Rigid-body sentence example
For a rigid body the kinetic energy will, in general, consist of three terms (AW1 2 +BW2 2 +CW3 2) in addition to the translational energy.
The previous treatment of the motion of a rigid body had in every case been purely analytical, and so gave no aid to the formation of a mental picture of the body's motion; and the great value of this work lies in the fact that, as Poinsot himself says in the introduction, it enables us to represent to ourselves the motion of a rigid body as clearly as that of a moving point.
For Diatomic Or Compound Gases Clerk Maxwell Supposed That The Molecule Would Also Possess Energy Of Rotation, And Endeavoured To Prove That In This Case The Energy Would Be Equally Divided Between The Six Degrees Of Freedom, Three Of Translation And Three Of Rotation, If The Molecule Were Regarded As A Rigid Body Incapable Of Vibration Energy.
At a later stage in our subject the conception of the ideal rigid body is introduced; this enables us to fill in some details which were previously wanting, but others are still omitted.
Hence a rigid body not constrained in any way is said to have six degrees of freedom.Advertisement
The six independent quantities, or co-ordinates, which serve to specify the position of a rigid body in space may of course be chosen in an endless variety of ways.
We proceed to sketch the theory of the finite displacements of a rigid body.
It follows from Eulers theorem that the most general displacement of a rigid body may be effected by a pure translation which brings any one point of it to its final position 0, followed by a pure rotation about some axis through 0.
It thus appears that an infinitesimal rotation is of the nature of a localized vector, and is subject in all respects to the same mathematical Jaws as a force, conceived as acting on a rigid body.
If we imagine a rigid body to be acted on at given points by forces of given magnitudes in directions (not all parallel) which are fixed in space, then as the body is turned about the resultant wrench will assume different configurations in the body, and will in certain positions reduce to a single force.Advertisement
In the same way, the work dne by a force acting on a rigid body in any infinitely small displacement of the body is the scalar product of the force into the displacement of any point on the line of action.
The work of a couple in any infinitely small rotation of a rigid body about an axis perpendicular to the plane of the couple is equal to the product of the moment of the couple into the angle of rotation, proper conventions as to sign being observed.
Considering a rigid body in any given position, we may eontemplate the whole group of infinitesimal displacements which might be given to it.
A general criterion for the case of a rigid body movable in two dimensions, with one degree of freedom, can be obtained as follows.
In the case of a rigid body we must suppose that those forces adjust themselves so as to preserve the mutual distances of the various particles unaltered.Advertisement
The increase of the kinetic energy of a rigid body in any interval of time is equal to the work done by the extraneous forces acting on the body.
Proceeding to the general motion of a rigid body in two dimensions we may take as the three co-ordinates of the body the rectangular Cartesian co-ordinates x, y of the mass-centre G and the angle C through which the body has turned from some standard position.
The equation of energy for a rigid body has already been stated (in effect) as a corollary from fundamental assumptions.
The motion of a rigid body in the most general case may be specified by means of the component velocities u, v, w of any point 0 of it which is taken as base, and the component angular velocities p, q, r.
Moving A xes of ReferenceFor the more general treatment of the kinetics of a rigid body it is usually convenient to adopt a system of moving axes.Advertisement
If we now apply them to the case of a rigid body moving about a fixed point 0, and make Ox, Oy, Oz coincide with the principal axes of inertia at 0, we have X, u, v=Ap, Bq, Cr, whence A (B C) qr = L,
This is the principle of the modification of motion by the lever, which consists of a rigid body turning about a fixed axis called a fulcrum, and having two points at the same or different distances from that axis, and in the same or different directions, one of which receives motion and the other transmits motion, modified in direction and velocity according to the above law.
Velocity Ratio of Components of Motion.As the distance between any two points in a rigid body is invariable, the projections of their velocities upon the line joining them must be equal.
If B atops rolling, then the two cylinders continue to move as though they were parts of a rigid body.
The mode of distribution of a force applied to a solid body requires to be considered when its stiffness and strength are treated of; but, in questions respecting the action of a force upon a rigid body considered as a whole, the resultant of the distributed force, determined according to the principles of statics, and considered as acting in a single line and applied at a single point, may, for the occasion, be substituted for the force as really distributed.Advertisement
If the small displacements of a rigid body be subject to one constraint, e.g.
I only use rigid body refinement and there is no TLS refinement in Refmac.