By Donald T. Greenwood
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Additional info for Advanced Dynamics
We can use the spherical coordinates θ and φ as independent generalized coordinates, that is, there are no constraints on the qs. Hence, the ﬁrst condition is satisﬁed. 210) z = R(t) cos θ These transformation equations do not satisfy the second condition, so the system is rheonomic. Another classiﬁcation of dynamical systems involves the categories catastatic or acatastatic. A catastatic system satisﬁes the conditions that: (1) all constraints, if any, are catastatic; and (2) the system is capable of being continuously at rest by setting all the ˙ equal to zero.
Note that F f is independent of the contact area. In the case vr = 0, that is, for no sliding, the force of friction can have any magnitude less than that required to initiate sliding. The actual force in this case is obtained from the equations of statics. Although the force required to begin sliding is actually slightly larger than that required to sustain it, we shall ignore this difference and assume that F f as a function of vr is given by Fig. 175). 176) This leads to the idea of a cone of friction having a semivertex angle φ.
The right-hand term also vanishes if ρc and r¨ p are parallel. Accelerating frames Consider a particle of mass m i and its motion relative to a noninertial reference frame that is not rotating but is translating with point P at its origin (Fig. 10). 116) where Fi is now the total force acting on the particle. 117) The term −m i r¨ p can be regarded as an inertia force due to the acceleration of the frame. Note that the same equation of motion is obtained if we assume that the frame attached to P is not accelerating, but instead there is a uniform gravitational ﬁeld with an acceleration of gravity −¨r p .
Advanced Dynamics by Donald T. Greenwood