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



31. An elastic thread of modulus X is wound round the smooth rim of a 

 homogeneous circular disc of mass m, one end being fastened to the rim, 

 and the other to the top of a smooth fixed plane of inclination a to the 

 horizontal, down which the disc moves in a vertical plane through a line of 

 greatest slope, which is the line of contact of the straight portion of the 

 thread with the plane. Initially the thread has its natural length I and is 

 entirely wound on the rim of the disc which is at rest at the top. Prove that 

 at any time t before the thread is entirely unwound the tension is 



a sin 2 [\t <J(3\/lm')}. 



32. Two equal cylinders of mass m, bound together by a light elastic 

 band of tension T, roll with their axes horizontal down a rough plane of 

 inclination a. Show that their acceleration down the plane is 



mg sm a 

 /A being the coefficient of friction between the cylinders. 



33. A waggon runs down a road inclined at an angle a to the horizon, 

 and the road is crushed uniformly by the wheels, prove that the accelera- 

 tion is 



( M + 2m) cos |3 4- 2rf/a 2 t 



the centre of inertia being midway between the wheels, M denoting the mass 

 of the body of the waggon, m the mass, mJP the moment of inertia, and a the 

 radius of each pair of wheels, and being an angle depending on the nature 

 of the road. 



34. A rod AB, whose density varies in any manner, is swung as a 

 pendulum about a horizontal axis through A. Prove that the couple resisting 

 bending is greatest at a point P determined by the condition that the centre 

 of inertia of the part PB is the centre of oscillation of the pendulum. 



35. A semicircular wire A CB whose line density varies as the distance 

 from the diameter AB rotates in its plane, which is vertical, with uniform 

 angular velocity o> about the fixed point A. Prove that the stress couple at 

 the middle point C of the arc AB vanishes when AB is vertical if 



36. A uniform rod of mass m has one extremity fastened by a pivot to 

 the centre of a uniform circular disc of mass M which rolls on a horizontal 

 plane, the other extremity being in contact with a smooth vertical wall at right 

 angles to the plane containing the disc and the rod. Prove that the inclination 

 6 of the rod to the vertical when it leaves the wall is given by the equation 



9 M cos 3 6 + 6m cos 6 - 4m cos a = 0, 

 the system starting from rest in a position in which 6= a. 



