Simple Harmonic Motion

In the preceding sections, the motion of a body when acted upon by a constant force was considered. The motion is one of constant acceleration. In this section, the motion of a body subjected to a variable force is considered. There is one common and important type of non-uniformly accelerated motion that can be analyzed easily. This motion is called periodic motion. A motion is said to be periodic if it repeats at regular intervals of time.

The motion of the hands of a clock, movement of a planet around the sun and the motion of the blades of a fan are examples of repetitive motion. Here, any point in the path of the body is crossed by the body in the same direction, at regular intervals of time. On the other hand, the motion of a body attached to a suspended spring, the motion of the prongs of an excited tuning fork, the motion of the plucked string of a musical instrument, the motion of the pendulum of a clock, the motion of the balance wheel of a watch are also examples of periodic motion but oscillatory. i.e., a to and fro motion. A given point in the path is crossed by the body in opposite directions at regular intervals. This is called oscillatory motion.

All oscillatory motions are periodic but not all periodic motions are oscillatory. On observing an oscillating system, like a loaded spring, one can conclude that two properties are involved. These are elasticity and inertia. When a spring is extended by a small force, it opposes the change in its shape. Restoring forces develop in the spring as soon the deforming force is removed. The spring bounces back to its original shape. This property is called elasticity. While returning to its normal state, it overshoots its position of equilibrium. This is because of inertia, which tries to keep the spring in a state of motion. The spring which has been compressed now, tries to attain the normal state. The process repeats itself and the body attached to the spring oscillates.

The restoring force in a spring (or in general, any elastic body) obeys Hooke's law. An English scientist Robert Hooke, in the year 1679, enunciated the fundamental law of elasticity in the form “ut tensio sic vis” which means, “As the tension, so is the extension”. Within the elastic limits, the deformation produced in a body is proportional to the restoring force.

or F = -K . y ------ 2.4 ___________ (1)

Here K is the force constant for the spring.

The negative sign indicates the tendency of the force to restore the body to its equilibrium state. In other words, F opposes increase in y. The body attached to the spring executes oscillations under the action of a force which is always directed towards the equilibrium position and always proportional to its displacement from the position. The motion is not only periodic, but also bounded i.e., the displacement on either side of the equilibrium position is confined within well-defined limits. The trigonometric functions namely, the sine and cosine functions are periodic as well as bounded. These functions are called harmonic functions. The displacement of a body subjected to elastic restoring forces can therefore be expressed in terms of sine and cosine functions or a combination of both. The musical instruments also involve motions of this type in their air columns or stretched strings.

Hence, this kind of motion in which a body oscillates on either side of its equilibrium position under the action of a force, which is proportional to the displacement and always directed towards the equilibrium position, in the absence of all frictional forces, is called simple harmonic motion. It is abbreviated as SHM.

In general, a body may vibrate under the action of restoring forces not directly proportional to displacement. However, such complicated motions can be considered as suitable combinations of two or more simple harmonic motions. Many types of motion, such as the oscillation of a pendulum, can be considered approximately simple harmonic, provided the amplitude is small. It must be noted that acceleration in SHM is not a constant and hence, the equations of motion of bodies with uniform acceleration cannot be applied in this case.