SOLUTION OF A DYNAMICAL PROBLEM. 



A PERFECTLY rough sphere is placed upon a perfectly rough 

 horizontal plane which is made to rotate with a uniform angular 

 velocity about a vertical axis : to determine the path described 

 by the sphere in space*. 



A sphere, resting on a perfectly rough horizontal plane, 

 receives a tangential impulse when the plane is made to move 

 in its own plane. This impulse gives a velocity to the centre 

 of the sphere and produces an angular velocity about a hori- 

 zontal axis. The centre of the sphere moves parallel to the 

 impulse, the axis of rotation is perpendicular to it; therefore 

 the point of contact moves parallel to the impulse and therefore 

 to the direction of motion of the centre. Therefore, as there is 

 no sliding, the centre moves in the same direction as that of the 

 motion of the plane supposed rectilineal. Moreover it is easily 

 seen that the velocity of the centre is to that of the point of 

 contact, or, which is the same thing, to that of the plane, as 



1 : 1 4- p > being the radius of the sphere, Jc its least radius 



of gyration. While the direction and velocity of the plane's 

 motion remain unaltered, no farther action occurs; when a 

 change takes place, a new tangential impulse is given to the 

 sphere, producing a new velocity of the centre parallel to its 

 own direction, and a new velocity of rotation about an axis at 

 right angles to it. The new velocity of rotation bearing to the 

 old the same ratio as the new velocity of the centre to the old, 

 the result is a compound velocity of the centre bearing the same 

 ratio as before to the velocity of the point of contact, and as 

 before parallel to it, and therefore still parallel to the direction 



* Walton's Problems in Theoretical Mechanics, p. 540. 



