244 MOTION OF A RIGID BODY IN TWO DIMENSIONS. [CHAP. XI. 



Further taking M and m for the masses of the cube and sphere, and x 

 for the distance described by the cube in time t, obtain the equations of 

 motion by resolving for the system down the plane and at right angles to it 

 and taking moments for the sphere about its point of contact with the 

 cavity. 



Finally obtain the equation 



\ {| (M+ m}-m cos 2 6} d 2 + (M+ m} cos a sin 6 gj(a -b}= const. 



8. Prove that, when the plane of Example 7 is rough and e is the angle 

 of friction between it and the cube, the value of 6 at time t is given by the 

 equation 



i -^ [{ ( M+ m) cos - m cos 6 cos (6 - e)} 2 ] - m0 2 sin e 



~ cLO 



+ (M+m) cos a cos (6 - e) g/(a -b)=0. 



9. Motion of a circular disc rolling on a given curve under gravity. 



Let c be the radius of 

 the disc, $ the angle the 

 normal at the point of 

 contact makes with the 

 vertical, p the radius of 

 curvature of the curve at 

 this point. The centre of 

 the disc describes a curve g&amp;lt; &quot; 



parallel to the given curve and at a distance c from it, and the instantaneous 

 centre of rotation of the disc is at the point of contact, so that, if o&amp;gt; is the 

 angular velocity of the disc, we have velocity of centre = CG&amp;gt; = (p + c) &amp;lt;. 



Hence obtain the equation of energy 



where k is the radius of gyration of the disc about its centre, and the centre 

 of inertia is the centre of the disc. Investigate the corresponding equation 

 when the curve is concave to the disc. 



Prove that the disc can roll inside a cycloid the radius of whose generat 

 ing circle is a and whose vertex is lowest with uniform angular velocity 



a 



Prove that when the disc is uniform and rolls outside a cycloid, the radius 

 of whose generating circle is Jc and whose vertex is highest, the motion is 

 determined by the equation 



3c0 2 cos 4 1 &amp;lt; = g (3 + cos &amp;lt;) sin 2 \ $, 

 and that the disc leaves the cycloid when cos $=f. 



10. A uniform rod slides between a smooth vertical wall and a smooth 

 horizontal plane. Assuming the rod to move in a vertical plane it is required 

 to determine the motion. 



