Question

A pair of straight teeth spur gears is transmitting power at 600 rpm and the pinion has 21 standard full depth involute teeth of module 7 mm. The pitch line velocity of the pinion is

• A. 0.09 \(m\cdot s^{-1}\)
• B. 0.22 \(m\cdot s^{-1}\)
• C. 2.32 \(m\cdot s^{-1}\)
• D. 4.62 \(m\cdot s^{-1}\)

Hint:

To speed up calculations, simplify the division first:
\(\frac{600}{60} = 10\).
Then multiply by the diameter: \(10 \times 0.147 = 1.47\).
Finally multiply by \(\pi \approx 22/7\): \(1.47 \times \frac{22}{7} = 0.21 \times 22 = 4.62\text{ m/s}\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The pitch line velocity of a gear represents the linear velocity of the pitch circle where the two mating gears make contact.

Step 2: Key Formula or Approach:
1. The pitch circle diameter (\(D\)) of a gear is:
\[ D = m \times T \] where \(m\) is the module and \(T\) is the number of teeth.
2. The linear pitch line velocity (\(v\)) is:
\[ v = \frac{\pi D N}{60} \] where \(N\) is the rotational speed in rpm.

Step 3: Detailed Explanation:
Given values for the pinion:
- Number of teeth (\(T\)) = \(21\)
- Module (\(m\)) = \(7\text{ mm} = 0.007\text{ m}\)
- Speed (\(N\)) = \(600\text{ rpm}\)
First, calculate the pitch circle diameter (\(D\)):
\[ D = 0.007 \times 21 = 0.147\text{ m} \] Now, calculate the pitch line velocity (\(v\)):
\[ v = \frac{\pi \times 0.147 \times 600}{60} \] \[ v = \pi \times 0.147 \times 10 \] \[ v = 1.47 \times \pi \approx 1.47 \times 3.14159 \approx 4.618\text{ m/s} \approx 4.62\text{ m}\cdot\text{s}^{-1} \]

Step 4: Final Answer:
The correct option is 4, which corresponds to 4.62 \(m\cdot s^{-1}\).