314 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



25. Weights P and W equilibrate on a wheel and axle of negligible mass. 

 A weight W is attached to P and after the lapse of one second another weight 

 W is attached to the ascending weight W. Prove that, after the lapse of 

 another second, the velocity of the ascending weight 2 W is 



a being the radius of the wheel, and b the radius of the axle. 



26. A particle of mass m is attached by inextensible threads to particles 

 of masses m and m&quot;. The particles are placed on a smooth table with the 

 threads in two perpendicular straight lines, and the particle m is struck by a 

 blow in the direction of the bisector of the angle between the threads so that 

 both threads are jerked. Prove that the initial velocities of m and m&quot; are 

 in the ratio m + m&quot; : m + m f . 



27. A particle of mass M is projected with velocity V in a direction 

 making an angle 6 with the horizontal, being attached to the point of pro 

 jection by an inextensible thread of length V 2 cosec 2 0/2&amp;lt;7. Prove that the 

 impulsive tension when the thread becomes tight is MVcosPQ cosectf, and 

 that immediately after the change of motion the tension is Mg(\ -2 sin 4 0). 



28. Three particles A, B, C of equal mass are placed on a smooth plane 

 inclined at an angle a to the horizontal, and J3, C are connected with A by 

 threads of length h sec a which make equal angles a with the line of greatest 

 slope through A on opposite sides of it. If A is struck by a blow along the 

 line of greatest slope so as to start to move downwards with velocity F, find 

 when the threads become tight, and prove that the velocity of A immediately 

 afterwards is 



F/(3 - 2 sin 2 a) + 2gh sin a/ V. 



29. Four equal particles are attached at the corners of a rhombus formed 

 of four threads each of length a, and the system is moving on a horizontal 

 plane with uniform velocity u in the direction of the longer diagonal AC 

 when the end A of that diagonal is suddenly fixed. Prove that the sides of 

 the rhombus begin to turn with angular velocity 2u sin a/a (I + 2 sin 2 a), 

 where 2a is the acute angle of the rhombus. 



30. A set of 2tt - 1 particles connected by inextensible threads are sus 

 pended from two fixed points in a horizontal line so as to hang symmetrically, 

 their weights being such that each of the two lowest threads makes an angle 

 a with the horizontal and each of the threads makes an angle a with the 

 one below it. Prove that, if the lowest particle (mass m) is struck by a 

 vertical impulse P, the horizontal component of the initial velocity of any 

 particle will vary inversely as its mass, and the vertical component of the 

 velocity of the rth from the lowest will be 



PBIU {(271- 2r - 1) sin a + 2 cos a cot na - sin (2r+ 1) a}. 

 2icos 2 a (V 



