GENERAL PRINCIPLES 35 



duration t, for a length /of the pendulum. In Paris, if / = 99'39 

 c.m., the value of g ==. 9'81 m. Thus in Paris the length of a 

 seconds pendulum is about 0'994 m. 



In fairly deep mines, t diminishes, therefore g increases, but the 

 most practically realisable pendulum is a heavy body attached to 

 a wire, fixed on a horizontal axis. It is what is called a com- 

 pound pendulum, the duration of its single oscillation being given 

 by the formula : 



M being the mass of the pendulum, I its moment of inertia (see 

 31), arid /' the distance irom the axis of suspension to the centre 

 of gravity. Tc find the equivalent simple pendulum we solve 

 the equation. 



from which, evidently, t' =.t and / = ^y> The musical metronome 



( 197) is a pendulum which has two divisions ; a heavy weight, 

 attached to the lower half, performs the movement, a smaller 

 weight, movable on the stem, permits of its regulation by modify- 

 ing the moment of inertia as will be considered later. 



The movements of a child playing in a swing are similar to 

 the movements of the pendulum. 



27. Movement of a Material System. In the movement of a 

 material system, the acting forces are either external forces 

 (gravity, reaction, pressure of air, etc.) or internal forces, equal 

 and opposite, acting between various points of the system. 

 These latter have a zero resultant, and if it is assumed which 

 is permissible that the mass of the system is concentrated at 

 the centre of gravity G, the movement, will be that of the point 

 G, under the action of the external forces only to the exclusion 

 of the others. This is the theorem of movement of the centre of 

 gravity. A body will therefore describe a parabola in space, 

 under the influence of its initial speed and the force of gravity. 

 In reality, it is the centre of gravity of that body which will 

 describe that parabola. An exploding bomb would be in this 

 category, the general centre of gravity describing a parabola, 

 whilst the fragments would have indeterminate movements. 



In the movement of a system it is useful to consider the product 

 mv of the mass by the velocity, a product which is called the 

 momentum. The force giving to the mass a speed, dv in the 

 infinitely short space of time dt gives 



F = m Jt 



