31G BOOK OF MATHEMATICAL PROBLEMS. 



IV. Miscellaneous. 



1511. A square is moving freely about a diagonal with angu- 

 lar velocity o>, when one of the angular points not in that diagonal 

 becomes fixed; determine the impulsive pressure on the fixed 



point, and prove that the instantaneous angular velocity is . 



1512. A uniform rod, of length a, freely moveable about one 

 extremity, is initially projected in a horizontal plane with angular 

 velocity w; prove that the equations of motion are 



= ?? cos e - o 2 cot 2 ; 

 a 



0, < being respectively the angles which the rod makes with the 

 vertical, and which its projection on a horizontal plane makes 



with the initial position. If the least value of be , the resolved 



o 



vertical pressure on the fixed point when 6 = Q is to the weight of 



o 



the rod as 31 : 1C. 



1513. A centre of force = //. . distance is at a point 0, and 

 from another point A at a distance a, are projected simultaneously 

 an infinite number of equal particles in a direction at right angles 



to OA, with velocities in arithmetical progression from ~- to 



~^- ; prove that, if after any lapse of time they become sud- 

 denly rigidly connected, the system will revolve with angular 

 velocity 



