By A S Ramsey
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Extra info for Dynamics, Edition: reprint 1929
24) The solution of the equation for vx (t) has been discussed in Chap. 2, with the result given by vx (t) = v0 e−α t cos ϕ . 25) admit an integration and enable one to find the functions x(t) and z(t) and the equation of the trajectory z(x), that we leave as an exercise. Example 4. A small body of mass m is sliding on a sphere of radius R, frictionless (Fig. 8).
13) Fres = −α v . For higher speeds, there is a quadratic term in velocity. In principle, there may be also cubic terms and so on. 14) where v = |v| is the magnitude of the velocity of a particle. 13), obviously, k(v) = α v. 14). 14) can be viewed as a more general form of friction force, also called a dissipative force. Of course, the function k(v) can be expanded in power series in velocity. 13) is the most important one. In this case the equations of motion are typically simpler. For higher speeds, the quadratic term begins to dominate and terms of higher orders may also be relevant.
The electromagnetic force can be relevant in mechanics, but it can be eliminated – just consider a body with no charge. All in all, we can see that all other fundamental forces except gravity can not, in principle, prevent the realization of the idea of a free particle. To better understand the issue of the existence of an inertial frame, we note that there are two kinds of forces, namely inertial and gravitational ones, which are proportional to the mass of the particle. So far, all the experiments have confirmed the equality of the gravitational and inertial masses with great precision.