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Chapter 18: Problem 28

Find the required quantities from the given proportions. For two connected gears, the relation $$\frac{d_{1}}{d_{2}}=\frac{N_{1}}{N_{2}}$$ holds, where \(d\) is the diameter of the gear and \(N\) is the number of teeth. Find \(N_{1}\) if \(d_{1}=2.60\) in. \(d_{2}=11.7\) in. and \(N_{2}=45 .\) The ratio \(N_{2} / N_{1}\) is called the gear ratio. See Fig. \(18.3 .\)

Short Answer

Expert verified
\(N_{1} \approx 10\)

Step by step solution

01

Understanding the Given Equation

The relationship between the diameters and the number of teeth of gears is given by the equation \( \frac{d_{1}}{d_{2}} = \frac{N_{1}}{N_{2}} \). Here, you need to find \( N_{1} \), given that \( d_{1} = 2.60 \) inches, \( d_{2} = 11.7 \) inches, and \( N_{2} = 45 \).
02

Setting Up the Equation

Substitute the given values into the equation \( \frac{d_{1}}{d_{2}} = \frac{N_{1}}{N_{2}} \). This gives us \( \frac{2.60}{11.7} = \frac{N_{1}}{45} \).
03

Solving for \( N_{1} \)

To find \( N_{1} \), multiply both sides by 45 to isolate \( N_{1} \) on one side of the equation: \( N_{1} = 45 \times \frac{2.60}{11.7} \).
04

Calculating the Value

Perform the calculation: first calculate \( \frac{2.60}{11.7} \), which is approximately 0.2222. Then, multiply by 45: \( N_{1} = 45 \times 0.2222 \approx 10 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion
The concept of proportion is a fundamental mathematical principle used widely in various fields, such as geometry, physics, and engineering. In the context of gears, proportion applies to the relationship between two sets of measured quantities, which ensures that one quantity changes in a stable manner relative to another. When dealing with connected gears, this relationship becomes crucial for predicting how movements translate from one gear to another.

Proportion can be expressed as an equation; a very common form used for gears is \( \frac{d_{1}}{d_{2}} = \frac{N_{1}}{N_{2}} \). This equation means that the ratio of the diameters of two gears \( (d_{1}/d_{2}) \) is equivalent to the ratio of their respective number of teeth \( (N_{1}/N_{2}) \).
  • If two gears have the same shape and teeth size, then the number of teeth is directly proportional to the diameter.
  • Understanding these proportional relationships allows for predicting gear behavior under different conditions, which is essential in mechanical design.
Gear Diameters
The diameter of a gear is an essential factor in determining the gear ratio and how the gears interact. Essentially, the diameter gives us the size of the gear and, more importantly, it is directly proportional to the number of teeth a gear possesses when gear design specifications remain constant.

Gear diameters are often given in inches or millimeters, and they influence the rotational speed as well as the torque the gear can handle. The relationship between diameter \( d \) and number of teeth \( N \) in gears is captured in the equation \( \frac{d_{1}}{d_{2}} = \frac{N_{1}}{N_{2}} \). Larger diameters will typically slow down rotation but increase torque, while smaller diameters increase speed but decrease torque.
  • Knowing the diameter is crucial when designing gear systems to fit specific space requirements or achieve desired rotational speeds.
  • Adjusting the diameter changes the mechanical advantage offered by the gear system.
Gear Teeth Calculation
Calculating the number of teeth on a gear is an application of the proportion relationship. When designing or selecting gears for a mechanism, it is vital to determine exactly how many teeth a gear should have to ensure proper functionality

In the context of the given exercise, you need to find the number of teeth for one gear \( N_{1} \) given the diameters and the number of teeth on the other gear \( N_{2} \). You start by using the formula \( N_{1} = N_{2} \times \frac{d_{1}}{d_{2}} \). This calculation allows you to derive \( N_{1} \), ensuring that the gear system functions smoothly and efficiently.
  • The gear teeth calculation is essential for applications where specific gear ratios are required.
  • Accurately calculating the number of teeth helps in preventing wear and tear from improper gear meshing.
  • Basic arithmetic and proportional reasoning skills are necessary to perform these calculations effectively.

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Most popular questions from this chapter

Solve the given applied problems involving variation. The \(x\) -component of the acceleration of an object moving around a circle with constant angular velocity \(\omega\) varies jointly as \(\cos \omega t\) and the square of \(\omega .\) If the \(x\) -component of the acceleration is \(-11.4 \mathrm{ft} / \mathrm{s}^{2}\) when \(t=1.00 \mathrm{s}\) for \(\omega=0.524 \mathrm{rad} / \mathrm{s},\) find the \(x\) -component of the acceleration when \(t=2.00 \mathrm{s}\).

Solve the given applied problems involving variation. In a physics experiment, a given force was applied to three objects. The mass \(m\) and the resulting acceleration \(a\) were recorded as follows: $$\begin{array}{l|c|c|c}m(\mathrm{g}) & 2.0 & 3.0 & 4.0 \\\\\hline a\left(\mathrm{cm} / \mathrm{s}^{2}\right) & 30 & 20 & 15\end{array}$$ (a) Is the relationship \(a=f(m)\) one of direct or inverse variation? Explain. (b) Find \(a=f(m)\)

Find the required ratios. The atomic mass of an atom of carbon is defined to be 12 u. The ratio of the atomic mass of an atom of oxygen to that of an atom of carbon is \(\frac{4}{3} .\) What is the atomic mass of an atom of oxygen? (The symbol u represents the unified atomic mass unit, where \(\left.1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{kg} .\right)\)

Solve the given applied problems involving variation. In electroplating, the mass \(m\) of the material deposited varies directly as the time \(t\) during which the electric current is on. Set up the equation for this relationship if \(2.50 \mathrm{g}\) are deposited in \(5.25 \mathrm{h}\)

A varies directly as \(x,\) and \(B\) varies directly as \(x,\) although not in the same proportion as \(A .\) All numbers are positive. Show that \(\sqrt{A B}\) varies directly as \(x\)
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