- #1

bagasme

- 79

- 9

Here in this thread I will derive formulas for relation between two wheels, either teethed (e.g. gears) or non-teethed.

In wheel relationship, we have three cases:

- Two wheels at the same axle
- Two wheels intersected in parallel (meshed)
- Two wheels connected by a belt

**1. At the same axle**

Figure 1 (source: fisikabc.com) shows two wheels connected to the same axle. Since both ##A## and ##B## are on same center, they have same angular accelleration:

$$\begin{align}

\omega_A & = \omega_B \nonumber \\

\frac {v_A} {r_A} & = \frac {v_B} {r_B} ~ (1)\nonumber

\end{align}$$

**2. Intersected in Parallel**

Figure 2 (source: fisikabc.com) show two wheels intersected each other in parallel. When wheel ##A## rotated counter-clockwise, wheel ##B## rotated clockwise, and vice versa. At the intersection point between two wheels, linear velocity of both wheels directed at the same direction, so:

$$\begin{align}

v_A & = v_B \nonumber \\

\omega_A \cdot r_A &= \omega_B \cdot r_B ~ (2)\nonumber

\end{align}$$

**3. Connected by Belt**

Figure 3 (source: fisikabc.com) shows two wheels connected by a belt. As wheel ##B## rotated, the belt rotated wheel ##A## at the same linear velocity direction as wheel ##B##. Thus the formula is same as when both wheels intersected directly (2).

**Relationship Involving Teethed Wheels (e.g. Gears)**

We will now deriving equations that relate between angular velocity, radius, and number of teeth. ##N_A## and ##N_B## are number of teeth of wheel A and B, respectively.

From equation (2):

$$\begin{align}

\omega_A \cdot r_A & = \omega_B \cdot r_B \nonumber \\

\frac {2 \cdot \pi \cdot r_A} {t} & = \frac {2 \cdot \pi \cdot r_B} {t} \nonumber \\

r_A &= r_B \nonumber \\

\frac {r_A} {n_A} &= \frac {r_B} {n_B} \nonumber \\

r_A \cdot N_B &= r_B \cdot N_A \nonumber \\

\frac {r_A} {r_B} &= \frac {N_A} {N_B} ~ (3) \nonumber

\end{align}$$

Combining equation (2) and (3):

$$\begin{align}

\omega_A \cdot r_A & = \omega_B \cdot r_B \nonumber \\

\frac {\omega_A} {\omega_B} &= \frac {r_B} {r_A} \nonumber \\

\frac {\omega_A} {\omega_B} &= \frac {N_B} {N_A} \nonumber \\

\end{align}$$

We can conclude that angular accelleration is related to radius, but inverse related to number of teeth.

I hope that this thread will be useful. Please let me know any flaws and suggestions.

Cheers, Bagas