247-249] INITIAL MOTIONS. 281 



this is the acceleration of m along the normal to its path. Hence, if p is the 

 initial radius of curvature of the path of w, 



u? _ m'l fu + v\ 2 



p" ~ m+m' \ I ) ' 

 giving 1/p = m' (u + v)*/{(m + m') lu*}. 



In like manner the initial curvature of the path of m' is 



249. Examples. 



1. Two bodies A and B of equal weight are suspended from the chains 

 of an Atwood's machine ; A is rigid, while B consists of a vessel full of 

 water in which is a cork attached to the bottom by a string. Supposing 

 the string to be destroyed in any manner, determine the sense in which A 

 begins to move. 



2. A particle is supported by equal threads inclined at the same angle a 

 to the horizontal. One thread being cut, prove that the tension in the 

 remaining thread is suddenly changed in the ratio 2 sin 2 a : 1. 



3. Particles of equal mass are attached to the points of trisection 0, D 

 of a thread ACDB of length 3, and the system is suspended by its ends from 

 points A, B distant (1+2 sin a) apart in a horizontal line. Prove that, if 

 the portion DB of the thread is cut, the tension of AC is instantly changed 

 in the ratio 2cos 2 a : l + cos 2 a, and that the initial direction of motion of D 

 is inclined to the vertical at an angle $ such that 



cot (p = tan a + 2 cot a. 



4. Three small equal rings rest on a smooth vertical circular wire at the 

 corners of an equilateral triangle with one side vertical, the uppermost being 

 connected with the other two by inextensible threads. Prove that, if the 

 vertical thread is cut through, the tension in the other thread is instantly 

 diminished in the ratio 3 : 4. 



5. A set of 2n equal particles are attached at equal intervals to a thread, 

 and the ends of the thread are attached to equal small smooth rings which 

 can slide on a horizontal rod. The rings are initially held in such a position 

 that the lowest part of the thread is horizontal, and the highest parts make 

 equal angles y with the horizontal, and the rings are let go. Prove that in 

 the initial motion (i) the acceleration of each particle is vertical, (ii) the 

 tension in the lowest part of the thread is to what it was in equilibrium in 

 the ratio m' : mncot 2 y+m' y where m is the mass of a particle and m' the 

 mass of a ring. 



6. Three particles A, B, C of equal masses are attached at the ends and 

 middle point of a thread so that AB=BC=a, and the particles are moving 

 at right angles to the thread, which is straight, with the same velocities, 

 when B impinges directly on an obstacle. Prove that, if there is perfect 

 restitution, the radii of curvature of the paths which A and C begin to 

 describe are equal to %a. 



