We know that when a body moves to and fro about its mean position along a fixed axis, with equal intervals of time, its motion is known as vibratory motion e.g., the motion of pendulum, mass attached to a spring, the motion of cardboard pieces along with the water wave’s etc. Simple harmonic motion is a vibratory motion which can be explained by the following examples.
Mass Attached to a Spring
To explain the simple harmonic motion, consider the motion of a mass attached with a spring. As shown in fig. 12.3, a spring is placed on a smooth and horizontal surface. Its one end is attached to a firm support and a mass ‘m’ is attached to its other end (fig. 12.3-a). There is no extension in the spring in this state. Such a state or position of a mass is called its equilibrium position. If an external force F ext is exerted on the mass towards right, the length of the spring increases by an amount x, and the mass moves from its initial position O to position A. according to hook’s law, the external force F ext acting on the spring is directly proportional to the increase in length x of the spring i.e.,
F ext oc x
Or F ext = kx …………. (12.1)
Here k is a constant, which is called the spring constant. The value of k can be obtained from Eq. 12. 1 as.
K = F ext /x
The ratio of the external force acting on a spring to the increase in its length is called the spring constant. Its unit is Nm-1.
Under the influence of the external force, the length of the spring increases. After releasing this force, spring moves towards its equilibrium position. This motion of the spring is due to the restoring force (Fig. 12.3-b), which is a characteristic of the spring. During the motion if the displacement of the mass m is x then restoring force will be:
F = -kx ……………. (12.2)
If now the mass m is let free, it starts moving towards the point O. using Newton’s second law, the acceleration a of the mass m can be found out as:
A = f/m = -kx/m ……….. (12.3)
Since k/m is a constant, therefore
A oc -x
This means that the acceleration of the body is directly proportional to its displacement x from the mean position O and is always directed towards the mean position O. as the mass ‘m’ moves towards the point O its displacement x goes on decreasing. Resultantly, the acceleration a of the body also decreases. On reaching the point the point O, x becomes zero and so the acceleration a of the mass m also reduces to zero. But it may be noted that its velocity is maximum at this point. Due to inertia, the mass m does not stop at the point O but continues its motion towards left till it reaches the point A. during this motion, the spring is now compressed. Now the restoring force and the acceleration due to it, are opposite to the motion of the mass m. this means that the acceleration of the mass m again starts decreasing as it passes the point O and finally becomes zero as it reaches the point A. after coming to rest at the point a the body again returns to the point O under the action of the restoring force. This process continues and the body m keeps on vibrating between the points a and A: the motion of a mass attached to a spring is known as simple harmonic motion.
Mathematically, it can be proved that the time period t of the simple harmonic motion of a mass attached to a spring can be given by the following equation:
T=2π …………………… (12.4)