744 



MECHANICAL I'll I LoSOl'H Y. DYNAMICa 



[TM8IOH. 



for instance, tho weight* hanging to tlio 

 rtring wore 1 Ibs, and 5 Ibs. respectively ; then thu 



tension of tho string would be 2 *** 5 lba. - 6J Ibi. ; 



that u, a woi-lit nf 5| Ibs. would, if suspended to a fixed 

 string, produce tho same tension an U actually produced 

 in the moving string. In like manner, the axis of tho 

 pulley sustains a pressure equal to 11} lb. As tho 

 masses are proportional to the weights, we may, in a 

 similar way, get a numerical expression for the accelera- 

 tive force /that is, by mihstittiting the weights for 

 tin- maBscs in the above value of/. Thus, taking the 

 weights as here assumed, we have 



Acceleration, / 





32 '2 - 5 36 feet per second, 



tho accelerating force that gives motion to the system 

 being Jth of the aocelerative force of gravity. 



As already noticed, tho moving force Mi/, which com- 

 bines quantity of matter with the force acting upon it, 

 is pretture or weight when motion is not the result of the 

 action of the force on the mass. The symbol g, therefore, 

 which stands for acceleration in Dynamics, should stand 

 for pressure or weight in Statics, as in the expression for 

 T given above ; namely, 



_, 2M g -M' g 2WW 

 = M ? + M'</ ~ W + W 



Tho learner, therefore, will not fall into the mistake of 

 regarding such an expression as W = Mjr equivalent to 

 the statement, that "weight is equal to mass multiplied 

 by 32 -2 feet." This would, of course, be absurd. The 

 symbol g here stands for "the weight impressed by 

 gravity on the unit of mass," whatever that unit may 

 be selected to be. The unit of mass being assumed or 

 fixed upon by common consent, then, in the above ex- 

 pression, M is the number of these units in the body re- 

 ferred to, and g the weight of one of them. 



The unit of mass is usually taken equivalent to a 

 cubic foot of distilled water, at the temperature 60 of 

 Fahrenheit's thermometer. The weight of this unit is 

 1000 ounces avoirdupois ; and the weight W of any body 

 whatever, containing in it M times the quantity of mat- 

 ter contained in one cubic foot of distilled water, will 

 always be W = M X 1000 ounces ; and the same would 

 obviously be the case if the unit of mass were anything 

 eke, g representing always the weight of that unit. 



Whenever, in any mechanical inquiry, the investigation 

 involves the mixed consideration of motion and statical 

 pressure, although the motion and the pressure may be 

 equally due to the same thing, and the same symbol 

 (g for instance) be employed indifferently for the one or 

 other of these different effects, yet as the effect is all 

 that we intend the symbol to represent, it is plain that 

 its signification in the one case must be different from 

 its signification in the other. And it is of importance 

 that the student always bear this in remembrance : he 

 will find some further reference to the matter under the 

 head of HYDROSTATICS. 



2. Two weights W, W (Fig. 140), connected by a string 

 passing over a small pulley C, are placed upon the two 

 inclined planes CA, CB: it is required to determine 

 the circumstances of their motion. 



Lot, as before, M, M' represent the two masses. The 

 moving force of W, in the direction W A, is tAg sin. ; 

 and the moving force of W in the direction WB is 

 t/l'y in. : these oppose one another ; so that the moving 

 force F, to which the motion is due, is the difference of 

 namely : 



F - (M sin . i v, M' sin. t) g. 



And consequently the accelerating force /, whirh is 

 equal to F divided by tho sum of the masses moved, in 



Msinie/) M' sin i' 

 f ' -M + M' 



This therefore is the expression for tho accelerating 

 force which urges W down the plane C A, and which, 

 consequently, draws W up the plane C B. 



And since the velocity generated in t seconds is ft, and 

 the space passed over \JV, the velocity acquired by W 

 downwards, or by W upwards, in ( seconds, is 



M ;n i - r . M' sin i" 





and tho space described is 



M sin i v> M' sin ' 

 2 (M + M') ** 



If the two planes wore vertical, tho problem wiuld 

 become identical with the last; we should then h.i\i- 

 sin. t and sin. i' each equal to 1 ; and we see that the 

 expressions for r, * would bo those already deduced. 



If only one of the planes were vertical, the problem 

 would be converted into this namely, to determine the 

 circumstances of the motion when one body W, hanging 

 vertically, draws another body W up an inclined plane. 

 In this case, sin. t = 1 ; and we have only to put tli 

 the preceding expressions, in order to obtain the neces- 

 sary particulars of the motion. 



Again, if one of the planes were vertical and the other 

 horizontal, the problem would be to determine the 

 motion when W, hanging vertically, draws W along a 

 horizontal plane. In this case, sin. t = 1, and sin. t' = 0; 

 so that the accelerating force would be 



f ~ ST-KM ' g ' 



TENSION OF THE STRING. For the tension of 

 the string, under the original conditions, we have, if W 

 be the descending weight, 



M sin. t-M'sin. i' 

 T=M(srsin. - M+ -, -g) 



MM' 

 -M + M'C'"'- ' + "> 00 



which becomes the same as the expression for T in the 

 last problem, when sin. i and sin. i' are each of them 1. 

 By substituting W for M</, and W for M'j, in each of 

 the foregoing formula', numerical expressions will be 

 obtained, as in the former problem. 



3. Two weights W, W are attached, tho latter to a 

 wheel, and the former to its axle : to determine their 

 motions. 



Let M be the mass of W acting at tho axle, and M' 

 the mass of W, acting at the rim of the wheel : let 

 also R be the radius of the wheel, and r that of the 

 axle. 



Then the moving forces M'</, M</ acting at the dis- 

 tances R, r from the centre of motion, if free, would 

 have the respective effects M'j R, and My r, which, 

 connected as they are, oppose each other. And iff, 

 f be the actual accelerations of W, W, the moving forces, 

 at the distances R, r from the centre of motion, are ac- 

 tually M'/- R, and M/ r ; so that 



M'0-R M0-r=M'/'R + M/-r .... (1) 



The accelerations f, / must bo to each other as tho 

 radii R, r, tho velocities themselves being always in this 

 constant ratio, since tho wheel and axle both turn in tho 



same time; therefore/ - /, therefore 





Mr)/ 



(M''R Mr) ? - 



M'Rr Mr 1 

 /* U-<+Mr*? 

 the accelerating force of the ascending weight W. 



Ty 



And since / - - /, we have 



