THE CYCL01DAL PENDULUM 265 



to AA f . The former is obviously the velocity of JV along AA f , and 

 it is at once seen to be of amount ka sin 6, or 



v = ka sin Jc (t e) 

 = k Va 2 s 2 , as before. 



In this motion, as in the motion of the simple pendulum, the 



quantity a is called the amplitude, while the time after which 



the motion repeats itself is called the period. 



EXAMPLES 



1. A point moves with simple harmonic motion of period 12 seconds, and has 

 an amplitude of 6 feet. Find its maximum velocity, and find its position and 

 velocity one second after an instant at which its velocity is a maximum. 



2. A particle moving with simple harmonic motion of period t is observed 

 to have a velocity v when at a distance a from its mean position. Find its 

 amplitude. 



3. A particle is free to move along a line AB and is acted on by an attract- 

 ive force directly proportional to its distance from a point P in AB, and con- 

 sequently moves with simple harmonic motion. Prove that its average kinetic 

 energy is equal to its average potential energy. 



4. A point moving with simple harmonic motion is observed to have 

 velocities of 3 and 4 feet per second when at distances of 4 and 3 feet respec- 

 tively from its mean position. Find its amplitude and period. 



5. A point moves with simple harmonic motion relative to one frame, and 

 the frame itself moves with simple harmonic motion relative to a second frame, 

 the directions of the two motions being parallel, and their periods the same. 

 Show that the motion of the moving point relative to the second frame is simple 

 harmonic motion, of the same direction and period as that of the frame. 



6. A weight w is tied to an elastic string of natural length a and modulus X, 

 and is allowed to hang vertically in equilibrium. The weight is now pulled 

 down vertically through a further distance h. Show that on being set free it 

 will describe simple harmonic motion of amplitude A, provided this does not 

 involve the string ever becoming unstretched. Find the period of the motion. 



THE CYCLOID AL PENDULUM 



210. We have seen that the motion of a simple pendulum is 

 simple harmonic motion only so long as the amplitude of the 

 motion is small It is, however, possible to constrain a particle to 

 move under gravity in such a way that its motion shall be simple 

 harmonic motion no matter how great the amplitude. 



