348 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



where I = c^ + (a + be}l(e - 1), 



p, being the coefficient of friction, and m the mass of a unit of length of the 



chain. 



232. A uniform chain falls in a vertical plane with uniform acceleration 

 f retaining an invariable form, while the chain advances along itself with a 

 velocity which at any instant is the same for all points of the chain. Prove 

 that the angle (f) which the tangent at any point of the chain makes with 

 the horizontal, considered as a function of the time t and of the arc s 

 measured up to this point from some definite point of the chain, satisfies 

 the two partial differential equations 



/ ^ x 



(f-A 



_ 

 ds dsdt~ 97 as 2 



233. A uniform flexible chain passes over and under two rough equal 

 pulleys of radius a whose centres are at a distance d apart in the same hori 

 zontal line ; part of the chain is coiled up on a horizontal platform at a 

 depth h below this line, the part between one pulley and the platform is 

 vertical, the part below the pulleys is a catenary of parameter c, and the 

 chain hangs from the second pulley to a platform at a lower level h , the 

 vertical parts being between the pulleys. Show that steady motion with 

 this configuration is possible the pulleys rotating with angular velocity 

 *J{g(h-h }}/a, and that the relation between c, d, and h can be found by 

 eliminating a between the equations 



hc sec a + a cos a, c?=2c sinh ~ l (tan a) 2a sin a. 



234. A uniform chain hanging under gravity receives a tangential impulse 

 at one end. Prove that the initial velocity at any point in the direction at 

 right angles to the directrix is proportional to the curvature at the point. 



235. A chain of variable density is in the form of an arc of a circle less 

 than a semicircle and subtending an angle 2a at the centre, and the line 

 density varies inversely as the square of the distance from the diameter 

 parallel to the chord joining the ends. The chain is set in motion by 

 equal tangential impulses T applied at its ends ; prove that the kinetic 

 energy generated is 27 T2 sin 2 a/J/, where M is the mass of the chain. 



236. The ends of a chain of variable density are held at the same level, 

 and the chain hangs in the form of an arc of a circle subtending an angle 

 20 (&amp;lt;TT) at the centre. If equal tangential impulses are applied at the ends 

 the initial normal velocities at the lowest point and at either end are in the 

 ratio 1 : cos 6. 



237. A uniform chain lying in a curve on a smooth horizontal plane is 

 set in motion by impulsive tension applied at one end in the direction of the 

 tangent. If the initial direction of motion of every element makes the 

 same angle with the tangent prove that the curve is an equiangular spiral. 



