# Mass-centre sentence example

- The point G determined by (I) is called the mass-centre or centre of inertia of the given system.
- It is easily seen that, in the process of determining the masscentre, any group of particles may be replaced by a single particle whose mass isequal to that of the group, situate at the mass-centre of the group.
- It is easily seen from (6) that if the configuration of a system of particles be altered by homogeneous strain (see ELASTICITY) the new position of the mass-centre will be at that point of the strained figure which corresponds to the original mass-centre.
- The mass-centre is accordingly that point the mean square of whose distances from the several particles is least.
- It follows, by the preceding kinematic theory, that the mass-centre G of thc system will move exactly as if the whole, mass were concentrated there and were acted on by the extraneous forces applied paralle to their original directions.Advertisement
- For example, the mass-centre of I system free from extraneous force will describe a straight lin with constant velocity.
- As a final example we may note the arrangement, often employed in physical measurements, where a body performs small oscillations about a vertical axis through its mass-centre G, under the influence of a couple whose moment varies as the angle of rotation from the equilibrium position.
- 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.
- It is usually convenient to take as our base-point the mass-centre of the body.
- In this case the kinetic energy is given by 2T = M0 (u1 +v2+w1) +AP2 +Bq2 +Cr2 2Fqr 2Grp 2Hpq, (13) where M0 is the mass, and A, B, C, F, G, H are the moments and products of inertia with respect to the mass-centre; cf.Advertisement
- Cji=L, Cq=M, Ct=N, (18) where C is the constant moment of inertia about any axis through the mass-centre.
- Free Motion of a Solid.Before proceeding to further problems of motion under extraneous forces it is convenient to investigate the free motion of a solid relative to its mass-centre 0, in the most general case.
- Motion of a Solid of Revolution.In the case of a solid of revolution, or (more generally) whenever there is kinetic syminetry about an asks through the mass-centre, or through a fixec point 0, a number of interesting problems can be treated almost directly from first principles.