146.] CENTRAL FORCES. 77 



(3) A number of particles are projected, from the same point in 

 the field of a force following Newton's law, with the same velocity, but 

 in different directions. Show that the periodic times are the same for 

 all the particles. 



(4) The mean distance of Mars from the sun being 1.5237 times 

 that of the earth, what is the time of revolution of Mars about the sun ? 



(5) A particle describes a conic under the action of a central force 

 following Newton's law; if the intensity /x of the force be suddenly 

 changed to /*', what is the effect on the orbit? 



(6) In Ex. (5), if the original orbit was a parabola and the intensity 

 be doubled, what is the new orbit ? 



(7) Regarding the moon's orbit about the earth as circular, what 

 would it become : (a) if the earth's mass were suddenly doubled ? 

 () if it were reduced to one-half ? 



(8) In Ex. (5), determine the effect on the major semi-axis (or 

 "mean distance") a and on the periodic time T, of a small change 

 in the intensity /A of the force. 



(9) If the mass of the sun be suddenly increased by a small 

 amount while the earth is at the end of the minor axis of its orbit, 

 what would be the effect on the earth's mean distance and on the 

 period of revolution T ? 



(10) Find the equation of the hodograph of planetary motion, 

 derive from it the expression for the velocity in terms of the radius 

 vector, and show that the velocity is a maximum in perihelion and 

 a minimum in aphelion. 



(n) Show that the greatest velocity of a planet in its orbit about 

 the sun is to its least velocity as Vi +e is to Vi e ; and find this 

 ratio for the earth, whose orbit has the eccentricity <? = 0.01677120. 



(12) Find the time exactly as a function of 0, for a parabolic orbit. 



146. Force any Function of the Distance. The general methods 

 have been given in Arts. 108-1 16. The equation of energy, 

 <6), Art. 112, gives, with u=i/r, 



u }d u i / \ 



r-+*5 (34) 



