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Chapter 12: Problem 66

A pinion having 18 teeth and a diametral pitch of 10 (in-1) is in mesh with a rack. If the rack moves 1 in, how many degrees will the pinion rotate?

Short Answer

Expert verified
When the rack moves 1 in, the pinion will rotate by approximately 63.66 degrees.

Step by step solution

01

Calculate Pinion Diameter

The diametral pitch of a gear system can be defined as the ratio of the number of teeth to the pitch diameter. So we can rearrange the formula to find the pitch diameter (D) of the pinion: \(D = \frac{Number\,of\,Teeth}{Diametral\,Pitch}\). Substituting the given values, \(D = \frac{18}{10} = 1.8\) inches.
02

Calculate Pinion Circumference

The circumference (C) of the pinion can be calculated using the formula \(C = \pi D\). So, substituting the pinion diameter, we get \(C = \pi * 1.8 = 5.65486\) inches. This means that for every 5.65486 inches the rack moves, the pinion completes one full rotation (360 degrees).
03

Calculate Pinion Rotation

Given that the rack moves 1 inch, we need to calculate what fraction of one full rotation (360 degrees) this corresponds to. This can be calculated by creating a ratio of the distance the rack moves (1 inch) to the circumference of the pinion, and then multiplying this ratio by 360 degrees to give the angle of rotation. So, \(Angle\,of\,Rotation = \left(\frac{1}{5.65486}\right) * 360 = 63.66\) degrees.

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Most popular questions from this chapter

A 78-tooth spur gear is in mesh with a 27-tooth pinion. The \(p_{d}=6\) and \(\phi=20^{\circ}\). Find the contact ratio.

A paper machine processes rolls of paper having a density of \(984 \mathrm{~kg} / \mathrm{m}^{3}\). The paper roll is \(1.50-\mathrm{m}\) outside dia \((O D) \times 0.22-\mathrm{m}\) inside dia \((I D) \times 3.23 \mathrm{~m}\) long and is on a simply supported, hollow, steel shaft with \(S_{u t}=400 \mathrm{MPa}\). Design a \(2.5: 1\) reduction spur gearset to drive this roll shaft to obtain a minimum dynamic safety factor of 2 for a \(10-\) year life if the shaft \(O D\) is \(22 \mathrm{~cm}\) and the roll turns at \(50 \mathrm{rpm}\) with \(1.2 \mathrm{hp}\) absorbed.

A \(20^{\circ}\)-pressure-angle, 27-tooth spur gear has a diametral pitch \(p_{d}=5\). Find the pitch dia. addendum, dedendum, outside dia, and circular pitch.

Design a two-stage compound spur-gear train for an overall ratio of approximately 78:1. Specify tooth numbers for each spur gear in the train.

A \(25^{\circ}\) pressure angle, 32 -tooth spur gear has a diametral pitch of 4 . Find the pitch diameter, addendum, dedendum, outside diameter, and circular pitch.
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