Problem:
The number of teeth in three meshed circular gears A,B,C are x,y,z, respectively. The angular speeds, in revolutions per minute, of A,B,C are in the proportion
Answer Choices:
A. x:y:z
B. z:y:x
C. y:z:x
D. yz:xz:xy
E. xz:yx:zy
Solution:
Since the teeth are all the same size, equally spaci3l and are meshied, they all move with the same absolute speed v(Ξ½ is the distance a point on the circumference moves per unit of time). Let Ξ±,Ξ²,Ξ³ be the angular speeds of A,B,C, respectively. If a,b,c represent the lengths of the circumferences of A,B,C, respectively, then
Ξ±=avβ,Ξ²=bvβ,Ξ³=cvβ
Therefore, Ξ±a=Ξ²b=Ξ³c or, equivalently,
a1βΞ±β=b1βΞ²β=c1βΞ³β.
Thus the angular speeds are in the proportion
a1β:b1β:c1β
Since a,b,c, are proportional to x,y,z, respectively, the angular speeds are in the proportion
x1β:y1β:z1β
Multiplying each term by xyz gives the proportion
yz:xz:xy.
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