EXAMPLES 283 



is moving with velocity F at inclination a ; and prove that in the air 

 the tension of the thread is 



F 2 



pounds. 

 W 2 ga r 



7. Two masses, m l and m 2 , are connected by a weightless spring of such 

 strength that when m l is held fixed, m 2 performs n vibrations a second. 

 Show that if ?n 2 be held fixed, m l will perform n "^m 2 /m l vibrations a sec- 

 ond, while if both masses are free, they will perform n-^l ^ vibra- 

 tions per second, the vibrations in every case being in the line of the 

 spring. 



8. A particle of mass m, moving in a smooth curved tube of any shape, 

 is in equilibrium under the tensions of two elastic strings in the tube, of 

 natural lengths I, I' and moduli of elasticity X, X', of which the other ends 

 are attached to fixed points of the tube. If the particle makes oscillations, 

 large or small, in the tube, show that the time of oscillation is 



9. A string passes through a small hole in a smooth horizontal table, 

 and has equal particles attached to its ends, one hanging vertically and the 

 other lying on the table at a distance a from the hole. The latter is pro- 

 jected with a velocity ^fga perpendicular to the string. Show that the 

 hanging particle will remain at rest, and that if it be slightly disturbed, 



\/ 

 \3< 



the time of a small oscillation will be 2 T 



10. A particle moves in a circular groove, under an attraction to a 



point P which is in the plane of the circle and distant b from its center. 

 The particle is projected with velocity Ffrom the point of the circle nearest 

 to P. Show that for the particle to perform complete revolutions, the value 



4 id) 



of F 2 must not be less than -- 



a 2 b z 



11. A smooth ellipse, semi-axes a and &, is placed with its major axis 

 vertical, and a particle is projected along the concave side of the arc, with 

 velocity due to a height h above the center. Find the point at which the 

 particle will leave the arc, and show that it will pass through the center of 



the ellipse if 



, 2 2 



h = 



