# Damped Simple Harmonic Motion

We have learnt that the total energy of a harmonic oscillator remains constant. Once started the oscillations continue forever with a constant amplitude (which is determined from initial conditions) and a constant frequency (which is determined from force constant K and mass m of oscillator). Simple harmonic motions of this type are called free and undamped. In actual practice, however, the energy of the oscillator gradually decreases with time and it eventually comes to rest. This happens because, in actual physical systems, friction is always present.

We know that the elastic restoring force is displacementÃ¢â‚¬â€œdependent (F = -Kx). The damping or frictional force is an additional restoring force which is velocity-dependent , where b is a positive constant and is a measure of the friction present in the system. The equation of motion of an oscillator of mass m moving under a linear restoring force Ã¢â‚¬â€œKx and also subjected to a damping force -b will be .

To find out how the displacement x of the damped oscillator changes with time t, we need to solve this equation. The solution of this equation is outside the scope of this book. Here we simply quote the solution which is valid for low damping.

x(t) = A exp (-bt/2m) cos(Î±t + Î´)

where is the angular frequency of the damped oscillator.

We know that Ï‰

^{2}= K/m. Therefore

Thus Ï‰ is the angular frequency in the absence of damping; Ï‰ is called the natural frequency of the oscillator.

Equation shows that the motion of the damped oscillator is periodic (its period is 2Ï€ /Î± ) but not simple harmonic because its amplitude A exp (-bt/2m) decreases exponentially with time. The energy of the oscillation therefore decreases with time, the loss of energy due to friction appearing as heat in the medium (see Fig. 12.24).

Equation tells us that Î± < Ï‰ , i.e. the frequency of the oscillator decreases due to damping. Thus the friction has three effects: (i) it changes the simple harmonic motion to periodic motion, (ii) it decreases the amplitude of the oscillation and (iii) it reduces the frequency of oscillation.

**Example**

A trolley of mass 3.0 kg is connected two identical springs, each of force constant 600 N m^{-1}, as shown in Fig. 12.25. The trolley is displaced from its equilibrium position by 5.0 cm and released. (i) What is the period of resulting oscillations? (ii) What is the maximum speed of the trolley? (ii) If the trolley eventually comes to rest due to damping forces, how much is the total energy dissipated as heat? Assume the damping forces to be weak.

**Solution** ** **

In general, the motion of a damped oscillator is not simple harmonic. If the damping forces are weak, the motion is very nearly simple harmonic and all the formulae of SHM supply.

Now m = 3.0 kg, k_{1} = k_{2} = k = 600 Nm^{-1}, A = 5.0 cm = 0.05 m

(i) The time period of oscillation is [refer to Example 12.18, Fig 12.16(b)]

(ii) The maximum speed is

(iii) If the trolley eventually comes to rest, the entire energy of oscillation is dissipated as heat due to damping forces. Hence, total energy dissipated as heat is

**Example **

What is the answer to questions (i) and (ii) in example 12.30 if the springs are replaced by rubber bands, each of force constant 600 Nm^{-1}?

**Solution**

(i) A rubber band exerts a restoring force only when it is stretched. When it is compressed, it simply sags and hence no restoring force is called into play. At any position of the trolley during its oscillation, only one rubber band is stretched. Hence the time period of the oscillation is given by

(ii)

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