An Elementary Treatise on Theoretical Mechanics, Parte 3Macmillan and Company, 1894 |
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Términos y frases comunes
acceleration angular momentum angular velocity axes ball central force centre centroid G co-ordinates coefficient coefficient of restitution components cone constant constraints cosines couple Ḥ curve d'Alembert's equation d'Alembert's principle denote determine differential direction displacement distance dt dt effective force elastic ellipse equal equations of motion Euler's Euler's equations expressed external forces fixed axis fixed point force F force-function forces acting free rigid body given gives Hence hyperbola impulse F instantaneous axis integration invariable plane kinetic energy left-hand member m₁ moment of inertia momental ellipsoid moving multiplied mv² orbit origin parallel particle of mass polhode principal axis pure rotation radius of inertia radius vector resultant force resulting couple right-hand member sphere surface system of impulses tangent tions variable vector Ḥ virtual displacement w₁ zero
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Página 135 - The moment of inertia of any mass M for a line is equal to the sum of the moments of inertia of the same mass for any two rectangular planes passing through the line. Thus, in particular, the moment of inertia for the axis of x in a rectangular system of co-ordinates is equal to the sum of the moments of inertia for the ^jr-plane and .ry-plane.
Página 153 - ... viz.: an ellipsoid, a hyperboloid of one sheet, and a hyperboloid of two sheets. This can also be shown analytically, as there is no difficulty in proving that the equation (20) has three real roots, say...
Página 49 - Kinematically, this equation means that the sectorial velocity remains constant. It can be put into the form ^5= _£-=*' dt 2 m whence, by integration, we find S-S0 = c'(t-t0).
Página 67 - All these facts led Newton to suspect that the force of terrestrial gravitation, as observed in the case of falling bodies on the earth's surface, might be the same as the force that keeps the moon in its orbit around the earth. This inference could easily be tested, since the acceleration g of falling bodies as well as the moon's distance and time of revolution were known. Let m be the mass of the moon, a the major semi-axis of its orbit, T the time of revolution, r the distance between the centers...
Página 46 - P". Proceeding in this way, we obtain a series of points P, P', P", P'", • • -, which in the limit will form a continuous curve whose direction at any point coincides with the direction of the resultant force at that point. Such a line is called a line of force. The lines of force evidently form the orthogonal system to the family of equipotential surfaces. The differential equations of the lines of force are therefore : dx _ dy _ ds ~ =
Página 48 - The product mvp of the momentum and its perpendicular distance from the origin is called the moment of momentum, or the angular momentum, of the particle about the origin. As the moment of mv is equal to the algebraic sum of the moments of its components, we have //!f ffr mvp = F1g.
Página 43 - T+V=T0+V0, (8) which expresses the principle of the conservation of energy for a particle : the total energy, ie, the sum of the kinetic and potential energies, remains constant throughout the motion whenever there exists a force-function. In other words, whatever is gained in kinetic is lost in potential energy, and vice versa. 509. As the force-function U is a function of the co-ordinates x, y, z alone, an equation of the form U = c, where c is a constant, represents a surface which is the locus...
Página 99 - A farticle of mass 1n, subject to gravity, is constrained to remain on the surface of a sphere of radius r. If the constraint is produced by a weightless rod or cord joining the particle to the center of the sphere, the rod or cord describes a cone, and the apparatus is called a conical or spherical pendulum Taking the center O of the sphere as origin (Fig.