Mechanic irodov

Author:mirtitles
Language: eng
Format: epub
Tags: mechanics, irodov, mir publishers, physics, moscow
Publisher: mir publishers moscow

And finally, the sought displacement is Ar c = f V c dt = {ml2M) f v' dt = (m/2A/)Ar\

Another method of solution is based on a property of the centre of inertia. In the reference frame fixed to the pulley axis the location of the centre of inertia of the given system is described by the radius vector

r c = lMT t + (M - m) r 2 + ror 3 ]/2Af,

where r lf r 2 , and r 3 are the radius vectors of the centres of inertia of the mass M, the ladder, and the man relative to some point 0 of the given reference frame. Hence, the displacement of the centre of inertia Ar c is equal to

Ar c = [M Ar x + (M — m) Ar 2 + m Ar 3 ]/2Af,

where Ar lt Ar g , and Ar 3 are the displacements of the mass M , the ladder, and the man relative to the given reference frame. Since Ar x = — Ar 2 and Ar 3 = Ar 2 + Ar\ we obtain

Ar c = (m/2M) Ar'.

C4.3. A system comprises two small spheres with masses m 1 and m 2 interconnected by a weightless spring. The spheres are set in motion at the velocities Vi and v 2 , as shown in Fig. 67, whereupon the system starts moving in the uniform gravitational field of the Earth. Ignoring the air drag and assuming that the spring is non-deformed at the initial moment of time, find:

(1) the velocity V c (t) of the centre of inertia of this system as a function of time;

(2) the internal mechanical energy of the system in the process of motion.

Solution. 1. In accordance with Eq. (4.11) the velocity vector increment of the centre of inertia is d Y c = g dt. Integrating this equation, we get V c (t) — V c (0) = gt, where V c (0) is the initial velocity of the centre of inertia. Hence

V c (0 = ("hVi + mjVi)/^ + ro 2 ) + gt.

2. The internal mechanical energy of a system is its energy E in the C frame. In this case the C frame moves with the acceleration g, so that each sphere experiences two external forces in that frame: gravity r&jg and the inertial force — m t g. The total work performed

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