222 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



141. A rough hollow circular cylinder is made to rotate uniformly about 

 its axis which is horizontal, and a particle within it is projected from the 

 lowest point in a direction contrary to that of the motion of the neighbouring 

 parts of the cylinder with such velocity that it comes to rest at an end of the 

 horizontal diameter. Prove that, provided the angular velocity is great 

 enough, the next position of instantaneous rest is given by the least positive 

 root of the equation 



3 M (e 2 * 6 - cos ff) = (2 M 2 - 1) sin 0, 



6 being the angle between the axial planes through the two positions of 

 instantaneous rest, and p the coefficient of friction. 



142. A particle is projected horizontally with velocity V along the interior 

 surface of a rough vertical circular cylinder. Prove that, at a point where 

 the path cuts the generator at an angle &amp;lt;/&amp;gt;, the velocity v is given by the 

 equation 



ag/ v 2 = sin 2 &amp;lt;f&amp;gt; {ay/ F 2 + 2/z log (cot $ + cosec &amp;lt;)}, 

 and the azimuthal angle and the vertical descent are respectively 



/Jir V 2 [fa V Z 



d(b and / cot d&amp;gt; o?d&amp;gt;. 

 * ag * 1*9 ^ 



143. A particle moves on the surface of a rough right circular cone of 

 vertical angle 2a under no forces but the reaction of the surface. It is 

 projected at a distance r from the vertex with velocity V perpendicular to the 

 generator. Show that, when its path crosses a generator at angle x, the velocity 



is 7 e -Mcotacos Xj and the time to that point ig ^ 

 being the coefficient of friction. 



144. A particle is projected vertically upwards in a medium in which the 

 resistance is k (velocity) 2 . If u is the initial velocity and T the whole time of 

 motion prove that *JJc(Zulg T] is positive and increases as k increases. 



145. A particle is projected vertically upwards in a medium in which 

 the resistance is J. 2 (velocity) 2 . Prove that, if U, V are the velocities with 

 which the particle leaves and returns to the point of projection, 



1 1 _ 1 



f-2-^2-^2- 



146. A particle falls from rest under gravity through a distance x in a 

 medium whose resistance varies as the square of the velocity ; v is the velocity 

 acquired by the particle, V the terminal velocity, and v the velocity that 

 would be acquired by falling through a distance x in vacuo ; prove that 



