---
title: "Path of contact and contact ratio numerical"
subject: "Theory of Mechanism and Machine I"
chapter: 4
key: "semester_6/theory_of_mechanism_and_machine_i:path-of-contact-arc-contact-ratio"
source: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/path-of-contact-arc-contact-ratio
site: BCT Engineering
---

# Path of contact and contact ratio numerical

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

- Asked in: year not given, 8 marks
- Video: [LENGTH OF PATH OF CONTACT, ARC OF CONTACT AND CONTACT RATIO](https://www.youtube.com/watch?v=Jj4HmrZSgZQ) — TECHNICAL CLASSES
- 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/path-of-contact-arc-contact-ratio
- Whole chapter: https://www.bctengineer.com/md/semester_6/theory_of_mechanism_and_machine_i/chapter-4

## Question

Two spur gears in mesh have 20 and 40 teeth, module 4 mm and 20 degree pressure angle. The addendum of each gear is equal to one module. The pinion is the driver. Derive the expression for the length of the path of contact and calculate (a) the length of path of approach and recess, (b) the length of path of contact, (c) the arc of contact, and (d) the contact ratio.

## Short answer

Derivation: contact starts where wheel addendum circle cuts line of action (approach) and ends where pinion addendum cuts it (recess). Path of approach KP = sqrt(R_a^2 - R_b^2) - R sinφ =10.117 mm; path of recess PL = sqrt(r_a^2 - r_b^2) - r sinφ =9.192 mm. Path of contact KL =19.31 mm. Arc of contact = KL/cosφ =20.55 mm. Contact ratio = arc/p_c =20.55/(π×4) =1.64.

## Answer

#### Derivation
Let $R_a$, $r_a$ be the addendum circle radii of the wheel and pinion, $R$, $r$ the pitch circle radii, and $R_b = R\cos\phi$, $r_b = r\cos\phi$ the base circle radii. The line of action $N_1N_2$ is tangent to the pinion base circle at $N_1$ and to the wheel base circle at $N_2$, and passes through the pitch point $P$. Contact starts where the wheel addendum circle cuts the line of action (point $K$) and ends where the pinion addendum circle cuts it (point $L$).

```text
   N1 ..........K....P....L......... N2
        approach | recess
```

In the right-angled triangle $O_2N_2K$ (wheel centre $O_2$): $KN_2 = \sqrt{R_a^2 - R_b^2}$. In triangle $O_2N_2P$: $PN_2 = R\sin\phi$. So

$$
\text{Path of approach } KP = KN_2 - PN_2 = \sqrt{R_a^2 - R_b^2} - R\sin\phi
$$

Similarly, from the pinion triangles $O_1N_1L$ and $O_1N_1P$:

$$
\text{Path of recess } PL = LN_1 - PN_1 = \sqrt{r_a^2 - r_b^2} - r\sin\phi
$$

$$
\text{Path of contact } KL = KP + PL = \sqrt{R_a^2 - R_b^2} + \sqrt{r_a^2 - r_b^2} - (R + r)\sin\phi
$$

Arc of contact $=\dfrac{\text{path of contact}}{\cos\phi}$; contact ratio $=\dfrac{\text{arc of contact}}{p_c}$.

#### Data
$r = 40$ mm, $R = 80$ mm, $r_a = 44$ mm, $R_a = 84$ mm, $\phi = 20^\circ$.

$$
r_b = 40\cos 20^\circ = 37.588\ \text{mm}, \quad R_b = 80\cos 20^\circ = 75.175\ \text{mm}
$$

#### (a) Path of approach and recess
$$
KP = \sqrt{84^2 - 75.175^2} - 80\sin 20^\circ = 37.479 - 27.362 = 10.117\ \text{mm}
$$

$$
PL = \sqrt{44^2 - 37.588^2} - 40\sin 20^\circ = 22.873 - 13.681 = 9.192\ \text{mm}
$$

#### (b) Path of contact
$$
KL = 10.117 + 9.192 = 19.31\ \text{mm}
$$

#### (c) Arc of contact
$$
\frac{KL}{\cos 20^\circ} = \frac{19.31}{0.9397} = 20.55\ \text{mm}
$$

#### (d) Contact ratio
Circular pitch $p_c = \pi m = 12.566$ mm.

$$
\text{Contact ratio} = \frac{20.55}{12.566} = 1.64
$$

A contact ratio of 1.64 means that two pairs of teeth are in contact for about 64% of the time and one pair for the rest. Since it is greater than 1, the drive is continuous.

**Answer: approach $= 10.117$ mm, recess $= 9.192$ mm, path of contact $= 19.31$ mm, arc of contact $= 20.55$ mm, contact ratio $= 1.64$.**

## Check yourself

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

1. The path of approach KP is given by:
   A. sqrt(R_a^2 - R_b^2) + R sin(phi)
   B. sqrt(r_a^2 - r_b^2) - r sin(phi)
   C. R_a - R
   D. sqrt(R_a^2 - R_b^2) - R sin(phi)
   Answer: D. Contact starts where the wheel addendum circle cuts the line of action: KP = sqrt(Ra^2 - Rb^2) - R sin(phi). The recess uses the pinion's radii. (Key point: Path of approach formula)

2. With path of approach 10.117 mm and recess 9.192 mm, the path of contact is:
   A. 0.925 mm
   B. 19.31 mm
   C. 20.55 mm
   D. 92.9 mm
   Answer: B. Path of contact KL = KP + PL = 10.117 + 9.192 = 19.31 mm. (Key point: Path of contact)

3. How is the arc of contact found from the path of contact?
   A. Multiply by cos(phi)
   B. Add the circular pitch
   C. Divide by cos(phi)
   D. Divide by sin(phi)
   Answer: C. Arc of contact = path of contact / cos(phi) = 19.31/0.9397 = 20.55 mm. (Key point: Arc of contact)

4. Contact ratio = 20.55 mm / 12.566 mm = 1.64. What must the contact ratio exceed for continuous drive?
   A. 2
   B. 1
   C. 0.5
   D. 1.64
   Answer: B. It is arc of contact divided by circular pitch and must exceed 1 so that a new pair engages before the previous pair leaves. (Key point: Contact ratio)

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