Question

Two connected gears are rotating. The smaller gear has a radius of 4 inches, while the larger gear has a radius of 7 inches. What is the angle through which the larger gear has rotated when the smaller gear has completed one full rotation? Answer in radians and degrees. 7 in 4 In

Answer 1

When there are connected gears, the arc lenght must be equal. That is:

When the smaller gear has completed one full rotation, we have

$\theta_2=2\pi$

since R=7in and r=4 in, we have that

$7\cdot\theta_1=4\cdot2\pi$

and we must isolate theta_1. Hence, the answer in radians is

$\theta_1=\frac{8\pi}{7}\text{ radians}$

Now, we must convert radians to degrees. This can be given by applyng the rule of three:

$\begin{gathered} \pi\text{ radian ----180 degre}es \\ \frac{8\pi}{7}\text{ radian ---- x} \end{gathered}$

therefore, x is equal to

$\begin{gathered} x=\frac{\frac{8\pi}{7}\cdot180}{\pi} \\ x=\frac{8\pi\cdot180}{7\cdot\pi} \\ x=\frac{8\cdot180}{7} \\ x=\frac{1440}{7} \\ x=205.71\text{ degr}ees \end{gathered}$

hence, in degrees the answer is 205.71 degrees