Capacitors, Magnetic Circuits, and Transformers is a free introductory textbook on the physics of capacitors, coils, and transformers. See the editorial for more information....

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 RFv 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. 1-30 the component of power represented by the term RFv2 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 RFv2 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


Mv2/2 represents the stored energy and represents the mechanical energy converted into thermal energy.

For this condition the solution of Eq. 1-29 expresses the velocity as a function of time as follows


Separation of the variables yields

from which


when t = 0, and v = 0, hence



Figure 1-6. Velocity of a mass moving against viscous resistance under constant applied force.

Equation 1-34 is plotted in Fig. 1-6 and shows the final value v = F/RF to be an asymptote that is approached as the time t approaches infinity.

Time constant

The reciprocal of the constant RF/M in the exponent of Eq. 1-34 is called the time constant τ. Hence

and Eq. 1-34 can be rewritten as


The time constant also expresses the time required for the velocity v to attain the asymptotic value F/RF if the acceleration were maintained constant at its initial value. This is indicated in Fig. 1-6 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. 1-34 or 1-35 as follows


where F/RF = the velocity as the time t approaches infinity.

Last Update: 2011-01-04