Question

the rotation of the smaller wheel in the figure causes the larger wheel to rotate. through how many degrees does the larger wheel rotate if the smaller one rotates 60.0°?
the larger gear rotates through approximately (square^{circ}). (do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Recall the relationship between arc - lengths

When two wheels are in contact, the arc - lengths they travel are equal. The formula for arc - length is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius, and $\theta$ is the angle in radians. Let $r_1 = 5.17$ cm be the radius of the smaller wheel, $\theta_1=60.0^{\circ}=\frac{\pi}{3}$ radians, and $r_2 = 8.12$ cm be the radius of the larger wheel, and $\theta_2$ be the angle of rotation of the larger wheel in radians. Since $s_1 = s_2$, we have $r_1\theta_1=r_2\theta_2$.

Step2: Solve for $\theta_2$

From $r_1\theta_1=r_2\theta_2$, we can express $\theta_2$ as $\theta_2=\frac{r_1\theta_1}{r_2}$. Substitute $r_1 = 5.17$, $\theta_1=\frac{\pi}{3}$, and $r_2 = 8.12$ into the formula: $\theta_2=\frac{5.17\times\frac{\pi}{3}}{8.12}=\frac{5.17\pi}{24.36}$.

Step3: Convert $\theta_2$ from radians to degrees

We know that to convert from radians to degrees, we use the formula $\text{Degrees}=\text{Radians}\times\frac{180^{\circ}}{\pi}$. So $\theta_2=\frac{5.17\pi}{24.36}\times\frac{180^{\circ}}{\pi}=\frac{5.17\times180^{\circ}}{24.36}$.

Step4: Calculate the value of $\theta_2$

$\theta_2=\frac{5.17\times180}{24.36}=\frac{930.6}{24.36}\approx38.2$.

Answer:

$38.2$