274 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



2. Prove that when the common normal at the points that come into 

 contact passes through the centres of inertia of the impinging bodies the 

 rules are equivalent even if the bodies are rough enough to prevent sliding. 



[This includes the case of the impact of rough spheres.] 



3. A uniform sphere of radius a and mass m moving without rotation 

 impinges directly on a smooth uniform cube of side 2a and mass m , the line 

 of motion of the sphere being at a distance b from the centre of inertia of 

 the cube. Prove that, if e = 0, the kinetic energy lost in the impact is to 

 that of the sphere before impact in the ratio 



4. A uniform rod, falling without rotation, strikes a smooth horizontal 

 plane. Prove that, for all values of e, the angular velocity of the rod im 

 mediately after impact is a maximum if the rod before impact makes with 

 the horizontal an angle cos&quot; 1 1/^/3. 



5. A sphere whose centre of inertia coincides with its centre of figure is 

 moving in a vertical plane and rotating about an axis perpendicular to that 

 plane when it strikes against a horizontal plane which is sufficiently rough 

 to prevent sliding. Prove that, for all values of e, the sphere will rebound 

 at an angle greater or less than if there were no friction according as the 

 lowest point of it at the instant of impact is moving forward or backward. 



6. A disc of any form of mass m, moving in its plane without rotation 

 and with velocity V at right angles to a fixed plane, strikes the plane so that 

 the distances of the centre of inertia from the point of impact and from the 

 plane are r and p. Prove that, if the plane is sufficiently rough to prevent 

 sliding, the impulsive pressure is 



where k is the radius of gyration of the disc about its centre of inertia. 



7. A ball spinning about a vertical axis moves on a smooth table, and 

 impinges directly on a vertical cushion. Prove that, if 6 is the angle of 

 reflexion, the kinetic energy is diminished in the ratio 



10 + 14 tan 2 &amp;lt;9 : 10e- 2 + 49 tan 2 0, 

 the cushion being sufficiently rough to prevent sliding. 



8. A circular disc of mass M and radius c impinges on a rod of mass m 

 and length 2a which is free to turn about a pivot at its centre, and the point 

 of impact is distant b from the pivot. Prove that, if the direction of motion 

 of the centre of the sphere makes angles a and /3 with the rod before and 

 after collision, then 



2 (3J/& 2 + ma?) tan ^=3 (3Mb 2 - ma 2 ) tan a, 

 the edges in contact being sufficiently rough to prevent sliding. 



9. A uniform sphere is let drop upon a uniform smooth hoop which is 

 free to turn in its own vertical plane about its highest point, the centre of 

 the sphere moving in the plane of the hoop. Prove that, for the sphere 

 to rebound horizontally, it must strike the hoop at an angular distance 

 tan~ 1 x /(2e/3) from the highest point. 



