262 MOTION UNDER, A VARIABLE FORCE 



ILLUSTRATIVE EXAMPLE 



A pendulum which beats seconds accurately at New York is found to gain 3 

 seconds a day when taken to Philadelphia. Compare the values of g at Philadel- 

 phia and New York. 



At Philadelphia the pendulum makes 24 x (60) 2 + 2 beats in 24 x (60) 2 

 seconds. Thus the time of a beat is 



24 x (60) 2 



seconds, 



24 x (60) 2 + 2 

 and this is equal to A/-i where a is the length of the pendulum and g is the 



value of gravity at Philadelphia. If go denote the value of gravity at New York, 

 we have 





24 x (60)2 



so that 



24 x (60) 2 / 



= 9o (1 + HEISTS) approximately. 

 Thus gravity is greater at Philadelphia than at New York by about one part in 



21,600. 



EXAMPLES 



1. Calculate the length of a pendulum to beat time to a march of 100 paces 

 a minute. 



2. A pendulum which beats seconds in London requires to be shortened by 

 one thousandth of its length if it is to keep time in New York. Compare the 

 values of gravity at London and New York. 



3. The value of g at a point on the earth's surface in latitude X is 



g = g (1 - .00257 cos 2 X), 



where g = 32.17 is the value of g in latitude 45 degrees. Show that the lati- 

 tude in which a short journey of given length will produce the greatest error 

 in the rate of a pendulum clock is latitude 45 degrees, and find the error per 

 mile in this latitude. (One minute of latitude = 6075 feet.) 



4. In a building, at height h feet above the ground, the value of gravity is 



g -. 000003 fc, 



where g is the value of gravity at the foot of the building. In New York, 

 go = 32.14. Find the error in the rate of a pendulum clock, produced by taking 

 it from the ground to the top of a building 300 feet high. 



