160 



WORK 



2. Gearing of a bicycle. As a second example, let us apply the principle of 

 virtual work to the mechanism of a bicycle. Let the length of the crank be a, 

 and let the bicycle be geared to 6 inches, so that each revolution of the pedals 

 causes the machine to move as far forward as it would in one revolution of a 

 wheel of 6 inches diameter. Let us find what pressure must be exerted on the 

 pedal by a rider in order that the machine may move forward against an 

 opposing frictional force of w pounds weight. 



Let us give the machine a small displacement, the cranks being supposed to 

 turn through an infinitesimal angle e, and the wheels and machine moving 

 forward accordingly. Since the gearing is to 6 inches, the distance moved by 

 the machine as a whole will be 1 be inches, while the distance moved by the pedal, 

 taking the machine itself as frame of reference, will be ae. Let W pounds weight 

 be the force exerted on the pedal when the machine is just on the point of motion, 

 so that the machine is in equilibrium under this force acting on the pedal, and 

 the backward pull of w pounds due to friction. The equation of virtual work is 



so that the required force is 



nr=*. 



.Thus the force is directly proportional to the gearing of the machine, but 

 inversely proportional to the length of the cranks. 



3. Four rods of equal weight w and length a are freely jointed so as to form a 

 rhombus ABCD. The framework stands on a horizontal table so that CA is vertical, 

 and the whole is prevented from collapsing by a weightless inextensible string of 



length I which connects the points B, D. It is 

 required to find the tension in this string. 



To find the tension by the principle of 

 virtual work, we must of course find a small 

 displacement such that work is done in oppo- 

 sition to the tension, or otherwise the tension 

 would not enter into the equations at all. 

 Since the string is inextensible, it is not 

 possible in actual fact to stretch it and so 

 perform work against its tension. We can 

 however imagine it to be stretched in spite of its actual inextensibility, or, 

 what comes to the same thing, we can imagine it replaced by an extensible 

 string of the same length and having the same tension. It is now easy to 

 arrange a displacement of the kind required. 



Let us imagine that the framework is displaced in such a way that A moves 

 vertically downwards towards (7, while C remains at rest. Let the displacement 

 be such that the angle D A C is increased from e to 6 + d0. The length I of the 

 string which corresponds to the angle 6 is given by 



I 2 a sin 8, 

 from which, by differentiation, we obtain 



dl = 2 a cos e d8, 



FIG. 93 



