188-190] SIMPLE PENDULUM. 179 



starting, and comes to rest after an interval JTT \l\ljg) from the 

 equilibrium position. The motion is then reversed. The time 

 from rest to rest is TT ^(l/ff)* This is known as the time of a 

 &quot;beat,&quot; the period 27r*/(l/g) is the time of a &quot;complete oscillation.&quot; 

 A pendulum which beats seconds is known as a &quot;seconds 

 pendulum;&quot; the time of a complete oscillation of such a pendulum 

 is two seconds. The length of the seconds pendulum at a place 

 is given by the equation 



Pendulum experiments afford the most exact method of deter 

 mining the value of g. 



190. Examples. 



1. Prove that, in the case of one-sided constraint, the particle cannot 

 leave the circle unless it has risen above the centre. 



2. Prove that, if in London &amp;lt;7 = 98ri7, the units being the centimetre 

 and the second, then the length of the seconds pendulum there is 99 413 

 centimetres. 



3. A balloon ascends with constant acceleration and reaches a height of 

 900 ft. in one minute. Show that a pendulum clock carried with it will gain 

 at the rate of 27*8 seconds per hour, approximately. 



4. If Zj is the length of a pendulum which gains n seconds in an hour, 

 and 1 2 the length of a pendulum which loses n seconds in an hour, n being 

 small, prove that the square root of the true length of the seconds pendulum 

 is the harmonic mean between V^ and J1 2 . 



5. The bob of a pendulum which is hung close to the face of a cliff is 

 attracted to the cliff with a horizontal force of intensity /. Show that the 

 time of a beat is 



where I is the length of the pendulum. 



6. A bead slides on a smooth circular wire of radius a, whose plane is 

 inclined at an angle a to the vertical. Find the period of its small oscillations 

 about the lowest point. 



7. Show that, if the suspending fibre of a simple pendulum is slightly 

 extensible, its vibrations of small amplitude take place in an arc having its 

 centre of curvature at a distance below the point of suspension equal to three 

 times the statical increase of length due to the weight of the bob, while the 

 period is simply that due to the stretched length of the fibre. 



8. Prove that on taking a pendulum down a mine, the time of vibration 

 is increased or diminished according as the mean density of the Earth at its 

 surface is greater or less than two- thirds of the Earth s mean density. 



122 



