221.] PLANE MOTION. I2r 



(8) A clock is gaining 3 minutes a day. How much should the 

 pendulum bob be screwed up or down ? 



(9) A clock regulated at a place where -=32.19 is carried to a 

 place where ^-=3 2. 1 4. How much will it gain or lose per day if the 

 length of its pendulum be not changed ? 



(10) The acceleration of gravity being inversely proportional to the 

 square of the distance from the earth's centre, show that the seconds 

 pendulum will lose about 22 seconds per day if taken to a height one 

 mile above sea level. 



(u) A seconds pendulum loses 12 seconds per day, if taken to the 

 top of a mountain. What is the height of the mountain ? 



(12) Show that for small oscillations the motion of a pendulum 

 is nearly simply harmonic, and deduce from this fact the equation 



221. When the oscillations of a pendulum are not so small 

 that the arc can be substituted for its sine as was done in Art. 

 217, an expression for the time t^ of one oscillation can be 

 obtained as follows. 



We have by (33), Art. 216, 



cos0. (33) 



Let the time be counted from the instant when the moving 

 point has its highest position (TV in Fig. 52), so that ^ =o. 

 Substituting v IdQ/dt and applying the formula 



cos 0=i2 sin 2 J 9 

 we find : 



whence 



d9 



(43) 



