---
title: "Involute tooth profile and its properties"
subject: "Theory of Mechanism and Machine I"
chapter: 4
key: "semester_6/theory_of_mechanism_and_machine_i:involute-profile-properties-law-of-gearing"
source: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/involute-profile-properties-law-of-gearing
site: BCT Engineering
---

# Involute tooth profile and its properties

Theory of Mechanism and Machine I · Chapter 4: Spur Gears

- Asked in: year not given, 8 marks
- Video: [Involute Gears Explained](https://www.youtube.com/watch?v=nrsCoQN6V4M) — Gergely Bencsik
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/involute-profile-properties-law-of-gearing
- Whole chapter: https://www.bctengineer.com/md/semester_6/theory_of_mechanism_and_machine_i/chapter-4

## Question

Explain the law of gearing. Show that the involute profile satisfies it and state the advantages of involute teeth. Describe how an involute is generated and write down its important properties.

## Short answer

Law of gearing: for constant velocity ratio, common normal at tooth contact must pass through a fixed pitch point on line of centres. Involute, generated by unwinding a string from a base circle, satisfies this because its normal is always tangent to base circle, giving a fixed line of action. Advantages: invariant ratio despite centre distance changes, constant pressure angle, easy manufacture with straight-sided rack cutter, interchangeability. Properties: normal tangent to base circle, radius of curvature equals unwound length, r_b = r cosφ, path of contact is a straight line.

## Answer

#### Law of gearing
For a constant angular velocity ratio, the common normal at the point of contact of the two teeth must always pass through a fixed point on the line of centres, called the **pitch point** $P$. Then

$$
\frac{\omega_1}{\omega_2} = \frac{O_2P}{O_1P} = \text{constant}
$$

(Proof: along the common normal the velocities of the contact point on both bodies must have equal components. This gives $\omega_1 \cdot O_1N_1 = \omega_2 \cdot O_2N_2$, where $O_1N_1$, $O_2N_2$ are perpendiculars from the centres to the common normal. Similar triangles give $O_1N_1/O_2N_2 = O_1P/O_2P$.)

#### Generation of involute
An involute is the curve traced by a point on a taut string as it is unwound from a circle, called the **base circle**. A point on a straight line that rolls without slipping on the base circle also traces the involute.

```text
        T      string (tangent)
        |\
        | \
        |  * P  (point on involute)
        |  /
   -----|-/-----
  /     T0      \
 |   base circle  |
  \              /
   --------------
```

The string length $TP$ equals the arc $T_0T$ on the base circle that has been unwound.

#### Involute satisfies the law of gearing
Take two base circles of radii $r_{b1}$, $r_{b2}$ and a common tangent $N_1N_2$ to both (the string). When one circle drives the other, the contact point of the two involutes always lies on this tangent. So the common normal at the contact point is always $N_1N_2$, a fixed straight line crossing the line of centres at the same pitch point. Hence the velocity ratio is constant:

$$
\frac{\omega_1}{\omega_2} = \frac{r_{b2}}{r_{b1}}
$$

This line $N_1N_2$ is the **line of action**; the angle it makes with the common tangent of the pitch circles is the **pressure angle** $\phi$.

#### Properties of involute
1. The normal to an involute at any point is tangent to the base circle.
2. The radius of curvature at any point equals the length of the tangent from that point to the base circle.
3. The involute starts on the base circle and exists only outside it.
4. Path of contact is a straight line, so the pressure angle and the direction of the tooth force remain constant.
5. $r_b = r\cos\phi$ (base circle radius from pitch circle radius).

#### Advantages
- Constant velocity ratio is kept even if the centre distance changes slightly (the base circles do not change).
- Constant pressure angle gives smooth transmission and steady bearing loads.
- Simple to manufacture: a rack cutter has straight sides, so one cutter serves all gears of the same module and pressure angle.
- Interchangeability of gears of the same module and pressure angle.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For a constant angular velocity ratio, the common normal at the contact point must:
   A. Always be perpendicular to the line of centres
   B. Always pass through a fixed point on the line of centres (the pitch point)
   C. Always be tangent to the pitch circles
   D. Always pass through the centre of the pinion
   Answer: B. The common normal must cut the line of centres at a fixed pitch point P, so omega1/omega2 = O2P/O1P stays constant. (Key point: Law of gearing)

2. An involute is generated by:
   A. A point on a circle rolling on a straight line
   B. A point on a rod sliding between two circles
   C. A point on a taut string unwound from a base circle
   D. A point moving at constant distance from the base circle
   Answer: C. A point on a taut string unwound from the base circle (or on a straight line rolling on it) traces an involute. (Key point: Generation of involute)

3. Why does an involute profile satisfy the law of gearing?
   A. The pitch circles never change
   B. The teeth never slide on each other
   C. The base circles are equal in size
   D. Contact always lies on the common tangent to the base circles, so the common normal is a fixed line through the pitch point
   Answer: D. The contact point of two involutes always lies on the tangent N1N2 to the base circles, a fixed line crossing the line of centres at the same pitch point. (Key point: Why involute meets the law)

4. Which is an advantage of involute teeth stated in the answer?
   A. Constant velocity ratio is kept even if the centre distance changes slightly
   B. Velocity ratio varies to give a smoother start
   C. Pressure angle changes along the path of contact
   D. Needs a different cutter for every gear size
   Answer: A. The base circles do not change with centre distance, so the ratio stays constant. The pressure angle is also constant and a rack cutter has straight sides. (Key point: Advantage of involute teeth)

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