gear with diameter of 16.0in rotates 45 revolutions every 2 minutes.

Question

find the linear and angular velocities of a point on the outer edge of the gear. Give answer's in both exact and approximate form.

Answer 1

First let's do angular velocity.  One full rotation is 2π radians.  Since the gear rotates 45 times in two minutes, that's 45 * 2π / 2 = 45π radians per minute.  Since radians are generally considered "unitless", you can say that the angular velocity is 45π / minute.  In approximate form, that's 141.37 / minute.

 

You can also state this in degrees.  One full rotation is 360°, so 45 rotations in 2 minutes is 45 * 360 / 2 = 8100° per minute.

 

By the way, the angular velocity doesn't depend on where you measure it.  Since it's a rigid object, the entire gear, on the edge or close to the center, has the same angular velocity.

 

Now the linear velocity.  The easiest way to do this is to multiply the angular velocity (in radians) by the radius.  So that's 45π / minute * 8 in = 360π inches / minute.  In approximate form, that's 1139.97 inches / minute.