### Circles and Vectors

An ac sine wave represents the most efficient possible way that an electrical quantity can alternate. It has only one frequency component. All the wave energy is concentrated into this smoothly seesawing variation. It is like a pure musical note.

#### Circular Motion Suppose that you swing a ball around and around at the end of a string, at a rate of one revolution per second (1 rps). The ball describes a circle in space (above Fig. A). If a friend stands some distance away, with his or her eyes in the plane of the ball’s path, your friend sees the ball oscillating back and forth (above Fig.B) with a frequency of 1 Hz. That is one complete cycle per second, because you swing the ball around at 1 rps.

If you graph the position of the ball, as seen by your friend, with respect to time, the result is a sine wave (following Fig.). This wave has the same fundamental shape as all sine waves. Some sine waves are taller than others, and some are stretched out horizontally more than others. But the general waveform is the same in every case. By multiplying or dividing the amplitude and the wavelength of any sine wave, it can be made to fit exactly along the curve of any other sine wave. The standard sine wave is the function y = sin x in the coordinate plane. Position of ball (horizontal axis) as seen from the side, graphed as a function of time (vertical axis).

You might whirl the ball around faster or slower than 1 rps. The string might be made longer or shorter. This would alter the height and/or the frequency of the sine wave graphed in following Figure. But the sine wave can always be reduced to the equivalent of constant, smooth motion in a circular orbit. This is known as the circular motion model of a sine wave.