0.29 Phy1260: circular motion -- the mathematics of circular motion  (Page 3/7)

Reduce the time interval

If we allow the time interval dT to be come shorter and shorter, we are averaging over smaller and smaller time intervals. In the limit, as dTapproaches zero, wAvg becomes w, which is the instantaneous angular velocity.

Angular velocity is a signed quantity

Angular velocity is also a signed quantity with the sign indicating the direction of rotation. By convention, counter clockwise rotation is viewed aspositive rotation. The sign of angular velocity is the same as the sign of the angular displacement that forms the basis for the angular velocity.

Units of angular velocity

The units of angular velocity are typically degrees per second or radians per second. You will learn later that radians is a dimensionless quantity.Therefore, when angular velocity is measured in radians per second, it often appears simply as

w = 10/sec

Radian measure

The most familiar measurement of angles, in the U.S. is in degrees. However, in some situations, it is more convenient to measure angles in radiansthan in degrees.

This becomes most apparent when we need to relate angular displacement or angular velocity with the distance traveled or the tangential speed of a point ona rotating object.

Simulate with a graph board

I recommend that you use your graph board to simulate an angle of one radian.

Using your graph board along with some string and pushpins, draw a Cartesian coordinate system. Then draw a circle with a convenient radius withits center at the origin of your coordinate system.

Make the arc match the radius

Cut a piece of string to the length of the radius of the circle. Then, beginning at the intersection of the circle and the horizontal axis, lay thestring along the circumference of the circle moving in a counter clockwise direction. Put a pushpin at the point where the string ends. Then stretch arubber band from that point back to the center of the circle.

Measure the angle

Using your protractor, measure the angle that the rubber band makes with the horizontal axis. That angle should be about 57.3 degrees, which is one radian.

Measure the number of radians in 360 degrees

Now, using the string whose length is equal to the radius of the circle as a measuring tool, determine how many strings of that length you can lay end-to-endaround the circumference of the circle.

You should find that about 6.28 (2*pi) such strings are required to go all the way around the circumference of thecircle.

An angle in radians is a ratio of lengths

An angle measured in radians is a ratio of two values, each of which have units of length. Therefore, such an angle has no dimensions.

An example measurement