By Ferdinand Beer
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Extra resources for Vector Mechanics for Engineers: Statics (8th Ed)
Example text
19) The time to reach the bottom of the cycloid, Eq. 20) z2 Applying the integral in Eq. 21) to Eq. 22) It is surprising that no matter where the particle starts its path along the trajectory, the time to reach the bottom of the cycloidal path is always the same. This seems counterintuitive when comparing two different starting points. x1 ; y1 / as in Fig. xi ; yi /. This argument does not prove the result, but it at least rationalizes it. As stated above, this problem was included as an illustration of the calculus of variations which is used as the tool applied to Hamilton’s principle to develop Lagrangian dynamics.
It will be seen that the solution is much easier. The inclined plane shown rests on a frictionless surface. The block of mass m is released from rest at the top of the frictionless incline. Using the coordinates in Fig. 10 use Lagrangian dynamics to find XR the acceleration of the inclined plane. Note that the block and plane move in opposite directions, x and X.
0 as it should. This problem will be worked later in this volume in a tidier fashion using Lagrangian dynamics (see Problem 14, Chap. 2). 10. 75) The motion is restricted to the range =2 < x < 3 =2. (a) Sketch the potential energy function for the region of interest. (b) What is the period T0 of the bounded motion for amplitudes small enough so that the motion can be considered to be simple harmonic? Explain. 20 1 Newtonian Physics Solution (a) The potential is a cosine function as shown in Fig.