225] 



ILLUSTRATIVE PROBLEMS. 



245 



Let AB be the rod, 2a its length, m its mass, and suppose the end A 

 moves vertically in contact 

 with the wall and the end B 

 horizontally in contact with 

 the plane. The instantaneous 

 centre / is the intersection 

 of the horizontal through A 

 and the vertical through B, 

 and the figure OBIA is a 

 rectangle, so that the centre 

 of inertia G, which is the 

 middle point of AB, is always 

 at a distance a from 0. 



The system of kinetic 



Fig. 64. 



reactions is therefore equiva- 



lent to a resultant kinetic 



reaction at G having components mad and mad 2 perpendicular to OG and 



along GO, and a couple mJPB in the sense of increase of the angle 6 which the 



rod BA makes with the vertical BI. 



The forces acting on the rod are its weight at G, the horizontal pressure 

 at A t and the vertical pressure at B. The lines of action of the two latter 

 forces meet in /. If then we take moments about / the unknown reactions 

 do not enter. 



Hence prove that the motion in 6 is the same as that of a simple pendulum 

 of length | a. 



By resolving horizontally and vertically find the pressures at A and B, and 

 show that the rod leaves the wall when cos #=f cos a, a being the initial 

 value of 6. 



11. When the plane and the wall of Example 10 are both rough, with the 

 same angle of friction c, prove that the value of 6 at time t is given by the 

 equation 



a (% + cos 2f)0-asin 2*0 2 =grsin (0-2*). 



12. A wheel whose centre of gravity is at its centre rolls down a rough 

 plane of inclination a dragging a particle of mass m which slides on the plane 

 and is connected with the centre of the wheel by a thread, so that the whole 

 motion takes place in a vertical plane, and the thread makes an angle ft with 

 the line of greatest slope down which the particle slides. Prove that the 

 system descends with uniform acceleration 



J/sinacos(/3-*)4- wi cos /3 sin (a *) 



a 2 ) cos O - * ) + ma 2 cos cos e ' 



where a is the radius of the wheel, M its mass, k its radius of gyration about 

 its axis, m the mass of the particle and e the angle of friction between it and 

 the plane. 



