EXAMPLES. 315 



31. Three particles of equal mass are attached at equal intervals to a 

 rigid rod of negligible mass, and the system being at rest one of the extreme 

 particles is struck by a blow at right angles to the rod. Prove that the 

 kinetic energy imparted to the system when the other extreme particle is 

 fixed and the rod turns about it is less than that which arises when the 

 system is free in the ratio 24 : 25. 



32. Two equal rigid rods AB, BC of negligible masses carry equal 

 particles attached at A, C and the middle points of the rods, and, the rods 

 being freely hinged at B and laid out straight, the end A is struck with an 

 impulse at right angles to the rods. Prove that the velocities of the particles 

 are in the ratios 9:2:2:1. 



33. Four particles of equal masses are tied at equal intervals to a thread, 

 and the system is placed on a smooth table so as to form part of a regular 

 polygon whose angles are each TT - a. Prove that if an impulse is applied 

 to one of the end particles in the direction of the thread attached to it 

 the kinetic energy generated is greater than it would be if the particles 

 were constrained to move in a circular groove and the impulse were 

 applied tangentially in the ratio cos 2 a + 4sin 2 a : cos 2 a+2 sin 2 a. 



34. A rod of length 2a is held in a position inclined at an angle a to 

 the vertical, and is then let fall on a smooth horizontal plane (no restitution). 

 Prove that the end of the rod which strikes the plane will leave it im- 

 mediately after impact if the height through which the rod falls is greater 

 than 



Y^a sec a cosec 2 a (1 + 3 sin 2 a) 2 . 



35. A particle of mass m impinges directly on a smooth uniform spheroid 

 of mass M and semiaxes a, b at rest, no energy being lost in the impact. 

 Show that, if 



1 < M/m < 6 - I0abf(a? + 6 2 ), 



the point of impact may be so chosen that the particle is reduced to rest. 



36. A circular cylinder rocks between two parallel rails whose distance 

 apart is less than the diameter of the cylinder. Prove that the greatest 

 heights of the axis above its equilibrium position diminish in geometrical 

 progression. 



37. Any number of equal uniform rods are jointed together so as to have 

 a common extremity and placed symmetrically so as to be generators of a 

 cone of vertical angle 2a, and the system falling with velocity V strikes sym- 

 metrically a smooth fixed sphere of radius c (no restitution). Prove that the 

 angular velocity with which each rod begins to turn is 



V (c cos a ~ a sin 3 a)/( a 2 sin 2 a + c 2 cot 2 a - ac sin 2a). 



