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Chapter 5: Problem 33

The gear motor can develop \(\frac{1}{10}\) hp when it turns at 300 rev \(/\) min. If the shaft has a diameter of \(\frac{1}{2}\) in., determine the maximum shear stress in the shaft

Short Answer

Expert verified
The maximum shear stress in the shaft is approximately 5368.04 psi.

Step by step solution

01

Convert Revolution per Minute to Radian per Second

It's necessary to convert RPM to rad/s prior to calculation because the equation's angular speed must be in radians per second. Every minute contains 60 seconds, and every revolution is \(2\pi\) radians. So, \(300 \cdot \frac{2\pi}{60}= 31.415\) rad/s.
02

Find the Torque using the Power-torque formula

Here, the value of power (P) is given in horsepower, which needs to be converted to lb-ft/s using the conversion factor 550, because 1 hp = 550 lb-ft/s. Now using the Power-torque formula \( P = T\cdot w\), rearrange for Torque (T): \(T= \frac{P}{w} = \frac{\frac{1}{10}\cdot 550}{31.415} = 1.75\) lb-ft.
03

Calculate Maximum Shear Stress using the Torsional Shear Stress formula

The shaft's diameter (d) is given as 0.5 inch, so the radius (r) = \(d/2 = 0.25\) inch. Now using the Torsional Shear Stress formula \(蟿 = \frac{Tr}{J}\), where \(J = \frac{\pi d^4}{32}\), we need to plug in the values in and get our shear stress in psi (pounds/inch^2): \(蟿= \frac{16 * T}{\pi d^3}= \frac{16 * 1.75 *12}{\pi * 0.5^3} = 5368.04\) psi.

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Key Concepts

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

Gear Motor
Gear motors are essential components in many mechanical systems. These devices integrate a gear train with an electric motor, allowing for torque output modification and control. The advantage of using a gear motor lies in its ability to produce a substantial amount of torque while maintaining a relatively small size.

When considering a gear motor, its benefits include:
  • Efficiency in torque management
  • Flexibility in speed adjustment
  • Compact design suitable for minimal space
Understanding how gear motors translate power from an electric input to mechanical output via a system of gears is fundamental. This process affects how efficient a gear motor is, and how suitable it will be for a specific application.
Torque Calculation
Torque, often referred to as rotational force, is a crucial factor when dealing with motors and rotational systems. It is the measure of the force that can cause an object to rotate about an axis. Calculating torque requires understanding its relationship with power and angular speed.

For calculating torque (T), you can use the formula:
\[ P = T \cdot \omega \]
where P is power and \(\omega\) is the angular speed in radians per second. From this, rearranging gives:
  • \[ T = \frac{P}{\omega} \]

This calculation is vital when determining how much twisting force is applied by the motor's shaft, influencing the design and choice of materials when building systems utilizing motors.
Horsepower Conversion
Horsepower is a unit of power often used in engines and motors, equivalent to 550 foot-pounds per second. Converting horsepower to other units is crucial for applications involving different measurements. The conversion process allows for the accurate calculation of other properties like torque.

To convert horsepower to lb-ft/s (pound-feet per second), use:
  • 1 horsepower = 550 lb-ft/s

This conversion factor is particularly useful when applying formulas involving power to derive other mechanical properties, like torque or speed. Understanding and correctly applying these conversions is essential in ensuring accurate performance predictions in mechanical systems.
Angular Speed
Angular speed, expressed in radians per second, measures how fast an object rotates. Unlike linear speed which deals with straight-line motion, angular speed involves circular motion and is pivotal in motor applications.

To convert from revolutions per minute (RPM) to radians per second (rad/s), the formula is:
  • \( RPM \times \frac{2\pi}{60} = \text{rad/s} \)

Understanding angular speed conversion is crucial because many mechanical formulas, such as those for calculating torque, require angular speed to be in radians per second. This conversion allows engineers and scientists to work with consistent and widely accepted units in technical calculations.

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Most popular questions from this chapter

A circular shaft having a diameter of 4 in. is subjected to a torque of 250 kip \(\cdot\) in. If the material is elastic perfectly plastic, with \(\tau_{Y}=16\) ksi, determine the radius of the elastic core.

the drive shaft \(A B\) of an automobile is to be designed as a thin-walled tube. The engine delivers \(150 \mathrm{hp}\) when the shaft is turning at 1000 rev / min. Determine the minimum thickness of the shaft's wall if the shaft's outer diameter is 2.5 in. The material has an allowable shear stress of \(\tau_{\text {allow }}=7 \mathrm{ksi}\)

The A-36 solid steel shaft is \(2 \mathrm{m}\) long and has a diameter of \(60 \mathrm{mm} .\) It is required to transmit \(60 \mathrm{kW}\) of power from the motor \(M\) to the pump \(P .\) Determine the smallest angular velocity the shaft if the allowable shear stress is \(\tau_{\text {allow }}=80 \mathrm{MPa}\)

The solid steel shaft \(D F\) has a diameter of \(25 \mathrm{mm}\) and is supported by smooth bearings at \(D\) and \(E .\) It is coupled to a motor at \(F,\) which delivers \(12 \mathrm{kW}\) of power to the shaft while it is turning at 50 rev / s. If gears \(A, B,\) and \(C\) remove \(3 \mathrm{kW}, 4 \mathrm{kW},\) and \(5 \mathrm{kW}\) respectively, determine the maximum shear stress in the shaft within regions \(C F\) and \(B C\). The shaft is free to turn in its support bearings \(D\) and \(E\)

A torque is applied to the shaft having a radius of \(80 \mathrm{mm} .\) If the material obeys a shear stress-strain relation of \(\tau=500 \gamma^{1 / 4}\) MPa, determine the torque that must be applied to the shaft so that the maximum shear strain becomes 0.008 rad.
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