314 BOOK OF MATHEMATICAL PROBLEMS. 



1502. A sphere is resting on a rough horizontal plane, half 

 its weight being supported by ail elastic string attached to the 

 highest point, the natural length of the string being the radius a, 

 and the stretched length the diameter of the sphere ; prove that 

 the time of small oscillations of the sphere parallel to a vertical 



pla s 



1503. A heavy uniform rod, resting in stable equilibrium 

 within a smooth prolate spheroid whose axis is vertical, is slightly 

 displaced in a vertical plane; prove that the length of the simple 



isochronous pendulum is a(e + ~; 2a being the length of the 



rod, and e the eccentricity of the generating ellipse. 



1504. A uniform beam rests with one end on a smooth hori- 

 zontal table, and has the other attached to a fixed point by means 

 of a string of length lj prove that the time of a small oscillation 



I'M tana 

 in a vertical plane is ir */ ------------ , a being the mean inclination 



of the rod to the vertical. 



1505. Two equal uniform rods AB, BC, freely jointed at 

 7?, are placed on a smooth horizontal table, at right angles to each 

 other, and a blow is applied at A at right angles to AB; prove 

 that the initial velocities of A, C are in the ratio 8:1. 



1506. Two equal uniform rods AB, BC, freely jointed at B, 

 are laid on a smooth horizontal table so as to include an angle a, 

 and a blow is applied at A at right angles to AB determine the 

 initial velocity of C. 



1507. Five equal uniform rods, freely jointed at their ex- 

 tremities, are laid in one straight line on a horizontal table, and a 

 blow applied at the middle point at right angles to the line; prove 

 that 



v _ o> _ --a 



Ha~9~~3 ' 



v being the initial velocity of the central rod, CD, O the initial an- 



