316 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



38. The corners A, B of a uniform rectangular disc A BCD are free to 

 slide on two smooth fixed rigid wires OA, OB at right angles to each other 

 in a vertical plane and equally inclined to the vertical. The disc being in a 

 position of equilibrium with AB horizontal, find the velocity produced by an 

 impulse applied along the lowest edge CD. 



Prove that, if AB = 2a, BC=4a, then AB will just rise to coincidence 

 with a wire if the impulse is such as would impart to a mass equal to that 

 of the disc a velocity %*J{ a ff(^~~ 2^/2)}. 



39. A uniform rigid semicircular wire is rotating in its own plane about 

 a hinge at one end, and is suddenly brought to rest by an impulse applied 

 at the other end along the tangent at that end. Prove that the impulsive 

 stress couple is greatest at a point whose angular distance from the hinge is 

 $, where &amp;lt; tan J&amp;lt; = 1. 



40. A heavy ring of radius a rolls with its plane vertical down a plane 

 of inclination a on which lie a series of pointed obstacles which are equal 

 and at equal distances from each other, and which are sufficiently high to 

 prevent the ring from ever touching the plane. Prove that, if the ring starts 

 from rest in a position in which it is in contact with two obstacles, and if 

 there is no slipping, its angular velocity o&amp;gt; as it leaves the (?i + l)th obstacle 

 is given by 



co 2 = 2&amp;lt;7 sin a sin y cos 4 y (1 - cos 4n y)/(l - cos 4 y), 



where 2y is the angle subtended at the centre by two adjacent obstacles 

 when the ring touches both. 



41. A circular disc, with n spikes projecting from it in its plane at equal 

 angular intervals, is projected with its plane vertical so as to strike a rough 

 horizontal plane (zero restitution) so that the line joining the point of contact 

 to the centre makes an angle TT/H with the vertical. Show that, if at the 

 instant the velocity of the centre is V and the angular velocity is &amp;lt;a, the 

 number of its spikes which strike the plane is the greatest integer in the 

 value of m given by the equation 



(1 - 2a 2 &amp;lt; - 2 sin 2 7r/n) m ( K 2 o&amp;gt; + a V] = 2/c J(ag) sin 7r/27i, 



where a is the radius of the circle on which the ends of the spikes lie, K is 

 the radius of gyration about the end of a spike and the radius of the disc is 

 less than a cos n/n. 



42. A uniform ball moving without rotation with velocity V strikes the 

 ground at an angle a with the vertical, and subsequently meets a bat whose 

 plane is vertical and perpendicular to the plane of the ball s motion, and 

 which is kept moving in the vertical plane of the ball s motion with a uniform 

 velocity in a direction making a given angle with the horizontal. Prove that 

 after striking the bat the ball will descend if the vertical velocity of the bat 

 is greater than 



| Fcos a (e + f tan a), 



gravity being neglected, and e being the coefficient of restitution between the 

 ball and the ground ; the bat and the ground are supposed to be sufficiently 

 rough to prevent sliding. 



