EXAMPLES. 317 



43. A sphere of radius a rolling on a rough table with velocity V comes 

 to a slit of breadth b perpendicular to its path. Prove that, if there is no 

 restitution, the condition that it should cross the slit without jumping is 



F 2 &amp;gt; ijp-ga (1 - cos a) sin 2 a (14 - 10 sin 2 a)/(7 - 10 sin 2 a) 2 , 

 where b = 2a sin a and VI ga cos a &amp;gt; F 2 + lO^a, 



44. A sphere centre with its centre of gravity at a point G distant c 

 from is dropped vertically upon a plane of inclination a to the horizontal 

 so that G is above and GO is normal to the plane. Prove that, if the 

 plane is rough enough to prevent sliding, the kinetic energy lost in the impact 

 is to that of the sphere before impact in the ratio 



(1 - e 2 ) cos 2 a + (/c 2 sin 2 a)/{K 2 + (a + c) 2 } : 1, 



where K is the radius of gyration of the sphere about an axis through G at 

 right angles to GO. 



45. A circular disc of mass J/, radius a, and moment of inertia MK* 

 about its centre, spinning with angular velocity Q. impinges normally on a 

 rough rod of mass m. Prove that the angular velocity immediately after 

 impact is (M+m)K 2 Q/{(M+m) K 2 + ma 2 }, there being no restitution. 



46. Two rough circular discs of masses M l , M%, radii a lt 2 , an( ^ ra( ^ u f 

 gyration k lt k 2 about their centres, spinning about their centres with angular 

 velocities Q lt G 2 impinge directly, the relative velocity of the centres before 

 impact being F. Prove that, if there is no restitution, the kinetic energy 

 lost in impact is 



2 1 



2 



47. A truck of mass M l which has n^ pairs of wheels, each pair having an 

 axle, the mass of the axle and pair of wheels being m 1} the radius of gyration 

 of the axle and pair of wheels about the axis of the axle being & 1? and the 

 radius of either wheel being a l5 impinges directly on another truck running 

 on the same metals and for which the corresponding quantities are Jf 2 , n 2 , 

 m 2 , & 2 , a z- P r o ve that, if 



J\ r 1 = M l +n 1 m l k l z /a 1 2 and N 2 = M 2 + n 2 m 2 k 2 2 /a 2 2 , 



the impulse between them is ^ 1 aV 2 F(l + e)/(^ 1 + ^ r 2 )) where F is the relative 

 velocity before impact and e is the coefficient of restitution. 



48. Assuming that in the impact of the two trucks the force of restitution 

 in any state of strain of a buffer is /3 times that during compression, prove 

 that the relative velocity after impact is reversed in sense and is in magnitude 

 v//3 times that before. Further, if the forces of compression for the two 

 buffers are the products of E 19 E 2 and the contractions of the buffers and the 

 forces of restitution are the products of Eft, E&amp;lt; and the contractions, the 

 periods of compression and restitution are T and 2W/3, where 



the notation N 19 N 2 being the same as in Example 47. 



