**Introduction**

Scientists and engineers break the ac cycle down into small parts for analysis and reference. One complete cycle can be likened to a single revolution around a circle.

**Sine Waves As Circular Motion**

Suppose that you swing a glowing ball around and around at the end of a string at a rate of one revolution per second. The ball thus describes a circle in space (Fig. 13-6*a*). Imagine that you swing the ball around so that it is always at the same level; it takes a path that lies in a horizontal plane. Imagine that you do this in a pitch-dark gymnasium. If a friend stands some distance away with his or her eyes in the plane of the ball’s path, what does your friend see? Only the glowing ball, oscillating back and forth. The ball seems to move toward the right, slow down, and then reverse its direction, going back toward the left (see Fig. 13-6*b*). Then it moves faster and faster and then slower again, reaching its left-most point, at which it turns around again. This goes on and on, with a frequency of 1 Hz, or a complete cycle per second, because you are swinging the ball around at one revolution per second.

If you graph the position of the ball as seen by your friend with respect to time, the result will be a sine wave (Fig. 13-7). This wave has the same characteristic shape as all sine waves. The standard, or basic, sine wave is described by the mathematical function *y* = sin *x* in the *(x, y)* coordinate plane. The general form is *y* = *a* sin *bx* , where *a* and *b* are real-number constants.

**Degrees**

One method of specifying fractions of an ac cycle is to divide it into 360 equal increments called *degrees* , symbolized ° or deg (but it’s okay to write out the whole word). The value 0° is assigned to the point in the cycle where the magnitude is zero and positive-going. The same point on the next cycle is given the value 360°. Halfway through the cycle is 180°; a quarter cycle is 90°; an eighth cycle is 45°. This is illustrated in Fig. 13-8.

**Radians**

The other method of specifying fractions of an ac cycle is to divide it into exactly 2 *π* , or approximately 6.2832, equal parts. This is the number of radii of a circle that can be laid end to end around the circumference. One *radian* , symbolized rad (although you can write out the whole word), is equal to about 57.296°. Physicists use the radian more often than the degree when talking about fractional parts of an ac cycle.

Sometimes the frequency of an ac wave is measured in radians per second (rad/s) rather than in hertz (cycles per second). Because there are 2 *π* radians in a complete cycle of 360°, the *angular frequency* of a wave, in radians per second, is equal to 2 *π* times the frequency in hertz. Angular frequency is symbolized by the lowercase italicized Greek letter omega *(ω)* .

**Fractions of a Cycle Practice Problems**

**Problem 1**

What is the angular frequency of household ac? Assume that the frequency of utility ac is 60.0 Hz.

**Solution 1**

Multiply the frequency in hertz by 2 *π* . If this value is taken as 6.2832, then the angular frequency is

ω = 6.2832 × 60.0 = 376.992 rad/s

This should be rounded off to 377 rad/s because our input data are given only to three significant figures.

**Problem 2**

A certain wave has an angular frequency of 3.8865 × 10 ^{5} rad/s. What is the frequency in kilohertz? Express the answer to three significant figures.

**Solution 2**

To solve this, first find the frequency in hertz. This requires that the angular frequency, in radians per second, be divided by 2π, which is approximately 6.2832. The frequency *f* _{Hz} is therefore

*f* _{Hz} = (3.8865 × 10 ^{5} )/6.2832

= 6.1855 × 10 ^{4} Hz

To obtain the frequency in kilohertz, divide by 10 ^{3} , and then round off to three significant figures:

*f* _{kHz} = 6.1855 × 10 ^{4} /10 ^{3}

= 61.855 kHz ≈ 61.9 kHz

Practice problems of these concepts can be found at: Alternating Current Practice Test

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