31G 



CO 







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300K OF MATHEMATICAL PROBLEMS. 



ing as the initial point of contact is without, upon 

 circle 



2o>" (x* + y 9 ) = $5gx sin a, 



being horizontal, and the axis of x the line of 

 greatest slope. If the initial point of contact be the centre of 

 this circle, the path will be a horizontal straight line. 



v 



1519. A rough hollow circular cylinder, whose axis is ver- 



| tical, is made to rotate with uniform angular^ velocity w about 

 a fixed generator, and a heavy uniform sphere is rolling on the 

 concave surface ; prove that the equation of motion is 



'cty\ , 10a + & , 



c: 



where < is the angle which the common normal to the sphere and 

 cylinder makes with the plane containing the axis of revolution 

 and the axis of the cylinder at a time t, and a, a + b are the radii 

 of the sphere and cylinder. 



1520. A rough plane is made to revolve uniformly with 

 angular velocity w about a horizontal line in itself, and a sphere is 

 projected so as to move upon it; determine the motion, and 

 prove that, if when the plane is horizontal the centre of the sphere 

 is vertically above the axis of revolution and moving parallel to it, 

 the contact will cease when the plane has revolved through an 

 angle 6 given by the equation 



1 1 cos B o> 2 / 5 a 



a being the radius of the sphere. 



1521. A uniform rod is free to move about one extremity in 

 a vertical plane, that plane being constrained to revolve uniformly 

 about a vertical axis through the fixed extremity with angular 

 velocity <>, and the greatest and least angles which the rod makes 

 with the vertical are a, (3; prove that 



s (cos a -1- cos fi) + 3g= 0, 



