# Theory of Machines Questions and Answers – Arc of Contact and Contact Ratio

This set of Theory of Machines Multiple Choice Questions & Answers (MCQs) focuses on “Arc of Contact and Contact Ratio”.

1. The distance travelled by a point on either pitch circle of the two wheels during the period of contact of a pair of teeth is called ____________________
a) path of contact
b) arc of contact
c) contact ratio
d) angle of action

Explanation: The arc of contact is defined as the distance travelled by a point on either pitch circle of the two wheels during the period of contact of a pair of teeth. Simply, it is traced out along the pitch circle while one pair of the teeth is in contact.

2. The contact ratio is the ratio of arc of contact to the ____________
a) circular pitch
b) dedendum
c) circular pitch
d) module

Explanation: Contact ratio is the of the arc od contact to the circular pitch. The number of teeth in contact = arc of contact / circular pitch = path of contact / (circular pitch x cos φ).

3. If the contact ratio is 2.7, two pairs of teeth are always in contact and three pairs of teeth are in contact for _________ % of the time.
a) 100
b) 50
c) 45
d) 70

Explanation: If the contact ratio is 2.7, two pairs of teeth are always in contact and three pairs of teeth are in contact for 70 % of the time.

4. For continuous transmission of motion, the contact ratio i.e. n must always be _______
a) equal to 1
b) equal to 0
c) more than unity
d) less than unity

Explanation: For a continuous transmission of motion, at least one tooth of the wheel must be in contact with another tooth of the second wheel. Hence, n must be greater than unity. If the contact ratio is less than unity, then it implies that the even a single tooth of the two gears are not in contact with each other.

5. Two involute gears have path of contact 40 mm and the pressure angle is 50°. Find the arc of contact.
a) 62.23 mm
b) 25.71 mm
c) 30.64 mm
d) 52.21 mm

Explanation: Arc of contact = Path of contact/cos φ = 40/cos 50° = 62.23 mm.
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6. Two gears in a mesh have the arc of contact = 27 mm. The pressure angle is 15°. Find the path of contact.
a) 104.32 mm
b) 6.98 mm
c) 27.95 mm
d) 26.08 mm

Explanation: Path of contact = Arc of contact x cos φ = 27 x cos 15° = 26.08 mm.

7. Two involute gears in a mesh have a circular pitch of 25 mm. The arc of contact is 55 mm. Find the number of pairs of teeth in contact.
a) 2.2
b) 0.45
c) 13.75
d) 1.3

Explanation: Contact ratio = Arc of contact/ Circular pitch = 55/25 = 2.2.

8. Two involute gear have a module of 5 mm, the arc of contact is 25 mm. Find the contact ratio of these two gears.
a) 5
b) 1.59
c) 0.2
d) 3

Explanation: Contact ratio = Arc of contact/ Circular pitch = Arc of contact / πm = 25/(5xπ) = 1.59.

9. The path of contact of two gears in a mesh is 50 mm and the pressure angle is 45°. The module is 8 mm. Find the contact ratio of these two gears.
a) 4.2
b) 1.9
c) 3.7
d) 2.8

Explanation: Arc of contact = Path of contact/cos φ = 50/cos 45° = 70.71 mm
Contact ratio = Arc of contact/ Circular pitch = Arc of contact / πm = 70.71/(πx8) = 2.8.

10. The path of approach = 13.92 mm and path of recess = 11.56 mm. The pressure angle is 22.5°. The module is 4 mm. Find the contact ratio of these two gears.
a) 2.19
b) 3.54
c) 3.12
d) 2.53

Explanation: Path of contact = Path of approach + Path of recess = 13.92 + 11.56 = 25.48 mm
Arc of contact = Path of contact/cos φ = 25.48/cos 22.5° = 27.58 mm
Contact ratio = Arc of contact/ Circular pitch = Arc of contact / πm = 27.58/(πx4) = 2.19.

11. Two involute gears in a mesh have a module of 10 mm and the pressure angle is 35°. The larger gear has 45 teeth whereas the pinion has 15 teeth. The addendum is equal to one module. Find the arc of contact.
a) 36. 832 mm
b) 32.460 mm
c) 39.626 mm
d) 27.239 mm

Explanation: φ = 35°, t = 15, T = 45, m = 10 mm and addendum = 1 module = 10 mm.
r = mt/2 = 75 mm, R = mT/2 = 225 mm
ra = r + a = 85 mm, Ra = R + a = 235 mm
Path of contact = (Ra2 – R2cos2 φ)0.5 + (ra2 – r2cos2 φ)0.5 – (R + r)sinφ = 32.46 mm
Arc of contact = Path of contact/cos φ = 32.46/cos 35° = 39.626 mm.

12. The pressure angle of two gears in a mesh is φ = 22.5°. The number of teeth on the pinion is 25 and the gear ratio is 2. The module is 7 mm and addendum = 1 module. Find the angle of action of these two gears.
a) 32.056°
b) 34.697°
c) 22.72°
d) 28.531°

Explanation: φ = 22.5°, t = 25 and T = 25 x 2 = 50, m = 7 mm and addendum = 7 mm.
r = mt/2 = 87.5 mm and ra = r + a = 94.5 mm
R = mT/2 = 175 mm and Ra = R + a = 182 mm
Path of contact = (Ra2 – R2cos2 φ)0.5 + (ra2 – r2cos2 φ)0.5 – (R + r)sinφ = 32.056 mm
Arc of contact = Path of contact/cos φ = 32.056/cos 22.5° = 34.697 mm.
Angle of action = Arc of contact / r = 0.3965 rad = 22.72°.

13. The pressure angle of two gears in a mesh is φ = 25°. The number of teeth on the pinion is 45 and the gear ratio is 2. The module is 6 mm and addendum = 1.1 module. Find the contact ratio of these two gears.
a) 1.7
b) 2.3
c) 4.2
d) 3.5

Explanation: φ = 25°, t = 45 and T = 45 x 2 = 90, m = 6 mm and addendum = 6.6 mm.
r = mt/2 = 135 mm and ra = r + a = 141.6 mm
R = mT/2 = 270 mm and Ra = R + a = 276.6 mm
Path of contact = (Ra2 – R2cos2 φ)0.5 + (ra2 – r2cos2 φ)0.5 – (R + r)sinφ = 29.068 mm
Arc of contact = Path of contact/cos φ = 29.068/cos 25° = 32.073 mm.
Contact ratio = Arc of contact / Circular pitch = 32.073 / πm = 32.073 / (πx6) = 1.7.

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