Simple Harmonic Motion

Vibratory motion is a common occurrence in the physical world. Here a weight is oscillating with one of the most important kinds of vibratory or periodic motion, simple harmonic. Each individual particle of a plucked violin string moves with simple harmonic motion. The periodic motion of a pendulum is approximately simple harmonic. To see exactly what is meant by simple harmonic motion, let's study the properties of a spring. We'll start with it neither stretched nor compressed. Now, if we displace its free end some definite distance from the original position, the force required to do so is proportional to the displacement. The same proportional relationship between force and displacement occurs on the opposite side, where the spring is compressed. Now, let's attach a body to the spring and assume that it rests on a frictionless surface. If we set it into motion, the spring, due to its elastic nature, will exert a variable force upon the body. The spring alternately pushes and pulls. The force exerted by the spring is always in a direction opposite to that of the displacement. F, the force exerted by the spring, is proportional to the magnitude of X, the displacement. To be correct, we must say that F is proportional to minus X because the force and the displacement are always in opposite directions. Another way of stating this relationship is F equals minus a constant times X. This is the special type of force which will cause the body to oscillate back and forth with simple harmonic motion. We'll return later to this relationship, F equals minus KX. Let us now consider some general features of simple harmonic motion. We are aided considerably by a circle diagram or circle of reference. This circle diagram, as we shall see, may be used to exactly describe the behavior of the body on the spring. The circle radius is made equal to the maximum displacement of the body from the central equilibrium point. This maximum displacement is called the amplitude. Now, let's take a point on the circle of reference and suppose that it revolves at a constant speed. The projection of its position upon a horizontal diameter will move back and forth. By a proper choice of the speed of the revolving point, this back and forth motion will become an exact replica of the body's simple harmonic motion. We should note that there is no circular motion of the body on the spring. It moves along a straight line. The period in simple harmonic motion is the time for one cycle, whether we start and finish at the central equilibrium position or at any other point. The frequency is the number of cycles per unit time. In order to see how the value of X, the instantaneous displacement, varies with the time, we may use the circle of reference. The maximum displacement, X sub M, is equal to the radius of the circle. The smallest value of X is zero. The magnitude of X, in general, varies with the time. We can plot the values of X by tracing the instantaneous positions of the body upon a strip of paper moving at a constant speed. We will then have the displacement X versus the time. This graph represents a sine curve. We'll now use the circle of reference to indicate the varying velocity of the body. V, the simple harmonic velocity along the diameter, is identical in magnitude with the horizontal component of V sub zero, the velocity of the point on the circumference. The magnitude of V sub zero remains constant, but the magnitude of V varies. In simple harmonic motion, the velocity is a maximum at the center and is zero at the ends. A very important feature of simple harmonic motion is the way in which the acceleration varies over different parts of the cycle. While V sub zero is, of course, constant in magnitude, its direction is not constant. Therefore, the point on the circumference has acceleration. We call it centripetal acceleration. It is always directed toward the center and its value is V sub zero squared over R. The horizontal component of this acceleration is identical with the simple harmonic acceleration A along the diameter. Let's watch the cycle of variation in the simple harmonic acceleration. We can see that it is zero at the center and maximum at the ends. At any point, the acceleration A can be expressed in terms of the displacement X. These two triangles are obviously similar. Therefore, the simple harmonic acceleration A is to the centripetal acceleration of the revolving point, V sub zero squared over R, as X is to R. Rewriting this equation, we find that A equals V sub zero squared over R squared times X. Now, we know that V sub zero squared over R squared is constant. Therefore, A is proportional to X. To be correct, we must say that A is proportional to minus X because the direction of A is opposite to the direction of X. So, we see that the circle of reference shows us that A is proportional to minus X. Now, we know that in any system, the resultant force F equals MA, mass times acceleration. In other words, F is proportional to A. Now, you'll remember that at the beginning we saw that the force of the spring acting on the body is proportional to minus X. Therefore, A is proportional to minus X. And that is exactly what the circle diagram indicates. Therefore, the circle of reference accurately describes the motion we call simple harmonic.