Answer to Question #316175 in Geometry for Jrs

Question

The distance between the centers of the two gears is 20

cm . The shaft of each gear is 5 cm and 9 cm respectively.

A. What are the central angles of the shafts

on the gears? 

B. What is the length of the arc length part

of the belt? 

C. What is the length of the non-contacting part of the belt? 

D. What is the total length of the belt?

Answer 1

A. Let shaft with length 5cm be 1 and that with length 9cm be 2

central angle of shaft 2 is given by $\sigma$

$\theta=cos^-\frac{4}{20}=cos^-(0.2)$

$=1.369rad$

$\sigma=2\pi-2\theta$

$$=2\pi-2(1.369)=3.545$$

Central angle of shaft 1 is given by,

$2\theta=2\times1.369$

$$=2.738 rad$$

B. Arc length of the belt $, l=r\theta$

$l_1=5\times2.738=13.69cm$

$l_2=9\times3.545=31.905cm$

C. Length of non- contacting part

Let the length of one side be x

2 sides will be 2x

x=$\sqrt{20^2-4^2}=\sqrt{384}$

$=19.596cm$

2x=$2(19.596)$

$$=39.192cm$$

D. Total length of belt

$=39.192+31.905+13.69$

$=84.787cm$