EXAMPLES. 327 



103. A chain is formed of n equal symmetrical rods, each of length 2a 

 and radius of gyration k about its centre of mass. One end is fixed and the 

 whole is supported in a horizontal line. Prove that, if the supports are 

 simultaneously removed, the free end begins to move with acceleration 



#[l+(-) n+1 sechlog(tamY40)], where 0=log(a/). 



104. A mass M rests on a smooth table and is connected with a particle 

 of mass m by an inextensible thread passing through a hole in the table. 

 Prove that, if m is released from rest in a position in which its polar coordi 

 nates are a, a referred to the hole as origin and the vertical as initial line, 

 then in the initial motion 



( M + m} r = mg cos a, a6 Q =-g sin a, 



a (M+ m) r iv = 3ra# 2 sin 2 a, 2 iv =g 2 sin a cos a (M+ 3m)/( J/ + m). 

 Also prove that the initial radius of curvature of the path of m is 



where 0=^0, y = 4 



105. One end of a uniform rod of length 2a and mass m is freely jointed 

 to a board of mass M at its centre of inertia and the board is placed on a 

 smooth table. The rod is held so as to make an angle a with the vertical, 

 and is let go. Prove that the initial radius of curvature of the path of its 

 middle point is 



a {M* cos 2 a + (M+ m) 2 sin 2 a} 1 / M (M+ m) 2 . 



106. A garden roller is at rest on a horizontal plane rough enough to 

 prevent slipping, the handle being so held that the plane through the axis of 

 the cylinder and the centre of inertia of the handle makes an angle a with 

 the horizon. Show that, if the handle is let go, the initial radius of curva 

 ture of the path described by its centre of inertia is c?i- 2 (sm 2 a + ft 2 cos 2 a)^, 

 where (n - 1 ) M (K 2 + a 2 ) = ma 2 , and c is the distance of the centre of inertia of 

 the handle from the axis of the cylinder, m its mass, and MK 2 the moment 

 of inertia of the cylinder about its axis, the cylinder being homogeneous and 

 of radius a. 



107. Two uniform rods of lengths 2a, 26 and masses A t B are freely 

 hinged at a common extremity and the other extremity of A is fixed. The 

 rods fall from a horizontal position of rest. Prove that the initial radius of 

 curvature of the further extremity of B is 



2ab (A +B) 2 l{ 



108. A rough plank of mass M is free to turn in a vertical plane about a 

 horizontal axis distant c from its centre of inertia, and a uniform sphere of 

 mass m is placed on the plank at a distance b from the axis on the side 

 remote from the centre of inertia, the plank being held horizontal. Prove 

 that when the plank is let go the initial radius of curvature of the path of 

 the centre of the sphere is 21&0/(5-110), where 0=(mb-Mc)l(mb + Ma), and 

 Mab is the moment of inertia of the plank about the axis. 



