Equations of Motion. 3 



is beins carried in the direction it is also bein<j: carried in 

 the direction of r by the velocity ii; consecjnently. at the end 

 of the time (// it will have fallen behind by an amount equal 

 to 



(?• + (/?•) dO — rdO or dr dO. 



From physics, s = h fl'- 



f=~ 



•' p ' 



where /=acceleration, s the space described, and / the time. 

 Substituting, 



dr do 



j=I -=2 uv* 



dt' 



which is the force due to this fall, multiplying by fvdo dr dz 

 we get for the total of this force 



2 uv prdo dr dz. 



The velocity along z as before contributes no comi^onent to 

 the forces in this direction. Equating the forces found and 

 simplifying, we get 



T^ dp ''V , c 



prdo ot 



Neither u nor v can contribute a component to the forces 

 acting in the direction of the z axis, therefore the equation 

 of these forces will be identical with the corresponding equa- 

 tion for rectangular coordinates. 



* If from an elevation on a uniformly rotating body a particle falls 

 towards the axis of rotation it con.stantly gains angular velocity with- 

 out the application of any force in the direction of f). This is true of 

 falling bodies on the surface of the earth. A rotating body, if contract- 

 ing, would increase its angular velocity. To maintain the motion 

 uniform under such conditions, a force must act in the direction op- 

 posite to the direction of motion. This force is 



(In 



2 uvp rdo dz dr ~rdo dr dz. 



rdo 



If such a force were acting the configuration of the element would be 

 maintained. If the motion were from instead of toward the center an 

 opposite force would be necessary to keep the velocity uniform or to 

 maintain the configuration of the element. 



