## Task $9 / 62$

a) A gear $K_{2}$ with the pitch circle diameter $d=2 r$ rolls on a fixed gear $K_{1}$ with the same pitch circle diameter. How many times does $K_{2}$ rotate about its axis during one full revolution around $K_{1}$?

b) A gear $K_{2}$ with the pitch circle diameter $d_{2}=2 r_{2}$ rolls on a fixed gear $K_{1}$ with a pitch circle diameter $d_{1}=2 r_{1}=3 d_{2}$. How many times does $K_{2}$ rotate about its axis during one full revolution around $K_{1}$?

c) A gear $K_{2}$ with the pitch circle diameter $d_{2}=2 r_{2}$ rolls on a fixed gear $K_{1}$ with the pitch circle diameter $d_{1}=2 r_{1}=\frac{1}{3} d_{2}$. How many revolutions does it need to make to rotate exactly once about its own axis?

![](https://cdn.mathpix.com/cropped/2024_06_06_1d2be4837fb839c2b31dg-037.jpg?height=545&width=505&top_left_y=858&top_left_x=296)
The fixed circle $K_{1}$ has the diameter $d_{1}=2 r_{1}$. The rolling circle $K$ has the diameter $d=2 r$. From the initial position $A$ (see figure), the circle $K$ rolls to the circumference point $B$. The contact point $A *$ of the circumference of $K$ in the initial position moves to the position $A^{\prime}$. The two circular arcs $A B$ and $A^{\prime} B$ have the same length due to the no-slip condition. The central angles corresponding to these arcs are denoted by $\beta$ and $\beta^{\prime}$. Then the total rotation $\varphi$ of the rolling circle $K$ is: $\varphi=\beta+\beta^{\prime}$. If the angles are measured in radians, then

$$
\overparen{A B}=r_{1} \overparen{\beta} \quad \text { and } \quad \overparen{A^{\prime} B}=r \widetilde{\beta^{\prime}}
$$

From $\overparen{A B}=\widetilde{A^{\prime}} B$ it follows that $r_{1} \overparen{\beta}=r \overparen{\beta^{\prime}}$ or $\beta^{\prime}=\frac{r_{1} \widehat{\beta}}{r}=\frac{d_{1} \widehat{\beta}}{d}$.

Thus, the resulting rotation is

$$
\widehat{\varphi}=\widehat{\beta}+\widehat{\beta^{\prime}}=\widehat{\beta}+\frac{d_{1} \widehat{\beta}}{d}=\widehat{\beta}\left(1+\frac{d_{1}}{d}\right)
$$

If the angles are measured in degrees, then $\overparen{A B}=\frac{\pi d_{1} \beta}{360^{\circ}}$ and $\overparen{A^{\prime} B}=\frac{\pi d \beta^{\prime}}{360^{\circ}}$. Since $\overparen{A B}=\overparen{A B^{\prime}}$, it follows that $d_{1} \beta=d \beta^{\prime}$, thus $\beta^{\prime}=\frac{d_{1} \widehat{\beta}}{d}$. Therefore,

$$
\varphi=\beta+\beta^{\prime}=\beta+\frac{d_{1} \beta}{d}=\beta\left(1+\frac{d_{1}}{d}\right)
$$

Substituting into the general form:

Question 1: $d_{1}=d, \widehat{\beta}=2 \pi$ or $\beta=360^{\circ}$, yields $\varphi=4 \pi=720^{\circ}$.

Question 2: $d_{1}=3 d, \widehat{\beta}=2 \pi=360^{\circ}$, yields $\varphi=8 \pi=1440^{\circ}$.

Question 2: $d=3 d_{1}, \widehat{\varphi}=2 \pi=360^{\circ}$, yields $\widehat{\beta}=\frac{6 \pi}{4}=270^{\circ}$.

Answer: In case a, the rolling circle rotates twice, in case b, four times around its center, and in case c, it must roll three-quarters of the fixed circle.