318 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



49. A sphere of mass ra falls vertically and impinges with velocity F 

 against a board of mass M which is moving with velocity U on a horizontal 

 plane. The upper face of the board is rough and the lower face smooth, and 

 the coefficient of restitution is e. Prove that, if the coefficient of friction 

 exceeds 2J/7/(7J/+2ra) (1 + e) V, the kinetic energy lost in the impact is 

 \m (1 - e 2 ) F 2 +mM U^(1M + 2m). 



50. A ball is let fall upon a hoop, of which the mass is I In of that of 

 the ball, and which is suspended from a point in its circumference, about 

 which it can turn freely in a vertical plane ; prove that, if e is the coefficient 

 of restitution, and a the inclination to the vertical of the radius passing 

 through the point at which the ball strikes the hoop, the ball rebounds in a 

 direction making with the horizontal an angle tan~ * {(1 + 1 ri) tan a e cot a}. 



51. A homogeneous sphere is allowed to fall on one end of a uniform 

 horizontal beam balanced on a horizontal axis through its centre of inertia. 

 Prove that the sphere will not rebound unless the mass of the beam is at 

 least three times as great as that of the ball, the coefficient of restitution 

 being unity. 



52. A plank of length 2a is turning about a horizontal axis through 

 its centre of gravity and a particle strikes the rising half, rebounds, and 

 strikes the other half, the coefficient of restitution being unity. Prove that, 

 if the motion indefinitely repeats itself, the inclination of the plank to the 

 horizontal must never exceed a where 7(7r+2a)tan a = wa 2 , / being the 

 moment of inertia of the plank about its axis, and m the mass of the particle. 



53. A wedge of mass M and angle a with a smooth face and a rough face 

 is placed with the smooth face on a table and a uniform sphere of mass m is 

 dropped upon it symmetrically. Prove that, if there is no restitution, the 

 kinetic energy is diminished by the impact in the ratio 



( M+ m) sin 2 a : M+m sin 2 a + f ( M+ m). 



54. Two equal rigid uniform discs, each in the shape of an equilateral 

 triangle, rest with two edges in contact. They are struck at the same instant 

 with equal blows P in opposite directions bisecting the common edge and one 

 other edge of each, so that they are pressed together and begin to slide one 

 over the other. Find the velocity v of the point of application of either 

 blow resolved in its direction, and prove that, if p is the coefficient of friction, 

 the kinetic energy generated in the system is (1 -pJ3)Pv, assuming no 

 restitution. 



55. A smooth oval disc is rotating with angular velocity a> on a smooth 

 horizontal plane about its centre of inertia which is fixed, when it strikes a 

 smooth rod of mass m at the middle point of the rod. Prove that the new 

 angular velocity is (I-mep 2 )a>/(I+mp 2 ), where / is the moment of inertia of 

 the disc about an axis through its centre perpendicular to its plane, p the 

 perpendicular from the centre of inertia to the normal at the point of contact, 

 and e the coefficient of restitution. 



