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| Frictional Effects |
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| An oscillator, in actual practice, almost always lies in a resisting medium, like air, oil etc., where part of its energy is dissipated in overcoming the opposing frictional or viscous forces and its amplitude, therefore, goes on decreasing progressively. Such forces, which are non-conservative in nature, have thus a damping, resistive or dissipative forces. In the absence of any such forces, the oscillations will continue, indefinitely, without any change in amplitude, as shown in the figure. |
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| Let us discuss in necessary detail the frictional effects on a harmonic oscillator. It has been shown by Mayevski that at ordinary velocities and most cases of interest to us fall in this category - the opposing, resistive or damping force is, to a first approximation, proportional to velocity and may thus be represented by |
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| where g is a positive constant, called damping coefficient of the medium and may be looked upon as the resistive force per unit velocity. |
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| So that, if there is no other force, other than this resistive or damping force, acting on the oscillating body or particle, of mass m, we have, in accordance with Newton's second law of motion, |
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| Here, m/g is usually denoted by, a constant, having the dimensions of time and called relaxation time. |
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| The constant 1/t
= g/m, or the resistive force per unit mass per
unit velocity, is often denoted by 2k, where k is called damping constant of the medium. |
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| where C is a constant of integration to be determined from the initial conditions. |
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| Thus, if at t = 0, v = v0, we have logev0= C. And, therefore, |
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 ………. (iv) |
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| clearly shows that the velocity decreases
exponentially with time, as shown by the curve in the figure between the
function e-t/t and t. |
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| We express this by saying that the velocity is damped, with time constant t.
As will be readily seen, at t = t, v = v0e-1 = v0 / e = v0 / 2.718 = 0.368 v0. |
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This enables us to define the time constant (or the relaxation time)  as the time in which the velocity of the oscillating particle falls to 1/eth (i.e., 0.368 or, roughly, one-third) of its initial value. |
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And, since the kinetic energy of the oscillating particle is given by  we have, on substituting the value of v from relation (iv) above,  Or, representing the initial kinetic energy  by T0, we have
T = T0 e-2t/t
indicating that the kinetic energy of the oscillating particle too falls exponentially with time, with a relaxation time half that for velocity, i.e.,  /2, which is only to be expected since K.E.  (velocity)2.
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| Putting dx/dt for v in relation (iv)
above, we have dx/dt = v0 e-t/t,
which, on integration, gives x = v0 te-t/t
+ C, where C is a constant of integration. |
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| Thus, the maximum value of x is the distance that would be covered by the particle in time t if its velocity remained constant at its initial value v0. |
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