pulleys have the same linear velocity. Since the pulleys have different radii, one pulley must be rotating faster, or have a greater angular velocity.

 

 

This result tells us that if the large pulley has a radius that is three times that of the small pulley, the small pulley must be rotating (spinning) three times faster than the large pulley.

 

 

Now let's examine a situation in which a wheel is being "brought up to speed." That is, you want to increase the wheel's angular velocity. To change a rotating object's angular velocity means that you want it to accelerate. From linear motion we know that net F = ma, that is, a force is required to change an object's translational velocity. Therefore, rotationally, we need a torque to be applied.

 

 

Recall that the formula to calculate torque is , which can be rewritten in terms of the sine of the angle between the line of action of the force vector and the radius. We usually expressed the formula as

 

 

The analogous equation for net F = ma is   where I is the moment of inertia of the rotating mass and alpha, , is the angular acceleration, or the rate at which the angular velocity changes. The angular acceleration is measured in rad/sec².