Question

two gears are adjusted so that the smaller gear drives the larger one, as shown in the figure. if the smaller gear rotates through an angle of 270°, through how many degrees will the larger gear rotate?

Explanation:

Step1: Recall arc - length formula

The arc - length formula is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius, and $\theta$ is the angle in radians. When two gears are in mesh, the arc - lengths they travel are equal. First, convert the angle of the smaller gear from degrees to radians. The angle of the smaller gear $\theta_1=270^{\circ}=270\times\frac{\pi}{180}=\frac{3\pi}{2}$ radians. Let $r_1 = 3.5$ cm be the radius of the smaller gear and $r_2 = 6.9$ cm be the radius of the larger gear. Let $\theta_2$ be the angle (in radians) of the larger gear.

Step2: Set arc - lengths equal

Since $s_1=s_2$, we have $r_1\theta_1=r_2\theta_2$. Substitute the known values: $3.5\times\frac{3\pi}{2}=6.9\times\theta_2$. Then $\theta_2=\frac{3.5\times\frac{3\pi}{2}}{6.9}=\frac{10.5\pi}{13.8}$ radians.

Step3: Convert $\theta_2$ to degrees

To convert $\theta_2$ from radians to degrees, use the conversion factor $\frac{180}{\pi}$. So $\theta_2=\frac{10.5\pi}{13.8}\times\frac{180}{\pi}=\frac{10.5\times180}{13.8}=\frac{1890}{13.8}\approx137^{\circ}$

Answer:

$137^{\circ}$