ILLUSTRATIVE EXAMPLES 



161 



giving the relation between the increments dl, dd in I and 0. The work done 

 against the tension of the string (T) in this displacement is Tdl. The height 

 of the center of gravity of the whole figure above C is initially A C, or a cos 0, 

 so that, as in 120, the work done against gravity is 



4wd(acos6). 

 Thus the total work performed by external forces in the displacement is 



4wd(acos6) + Tdl, 

 or, on substituting the values of dl and d(a cos 6), 



- 4 wa sin cZ0 -f T-2acos0d0. 

 For equilibrium this must vanish. We must therefore have 



T = 2 w tan 0, 

 giving the required tension. 



4. A rod of length I and weight w is suspended by its two ends from two points 

 at the same height and distant I apart, by two strings each of length a. Find the 

 couple required to hold the rod in a position 

 in which it makes an angle 6 with its equi- 

 librium position. 



In equilibrium the strings are vertical, 

 the two ends A, B of the rod lying exactly 

 underneath the two points of suspension 



P, Q- 



As the rod is turned from its equilibrium 

 position, we can imagine its middle point to 

 rise gradually along the vertical line through 

 the original position of this middle point. 

 When the rod has been turned through any 

 angle 6, let the height through which this 

 point has risen be x. 



Then the projection of the length PA' 

 on a vertical line will be a x, while its projection on a horizontal plane, 



/j 



being equal to the horizontal projection of A A', will clearly be I sin - . 



Thus, expressing that the length of the displaced string PA' remains equal 

 to its original value a, we have 



a 2 = (a-z) 2 + Z 2 sin 2 -. (a) 



To find the couple required to hold the rod ai an angle 6, let us suppose that 

 the rod is held in equilibrium in this position by a couple G, and that a small 

 displacement occurs in which is increased to + dd. The work done against 

 the couple is equal, by 121, to V (?d0, the negative sign being taken, since 

 the couple aids, instead of opposing, the motion. The work done against gravity 

 is equal to w dx. Thus the equation of equilibrium is 



- G d + w dx = 0, 



