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Mass and Viscous Friction
Consider a constant force F applied horizontally to a body of constant mass M moving along a stationary horizontal surface. Assume that the force required to overcome the frictional resistance alone is proportional to the velocity of the body. Such an assumption is valid to a good degree of approximation in the case of lubricated surfaces, once the moving body has attained an appreciable velocity. Then
where R_{F}v is the component of force required to overcome frictional resistance and M dv/dt is the component of force required to overcome the inertia of the body. The mechanical power input to this system is
as v and F are both in the same direction, the angle between these vectors thus being zero. Hence
In Eq. 130 the component of power represented by the term R_{F}v^{2} is the one converted into heat. The term Mv dv/dt represents the power expended in storing kinetic energy in the mass M. If v is constant, dv/dt = 0 and there is no energy storage because all of the power p is converted into heat. If dv/dt is positive the mass M undergoes acceleration and stores kinetic energy. On the other hand if dv/dt is negative the mass M decelerates and gives up its stored kinetic energy and the power R_{F}v^{2} being converted into heat is greater than the applied power p. The energy input into this system is expressed by
Further, if the body starts from rest, i.e., v = 0, when t = 0 under the influence of the constant force F we have
Mv^{2}/2 represents the stored energy and represents the mechanical energy converted into thermal energy. For this condition the solution of Eq. 129 expresses the velocity as a function of time as follows
Separation of the variables yields
from which
when t = 0, and v = 0, hence
and
Equation 134 is plotted in Fig. 16 and shows the final value v = F/R_{F} to be an asymptote that is approached as the time t approaches infinity. Time constant The reciprocal of the constant R_{F}/M in the exponent of Eq. 134 is called the time constant τ. Hence
and Eq. 134 can be rewritten as
The time constant also expresses the time required for the velocity v to attain the asymptotic value F/R_{F} if the acceleration were maintained constant at its initial value. This is indicated in Fig. 16 by the tangent to the curve at t = 0. Since the friction absorbs some of the applied energy in irreversible form, i.e., in the form of heat, the acceleration is not constant but decreases with time. Hence the velocity when t = τ sec is determined from Eq. 134 or 135 as follows
where F/R_{F} = the velocity as the time t approaches infinity.


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