26 KINETICS OF A PARTICLE. [49. 



of the equation is that the increase in the kinetic energy is equal 

 to the work done by the resultant force. This is the principal of 

 work or of kinetic energy (or of vis viva) for the case of the rec- 

 tilinear motion of a particle. 



Thus, for a falling body, F is constant and equal to the weight 

 mg of the body; hence, equation (12) gives, if s be counted, 

 positive downwards, 



1 - \rnvg = mg(s S Q ), 



where the right-hand member represents the work done by the 

 weight of the body, i.e. by the attractive force of the earth 

 during the fall of the body through the distance s S Q . 



For a body thrown vertically upwards with an initial velocity 

 v, we have 



J mi? mv<?= mgs y 



if s be counted from the starting point and positive upwards. 

 The kinetic energy here decreases, the initial kinetic energy, 

 \ mv^, being, so to speak, consumed by the work done against 

 the force of gravity. 



49. Inclined Plane. When a particle of mass m is moved 

 uniformly up a smooth inclined plane from P Q to P l (Fig. 8), 



the work done against gravity 

 FV^ is equal to the work that would 

 8^^^ have to be done in raising the 



\h particle m through the vertical 



height PP l of P l above the 



Po^p \ _j\ * L_ _ j n i t i a ] point /V For, putting 



P Q P l =s, PP^=h t and denot- 

 ing the inclination of the plane 

 Pig. s. to the horizon by 0, we have 



for the work, 



mg-sin 6 s=-mg s sin Q = mg - h. 

 If the plane be rough, the coefficient of friction being /*, the 



