1 9 o RESEARCHES ON FUNGI 



APPENDIX 



THE MOTION OF A SPHERE IN A VISCOUS MEDIUM 

 Contributed by Dr. Guy Barlow. 



(The notation is the same as that employed in Chapters XV. and XVII.) 



As shown by Stokes, the resisting force on a sphere of radius a when moving 

 with velocity v is given by 



F = 6vfmv (1) 



Since the force is directly proportional to the velocity, it is evident that the 

 component of this force in any direction is also directly proportional to the 

 component of the velocity in that direction. The motion of the sphere when 

 projected under gravity can therefore be regarded as compounded of the 

 independent horizontal and vertical motions, and these may be conveniently 

 investigated separately. 



1. Fall from rest under gravity. 



The equation of motion is — 



dv 

 m -r = nig - oir/xav, 

 at 



where in is the mass of the sphere and v its velocity downwards at time /. 

 The density cr of the medium is here neglected. 

 This ecpiation may be written 



a=»- < 2 > 



where 



"When the steady terminal state is reached, -£~o, v = Y, hence from (2), 



dt 

 g 



(3) 



Substituting value of c and putting m = -Tra z p we obtain Stokes' expression 



for the terminal velocity 



V= 2 K l£ a2 (4) 



gpa< 

 9 M ' 



Equation (2) may now be written — 



Integration with initial condition v = o when t = o gives 



v=Y(l-e-^) (5) 



