GENERAL THEOEY OF MOMENTS OF IKEKTIA 301 



EXAMPLES 



1. The line of hinges of a door makes an angle a. with the vertical, and the 

 door swings about its position of equilibrium. Show that its motion is the same 

 as that of a certain simple pendulum, and find the length of this pendulum. 



2. A target consists of a square plate of metal of edge a and of mass 3f, 

 hinged about its highest edge, which is horizontal. When at rest it is struck by 

 an inelastic bullet of small mass ra moving with velocity v, at a point at depth h 

 below the line of hinges. Find the subsequent motion of the target. 



3. A homogeneous sphere is projected without rotation up a rough inclined 

 plane of inclination a and coefficient of friction /*. Show that the time during 

 which the sphere ascends the plane is the same as if the plane were smooth, and 

 that the time during which the sphere slides stands to the time during which it 

 rolls in the ratio 2 tana : 7/x. 



4. A sphere of radius a is held at rest at a point on the concave surface of 

 a spherical bowl of radius 6. It is suddenly set free and allowed to roll down 

 the surface. Show that the line joining the centers of the two spheres swings 

 in the same way as a simple pendulum of length |(6 a). 



^.--5. A sphere of radius a is held at rest at the highest point of the rough con- 

 vex surface of a sphere of radius 6. It is then set free and allowed to roll down 

 this sphere. Show that the spheres will separate when the line joining their 

 centers makes an angle cos- 1 ^ with the vertical. Examine the case of 6 = 0. 



6. A circular hoop, which is free to move on a smooth horizontal plane, has 

 sliding on it a small ring of I/nth its mass, the coefficient of friction between 

 the two being /*. Initially the hoop is at rest, and the ring has an angular 

 velocity w round the hoop. Show that the ring comes to rest relative to the 



hoop after a time . 



ft* 



GENERAL THEORY OF MOMENTS OF INERTIA 

 Coefficients of Inertia 



246. Suppose that a rigid body is rotating about an axis of 

 rotation of which the direction cosines, referred to any three fixed 

 coordinate axes, are I, ra, n. Let us 

 take any point O on the axis of rota- 

 tion for origin, and let L be any par- 

 ticle of mass m lt distant p from the 

 axis of rotation. Let the coordinates 

 of L be x, y, z, and let LN(=p) be 



the perpendicular from L on to the 



axis of rotation. " y FIG. 148 



