264-266] MOTION OF A CHAIN. 301 



7. Four equal uniform rods freely hinged together so as to form a 

 rhombus of side 2a rotate about one diagonal which is fixed in a vertical 

 position, the highest point being fixed and the lowest being free to slide on 

 the diagonal. Find the angular velocity in the steady motion in which each 

 rod makes an angle a with the vertical, and prove that the period of the 

 small oscillations about this state of steady motion is the same as for a 

 simple pendulum of length f a cos a (1 -f 3 sin 2 a)/(l + 3 cos 2 a). 



MOTION OF A STRING OR CHAIN. 



265. Suppose in the first place that a uniform inextensible 

 chain is in rectilinear motion in such a way that every particle 

 moves along the line of the chain. Then, if V is the velocity of 

 any particle, the condition of inextensibility requires that every 

 particle moves with the same velocity F; and since this condition 

 holds at all times every particle moves with the same acceleration. 

 In such cases any part of the string which is in continuous recti 

 linear motion moves like a particle at its centre of inertia under 

 the action of the resultant of the forces acting on the part, every 

 particle having the velocity and acceleration of the centre of 

 inertia. The special interest presented by problems of this kind 

 attaches to the magnitude of the tension at a place where the 

 character of the motion changes discontinuously. 



The principle to be adopted in such cases is that the increase 

 of momentum of any system in any time is equal to the impulse 

 of the resultant force acting on the system during that time (cf. 

 Article 113). Taking, as the system, a small element of the 

 chain whose motion is changing discontinuously, this element 

 passes in a very short interval from one state of motion to another, 

 and the product of the interval and the resultant of the tensions 

 on the ends is equal to the momentum generated in the element. 

 The method of application of this principle will be made clearer 

 by means of an example. 



266. Illustrative Problems. 



I. A chain is coiled at the edge of a table with one end just hanging over. 

 To find the motion. 



At any time t let x be the length which has fallen over the edge, T the 

 tension at the edge in the falling portion. There is no tension in the part 

 coiled up. Let m be the mass per unit length of the chain. 



