40 THE HUMAN MOTOR 



Chauveau) ; the idea, although correct, is a contradiction of 

 terms : work, pre-supposing movement, and statics implying 

 absence of movement. 



30. Principle of Vis Viva. Let there be a body with a weight 

 P falling from a height h. The weight will perform work 

 T = P X h. It is known that P = mg and that h = \ gt 2 ; there 

 fore it follows that : 



At the time /, the speed is v = gt, therefore 



g*t 2 = v*, 

 and finally 



T = Ph = % mv 2 . 



Whether it is question of a point or of a material system, the 

 equation T = ^mv 2 applies if in the working of the system the 

 sum Te + TJ, of external and internal work is not equal to zero. 



The product mv 2 has been named "vis viva" (Leibnitz), or energy 

 of motion, and the semi-product -| mv 2 " live power " (Belanger) 

 or actual energy. Thus the work done in a material system 

 endows it with a certain actual energy or " live power " which 

 becomes, in a way, potential work in the system. If, for example, 

 a spring is coiled by depressing it with a weight P, the weight 

 will perform work P x r, by a contraction r of the spring, and if 

 the weight is removed, the spring will expand performing work 



mv 2 = Pr. 



The stored-up, latent, or potential energy does work in 

 becoming actual or kinetic energy. ( l ) 



If a fly-wheel in motion receives energy, its speed will -change 

 from v to v' and its actual or kinetic energy from ^mv 2 to -J- mv' 2 . 

 Thus the work developed is found in the increase" of energy. It 

 is written : 



T = 1 mv' 2 1 mv 2 = \ m (V 2 v 2 ). 



31. Given that the speed of a body rotating round a fixed 

 axis is tor ( 2), its " vis viva " will be w*to 2 r 2 , and its energy ^ 

 mo> 2 r 2 . The radius r varies according to the position of each 

 point in relation to the axis. A value of r designated by p is 

 calculated for the given body. The whole of the bulk in rotation 

 is imagined as concentrated at a circumference having the radius 

 P, and then co 2 Mp 2 , M being the total mass, p is called the radius 

 of gyration of the body and Mp 2 its moment of inertia. Thus the 

 energy of a rotating body is the semi-prodnct of the square of 

 the angular speed by the moment of inertia. 



(*) Expression already used by Lazare Carnot (loc. cit., p. 247). 



