EXAMPLES. 253 



37. A homogeneous sphere of mass M and radius a rests on a horizontal 

 plane in contact with a vertical wall, and a second homogeneous sphere of 

 mass m and radius b(<a) is placed in contact with it and the wall, the 

 centres being in a vertical plane at right angles to the wall. Prove that, if 

 all the surfaces are smooth, the spheres will separate when the line joining 

 their centres makes with the horizontal an angle 6 given by the equation 



(a + b) {(M - in) sin 3 6 + 3m sin 6} = 4m ,J(db). 



38. A smooth circular cylinder of mass M and radius c is at rest on a 

 smooth horizontal plane, and a heavy straight rail of mass m and length 2a 

 is placed so as to rest with its length in contact with the cylinder and to have 

 one extremity on the ground. Prove that the inclination of the rail to the 

 vertical in the ensuing motion (supposed to be in a vertical plane) is given by 

 the equation 



+ sin 2 0) a 2 + - cos =ga (cos a - cos 0), 



where a is the initial value of 6. 



39. The outer surface of a uniform spherical shell of mass M is rough and 

 of radius a, and the inner (concentric) surface is smooth and of radius 6. 

 A particle of mass m moves inside the shell while the shell rolls on a hori- 

 zontal plane. Show that the angular distance 6 of the particle from the 

 vertical diameter at time t is given by the equation 



i (IM+m sin 2 0) #*= ( M +m} (cos 0-cos a) (#/&), 

 where a is the greatest value of 6. 



40. A circular cylinder of radius a and radius of gyration k rolls inside 

 a fixed horizontal cylinder of radius b. Prove that the plane through the 

 axes moves like a simple pendulum of length 



When the second cylinder can turn about its axis, and when the first 

 cylinder is of mass m and the moment of inertia of the second about its 

 axis is J/7T 2 , prove that the length of the equivalent simple pendulum is 

 (&-)(! + n)/w, where n = a 2 /F + m6 2 /J/7i r2 ; prove also that the pressure 

 between the cylinders is proportional to the depth of the point of contact 

 below a plane which is at a depth 2n6cosa/(l + 3w) below the fixed axis, 

 where 2a is the angle of oscillation. 



41. A perfectly rough inelastic sphere is dropped on the lowest generating 

 line of a horizontal circular cylinder which is revolving freely about its axis, 

 which is fixed, with angular velocity Q. Prove that the plane through the 

 axis of the cylinder and the point of contact will move like a simple 

 pendulum of length 



~ C) 



where a and c are the radii, K and k the radii of gyration, M and m the 



