Oscillation of a Mass-Spring System

Oscillation of a Mass-Spring System:

Let us consider a mass-spring system consisting of a body of mass m attached with a spring of spring constant k. When the body is suspended from the spring, the spring stretches and the body hangs at some equilibrium position as shown in the figure. If the body is slightly pulled and released the mass-spring system executes up and down oscillatory motion.

Oscillation of Mass-Spring System

The oscillations are produced because of the property of elasticity of the spring and the property of inertia of the body. When the body is slightly pulled, the spring is stretched and by virtue of the property of elasticity, a restoring force comes into play which tries to bring back the mass-spring system to its equilibrium position. According to Hooke’s law, this restoring force is directly proportional to the strain and acts in such a direction as to restore the equilibrium.

When the system is left after giving a slight downward displacement, the body moves further in the downward direction due to the property of inertia. At every instant of the downward journey of the body, a restoring force acts in the upward direction which tries to decrease the speed of the down-going body till the speed is reduced to zero. After this the restoring force, which still acts in the upward direction reverses the direction of motion and the body is thus accelerated to move up. The body moves toward its mean position with increasing speed, the negative displacement decreases and the upward force reduces. When the body reaches its equilibrium position, it possesses its maximum speed, and the restoring force is reduced to zero. However, due to the property of inertia of motion, the body does not stop at its equilibrium position but overshoots on the other side. When the body moves upward, the spring is compressed and a restoring force once again comes into play. Since now the displacement is positive, the restoring force acts in the downward direction and retards the upward motion till the speed of the body is again reduced to zero. After this, the restoring force, which still acts in the downward direction, accelerates the body in the downward direction towards its equilibrium position and the body crosses its equilibrium position with a large speed. The mass-spring system once disturbed, thus, continues to oscillate up and down.

The oscillation of the mass-spring system is a typical example of a class of oscillations known as simple harmonic oscillations. Since the force decides the dynamics of motion and the acceleration gives the kinematics of motion, let us investigate what kind of force and acceleration act on the body executing simple harmonic motion. We find that the restoring force F acting on the mass-spring system that brings the system back to its equilibrium position is given by-

F = -ky

where y is the displacement from the equilibrium position and k is a constant of the spring, known as the spring constant or force constant. Since acceleration, a is given by a = F/m, we have-

a = – (k/m) y
or a ∝ -y

Thus, we find that the acceleration acting on the body of the mass-spring system is directly proportional to its displacement from the equilibrium position and it is always directed towards the equilibrium position.

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