302 BOOK OF MATHEMATICAL PROBLEMS. 



1444. A uniform heavy chain of length I is in motion on the 

 arc of a smooth curve in a vertical plane, and the tangent at the 

 point of greatest tension makes an angle < with the vertical; 

 prove that the difference between the depths of the extremities 

 is I cos </>. 



1445. A uniform inextensible string is at rest in a smooth 

 groove which it just fits, and a tangential impulse P is applied at 

 one extremity; prove that the normal impulse, referred to a unit of 



Ps 



length of the string, at a distance s from the other extremity is , 



ap 



a being the whole length of the string, and p the radius of 

 curvature at the point under consideration. 



1446. A straight tube of uniform bore is revolving uniformly 

 in a horizontal plane about a point at a distance c from the tube, 

 and within the tube is a smooth uniform chain of length 2a, which 

 is initially at rest with its middle point at the distance c from the 

 axis of revolution; prove that the chain in a time t will describe 

 a space 



along the tube, and that the tension at any point of the chain is 



m i 9 \ 2 



(-*>*; 



where x is the distance of the point from the middle point, m the 

 mass of the chain, and to the angular velocity. 



1447. A circular tube, of radius a, revolving with uniform 

 angular velocity <o about a vertical diameter contains a heavy 

 uniform chain, which subtends an angle 2a at the centre ; prove 

 that the chain will remain in relative equilibrium if the radius 

 through its middle point makes with the vertical an angle 



aw cos a/ ; 



