Technical Mathematics, Sixth Edition

Chapter 14. Radian Measure, Arc Length, and Rotation

OBJECTIVES When you have completed this chapter, you should be able to Convert angles between radians, degrees, and revolutions. Write the trigonometric functions of an angle given in radians. Compute arc length, radius, or central angle. Compute the angular velocity of a rotating body. Compute the linear speed of a point on a rotating body. Solve applied problems involving arc length or rotation.

Many objects in technology spin at constant speed: wheels, CDs, hard drives, gears, pulleys, motors, shafts, and the earth itself, rotating about its axis. In this short chapter we will develop the math to deal with these applications. We also need to be able to find the distance between two points on a circle, such as two locations on a band of a computer disk, or two points on a great or small circle on the globe. We should be able to find out, for example, how far the rack of Fig. 14-1 will move when the pinion (the small gear) rotates, say, 300°.

Our main tool for both arc length and rotation will be radian measure. We introduced radian measure earlier, but here we give it full treatment. Up to now we have usually used degrees as our unit of angular measure, but we will see that the radian is more useful in many cases. The reason is that the radian is not an arbitrary unit like the degree (why 360 degrees in a revolution rather than, say, 300?). The radian uses a part of the circle itself, (the radius) as a unit.

This chapter also ...