178 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



188. Simple Circular Pendulum. A particle constrained 

 to describe a circle in a vertical plane is called a "simple circular 

 pendulum." When the constraint is applied by means of an 

 inextensible thread of negligible mass with one extremity fixed, or 

 when the particle is within a hollow cylinder with a horizontal 

 axis, the particle can leave the circle. For the present we shall 

 suppose the particle to be within a circular tube, fixed in a 

 vertical plane. 



Let I be the radius of the circle. This is called the length of 

 the pendulum. 



Let 6 be the angle the radius through 

 the particle makes with the vertical at time t. 

 The kinetic energy is JmW, where m is 

 the mass of the particle. 



The potential energy of the particle in the 

 field is mgl (1 cos 0), the standard position 

 Fig. 49. of the particle being at the lowest point. 



Hence the equation of energy may be 

 written, after division by ml, 



^l& = g cos 6 + const, 

 where the constant depends on the velocity at the lowest point. 



189. Small Oscillation. Differentiating the equation of 

 energy last written with respect to the time, we have 



16 ' = g sin 6. 



This equation might have been obtained by resolving along the 

 tangent to the circle. 



If 6 is very small throughout the motion we may put 6 for 

 sin 6, and thus 



w = - g e. 



This equation shows (Article 47) that the motion in 6 is simple 

 harmonic with a period 2-Tr *J(l/g). 



The pendulum swings from side to side of the vertical. 

 Suppose it to start from rest in a position near the equilibrium 

 position but slightly displaced to the right. It falls to the equili- 

 brium position in time JTT */(l/g), passes through it, and proceeds 

 to the left until its displacement is numerically equal to that at 



