120 KINEMATICS. [219. 



219. Let n be the number of small oscillations made by a 

 pendulum of length /in the time 7! Then, by (38), 



T I/ 



~ =7r \r 



n \ sr 



(40) 

 g 



If 7 1 and one of the three quantities n, /, g in this equation be 

 regarded as constant, the small variations of the two others can 

 be found approximately by differentiation. For instance, if the 

 daily number of oscillations of a pendulum of constant length 

 be observed at two different places, we have, since T and / are 

 constant, 



Tj TT^Jdg 



-- -dn = --- *, 

 n* 2 gl 



or, dividing by (40), 



^ = 1* (41) 



n 2g 



220. Exercises. 



(1) Find the number of oscillations made in a second and in a day 

 by a pendulum i metre long, at a place where -=981.0. 



(2) Find the length of the seconds pendulum at a place where 

 #=32.12. 



(3) To determine the value of g at a given place, the length of a 

 pendulum was adjusted until it would make 86 400 oscillations in 24 

 hours. Its length was then found to be 3.3031 feet. What was the 

 value of g ? 



(4) A chandelier suspended from the ceiling of a theatre is seen to 

 vibrate 24 times a minute. Find its distance from the ceiling. 



(5) A pendulum adjusted so as to beat seconds at the equator 

 (^=978.1) is carried to another latitude and is there found to make 

 100 oscillations more per day ; find the value of g at this place. 



(6) Investigate whether the approximate process of Formula (41) is 

 sufficiently accurate for the solution of Ex. (5). 



(7) If the length of a pendulum be increased by a small amount dl t 

 show that the daily number of oscillations, , will be decreased so that 



2 / 



