The motion of a simple pendulum, swinging in air, dies out eventually.This is because the air drag and the friction at the support oppose the motion of the pendulum and dissipate its energy gradually. The pendulum is said to execute damped oscillations.
In damped oscillations, although the energy of the system is continuously dissipated, the oscillations remain apparently periodic. The dissipating forces are generally the frictional forces. To understand the effect of such external forces on the motion of an oscillator,let us consider a system as shown in figure where a block of mass m oscillates vertically on a spring with spring constant k.
The block is connected to a vane through a rod . The vane is submerged in a liquid. As the block oscillates up and down, the vane also moves along with it in the liquid. The up and down motion of the vane displaces the liquid, which in turn, exerts an inhibiting drag force (viscous drag) on it and thus on the entire oscillating system. With time, the mechanical energy of the block spring system decreases, as energy is transferred to the thermal energy of the liquid and vane.
Let the damping force exerted by the liquid on the system be Fd. Its magnitude is proportional to the velocity v of the vane or the block. The force acts in a direction opposite to the direction of v. This assumption is valid only when the vane moves slowly. Then for the motion along the x-axis Fd = –b v
where b is a damping constant that depends on the characteristics of the liquid and the vane. The negative sign makes it clear that the force is opposite to the velocity at every moment.
When the mass m is attached to the spring and released, the spring will elongate a little and the mass will settle at some height. This position is the equilibrium position of the mass. If the mass is pulled down or pushed up a little, the restoring force on the block due to the spring is
FS = –kx, where x is the displacement of the mass from its equilibrium position. Thus the total force actingon the mass at any time t is F = –k x –b v.
If a(t) is the acceleration of the mass at time t, then by Newton’s second law of motion for force components along the x-axis, we have m a(t) = –k x(t) – b v(t)
If the oscillator is damped, the mechanical energy is not constant but decreases with time.SHM related topics
Simple Pendulum
What is periodic and Oscillatory Motion is ? Displacement in Oscillatory motion
Simple Harmonic Motion
Velocity and acceleration of SHM
Energy of particle in SHM
Wave Motion an introduction
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