83 ES. 



C. A cylinder n-sts with its ba-e I inclined 



plane; a string attar-hoi to its highest point over 



a pully at the tp of the inclinea plant-. hanjs vertically 

 and.- ;i wei-hf; the nortion of t t \\vni the 



is horizontal. Determine the con- 

 ditions of equilibrium. 



Results. Let W be the weight of the cylinder. I!" the 



ht attached to tl. a the inclination of the plane 



t the horizon; then J-F'= JKtan a, and tan a must imt ex- 



thr ratio of the diameter of the base of the cylinder to 



the height of the cylinder. 



7. A cone of given weight W is placed with its 



on an inclined plane, and supported by a weight JI" which 

 hangs by a string fastened to the vertex of the cone and 

 passing over a pully in the inclined plane at the 

 height as the vertex. Determine the conditions of equilibrium. 



Results. Let a be the inclination of the plane to the 

 horizon, 6 the semi- vertical angle of the cone; then 



^ 

 W = JFtan a, and tan 6 must not be less than '- sin 2a. 



o 



8. A smooth hemispherical shell whose base is cl 

 includes two equal spheres whose radii are one third of that 

 of the shell. The shell is fixed with its base vertical; find 

 the mutual pressures at all the points of contact. 



Results. Let R l be the pressure between the upper sphere 

 and the shell, R a that between the two spheres, lt % that be- 

 tween the lower sphere and the base of the shell, E 4 that 

 en the lower sphere and the curved put of the shell; 

 then 



W * _W p _3TF 7> _4TT 



7 ''~V3' '~V3~' '~V3~' 4 "73* 



9. A rectangular table is supported in a horizontal posi- 

 bjf four legs at its four angles: a given weight IT ! 



placed upon a given point of it, shew that the pressure on 

 leg is indeterminate, and find the greatest and least value 

 it can have for a given position of the weight. 



