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



masses of the cylinder and sphere respectively. Prove also that the angular 

 velocity of the pendulum at the lowest point is 



a Q 



a c ~ c 2 ma 2 ' 



42. A garden roller stands at rest on a level path with the handle 

 vertical; the handle is pulled down into a horizontal position held at rest 

 and then released. Prove that the angular motion of the handle about the 

 axis of the roller is given by the equation 



h cos 2 </> 



where R is the radius of the roller, K its radius of gyration about its axis, 

 M its mass, and m is the mass of the handle, h the distance of its centre of 

 gravity from the axis, and I the length of the simple equivalent pendulum of 

 the handle when the roller is held fixed. 



43. A uniform circular hoop of radius a is so constrained that it can 

 only move by rolling in a horizontal plane on a fixed horizontal line, and a 

 particle whose mass is 1/X of that of the hoop can slide on the hoop without 

 friction. Prove that, if initially the hoop is at rest, and the particle is 

 projected along it from the point furthest from the fixed line with velocity v, 

 then the angle turned through by the hoop in time t will be 



where \jf is the angle the diameter through the particle has turned through in 

 the same interval. Prove also that 



44. A uniform rod swings in a vertical plane hanging by two cords 

 attached to its ends and to points A, B in a horizontal line, AB being equal 

 to the length of the rod, and the cords not being crossed. Prove that, if the 

 cords attached to A and B are of lengths a and a + \ respectively, where X is 

 small, the angular velocity of the cord attached to A when inclined to the 

 vertical at an angle 6 is greater than it would be if X were zero by 



X */(#/2a 3 ) (cos 6 - cos a)* (tan 2 B - 1 sec 6 sec a) 



approximately, a being the value of 6 in a position of rest and not being 

 nearly equal to a right angle. 



45. A uniform rod which is free to turn about a point fixed in it touches, 

 at a distance c from the fixed point, the rough edge of a disc of mass m, 

 radius a, and radius of gyration Jc about its centre. The system being at 

 rest on a smooth horizontal plane, an angular velocity Q is suddenly commu- 

 nicated to the rod so that the disc also is set in motion. Prove that in the 

 subsequent motion the distance r of the point of contact from the fixed 

 point satisfies the equation 



where MK 2 is the moment of inertia of the rod about the fixed point and the 

 edge is rough enough to prevent slipping. 



