269-271] MOTION OF A CHAIN. 307 



The equations (1) and (2) of that Article then give us 



8w w_90 



8* p~3* 

 so that the chain moves uniformly along itself. 



The equations of motion (3) of the same Article are satisfied by 



T mgc sec -f mw 2 , 

 provided the form of the curve is the catenary s = c tan &amp;lt;. 



*271. Examples. 



1. Prove that any curve which is a form of equilibrium for a uniform 

 chain under conservative forces is a form which the chain can retain when 

 moving uniformly along itself under the same forces and that the tension is 

 greater in the steady motion than in equilibrium by mw 2 , where m is the 

 mass per unit length of the chain, and w is the velocity with which the chain 

 moves along itself. 



2. A uniform chain moves over two smooth parallel rails distant 2a apart 

 at the same level and is transferred from a coil at a distance k vertically below 

 one rail to a coil at a distance h + b vertically below the other. Prove that 

 the portion between the rails can be a common catenary provided the velocity 

 of the chain along itself is &amp;lt;J(gb). 



3. A uniform chain moves in a plane under no forces in such a way that 

 the curve of the chain retains an invariable form which rotates about a fixed 

 point in the plane with uniform angular velocity &&amp;gt;, while the chain advances 

 relatively to the curve with uniform velocity V. Prove that the general (p, r) 

 equation of the curve must be of the form 



Q9 + 2r/o&amp;gt;)r 2 = ap + &, 

 where a and b are constants. 



4. A uniform chain falls in a vertical plane under gravity. Prove that 

 the square of the angular velocity of the tangent at any element is 



the notation being that of Article 269. 



5. A uniform chain hangs in equilibrium over a smooth pulley with one 

 end fixed to the extremity of the vertical diameter and portions hanging 

 vertically on both sides. Prove that if the end is set free the distance y of 

 the lowest point from the horizontal diameter during the first part of the 

 motion satisfies the equation 



where I is the length of the chain and 2c is the circumference of the circle. 



6. A uniform chain of length 2Z and mass 2Z/x has its ends attached to 

 two points A, C and passes over a smooth peg B between A and C and in 

 the same horizontal line with them, the points J, B, C being so close together 



202 



