150 MOTIONS OF FREE PARTICLES. [CHAP. IX. 



Thus, if m is the mass of the body, the equation of motion of the 

 body is mx = px. 



The mass of the spring being neglected, the kinetic energy is 

 ^mx*, and the potential energy is ^we 8 , and thus we have the 

 equation of energy ^mx 2 + Jyu# 2 const., from which the equation 

 of motion might be deduced by differentiating with respect to the 

 time. 



168. Examples. 



1. A particle of mass m is attached to the middle point of an elastic 

 thread, of natural length a and modulus X, which is stretched between two 

 fixed points. Prove that, if no forces act on the particle other than the 

 tensions in the parts of the thread, it can oscillate in the line of the thread 

 with a simple harmonic motion of period IT *J(ma/\). 



2. A particle of mass m is attached to one end of an elastic thread, of 

 natural length a and modulus X, the other end of which is fixed. The 

 particle is displaced until the thread is of length a + 6, and is then let go. 

 Prove that, if no forces act on the particle except the tension of the thread, it 



will return to the starting point after a time 2 ( ?r + 2 ^- j */-%- 



3. Prove that, if a body is suddenly attached to an unstretched vertical 

 elastic thread and let fall under gravity, the greatest subsequent extension is 

 twice the statical extension of the thread when supporting the body. 



4. Prove that, if a spring is held compressed by a given force and the 

 force is suddenly reversed, the greatest subsequent extension is three times 

 the initial contraction. 



5. An elastic thread of natural length a has one end fixed, and a particle 

 is attached to the other end, the modulus of elasticity being n times the 

 weight of the particle. The particle is at first held with the thread hanging 

 vertically and of length a', and is then let go from rest. Show that the time 

 until it returns to its initial position is 



2 (TT - 6 + & + tan 6 - tan ff] J(ajng), 

 where 6, 0' are acute angles given by 



sec0 = na'/a-n-l, sec 2 & = sec 2 Q - 4n, 

 and a' is so great that real values of these angles exist. 



169. Gravitation. The case of a central force varying 

 inversely as the square of the distance is the case of gravitation, at 

 least when the mass of one gravitating body is great compared with 

 that of the other. Thus for a body near the Earth's surface the 

 action of the Earth on the body produces an acceleration which is 

 nearly constant and in a vertical direction. A correction to this 

 statement will be made by saying that the force in question is 

 directed to the centre of the Earth, and varies inversely as the 



