Simple Harmonic Motion (SHM)

A body is said to be vibrating if it moves back and forth or to and fro about a point. Another term for vibration is oscillation. A special kind of vibratory or oscillatory motion is called the simple harmonic motion (SHM), which is the main focus of this chapter. We will discuss important characteristics of SHM and systems executing SHM. We will also introduce different types of waves and will demonstrate their properties with the help of ripple tank.

10.1 SIMPLE HARMONIC MOTION (SHM)

In the following sections we will discuss simple harmonic motion of different systems. The motion of mass attached to a spring on a horizontal frictionless surface, the motion of a ball placed in a bowl and the motion of a bob attached to a string are examples of SHM.

MOTION OF MASS ATTACHED TO A SPRING

One of the simplest types of oscillatory motion is that of horizontal mass- spring system (Fig.10.1). If the spring is stretched or compressed through a small displacement $x$ from its mean position, it exerts a force $F$ on the mass. According to Hooke's law this force is directly proportional to the change in length $x$ of the spring i.e.,

where $x$ is the displacement of the mass from its mean position $O$ and $k$ is a constant called the spring constant defined as

The value of $k$ is a measure of the stiffness of the spring. Stiff springs have large value of $k$ and soft springs have small value of $k$ .

It means that the acceleration of a mass attached to a spring is directly proportional to its displacement from the mean position. Hence, the horizontal motion of a mass- spring system is an example of simple harmonic motion.

The negative sign in Eq. 10.1 means that the force exerted by the spring is always directed opposite to the displacement of the mass. Because the spring force always acts towards the mean position, it is sometimes called a restoring force.

A restoring force always pushes or pulls the object performing oscillatory motion towards the mean position.

Initially the mass $m$ is at rest in mean position O and the resultant force on the mass is zero (Fig.10.1- a). Suppose the mass is pulled through a distance $x$ up to extreme position A and then released (Fig.10.1- b). The restoring force exerted by the spring on the mass will pull it towards the mean position O. Due to the restoring force the mass moves back, towards the mean position O. The magnitude of the restoring force decreases with the distance from the mean position and becomes zero at O. However, the mass gains speed as it moves towards the mean position and its speed becomes maximum at O. Due to inertia the mass does not stop at the mean position O but continues its motion and reaches the extreme position B.

As the mass moves from the mean position O to the extreme position B, the restoring force acting on it towards the mean position steadily increases in strength. Hence the speed of the mass decreases as it moves towards the extreme position B. The mass finally comes briefly to rest at the extreme position B (Fig. 10.1- c). Ultimately the mass returns to the mean position due to the restoring force.

This process is repeated, and the mass continues to oscillate back and forth about the mean position O. Such motion of a mass attached to a spring on a horizontal frictionless surface is known as Simple Harmonic Motion (SHM).