DYNAMICS OF A RIGID HOLY. 



1511. A uniform rod, moveable about one extremity, m 

 in such a manner as to make always nearly the same angle a with 

 the vertical; prove that the time of its small oscillations is 



2a cos a 



//2a 

 V \fy 



3 cos' 

 a being the length of the rod. 



1.115. A uniform rod is suspended by two strings of equal 

 lengths, attached to its extremities and to two fixed points in the 

 same horizontal line whose distance is equal to the length of the 

 rod ; an angular velocity is communicated to the rod, about a 

 vertical axis through its centre, such that it just rises to the hori- 

 zontal plane in which are the fixed points; find the impulsive 

 couple, and prove that the tension of either string is instanta- 

 neously increased in the ratio 7:1. 



1 :> 1 G. If two equal uniform rods A B, EC, freely jointed at B, 

 rotate uniformly about a vertical axis through A, which is ; 

 witli angular velocity w, the equations to determine the angles 

 o, /?, which the rods make with the vertical are 



(8 sin a + 3 sin ft) cos a _ (3 sin a + 2 sin /?) cos /? _ 3.7 

 3 sin a sin/? ~~ iW ' 



2a being the length of either rod. 



1.117. A perfectly rough hori/ontal plain* is made to rotate 

 with constant angular velocity about a vertical axis which 

 th- plane in 0; a sphere is projected on it at a point /'. so that 

 its centre is initially in the same state of motion as if the sphere 

 had been placed freely on the plane at a point Q; ] tho 



rih.- 11 1. if. .n nly a circle of radius OQ, and whose 

 centre R is such that OR is parallel and equal to (J/'. 



l.~>18. A rough plain-, inclino! at an angle a to the hoi 

 is made to revolve with uniform angular velocity <u about a 



icre without motio; '-d upon it; prove that 



ith of the centre will be a prolate, a common, or a cu 



