/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 74 The square-threaded bolt is used... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 74

The square-threaded bolt is used to join two plates together. If the bolt has a mean diameter of \(d=20 \mathrm{~mm}\) and a lead of \(l=3 \mathrm{~mm}\), determine the smallest torque \(M\) required to loosen the bolt if the tension in the bolt is \(T=40 \mathrm{kN}\) The coefficient of static friction between the threads and the bolt is \(\mu_{s}=0.15\)

Short Answer

Expert verified
The minimum torque needed to loosen the bolt is calculated in the last step. Make sure to express the answer in an appropriate unit such as N.m.

Step by step solution

01

Calculate Normal Force (N)

The Normal Force (N) on the threads of the bolt due to the tension (T) can be found using the equation \(N = \frac{T}{\pi \times d}\). Substitute \(T = 40 kN = 40000 N\) and \(d = 20 mm = 0.02 m\), and calculate N.
02

Calculate Friction Force (F)

Friction Force (F) is calculated by multiplying the coefficient of static friction (\(\mu_{s}\)) with the Normal force (N). Substituting \(\mu_{s} = 0.15\) and the computed value of N, calcualte F.
03

Calculate Lead Angle (L_a)

The Lead Angle (\(L_a\)), expressed in radians, is calculated using the equation \(L_a = \arctan (\frac{l}{\pi \times d})\), where \(l = 3 mm = 0.003 m\) is the lead and \(d = 0.02 m\). Compute \(L_a\).
04

Calculate minimum required Torque (M)

The Torque (M) can be calculated using the equation \(M = \frac{d \times F}{2 \times \cos (L_a)}\). Substitute the values of d, F and \(L_a\) to determine M.

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Key Concepts

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

Statics
At its core, statics is the branch of mechanics that deals with bodies at rest or in constant motion, where the net force and the net torque on every part of the system is zero. In the context of the problem involving the square-threaded bolt, statics principles are used to determine forces acting on the bolt, specifically the tension and how it translates to the normal force exerted by the bolt's threads.

Understanding the equilibrium of forces is crucial because it allows you to deduce that the tension in the bolt creates an inward radial force that is balanced out by the normal force of the threads. This normal force is a key component in the calculation of the required torque to loosen the bolt because it introduces friction, which must be overcome.
Friction Force

Understanding Friction in Bolts

Friction is the resistance encountered when one body moves relative to another body with which it is in contact. In the bolt exercise, this concept is essential for understanding how the friction force affects the tightening and loosening of bolts. The friction force is a result of the interaction between the threads of the bolt and the plate surfaces, and it can prevent the bolt from moving unless an adequate torque is applied.

The coefficient of static friction (\(mu_s\)) represents how much frictional force must be overcome to start the motion. By calculating the friction force (\(F\)), you account for how much torque is needed to initiate the loosening of the bolt since it directly depends on the normal force which in turn, is dependent on the applied tension.
Lead Angle

Role of Lead Angle in Torque Calculation

The lead angle (\(L_a\)) is a lesser-known but equally important factor in the mechanics of screws and bolts. It is the angle between the helix of the thread and a plane perpendicular to the bolt's axis. In simpler terms, it describes the steepness of the thread's helix. This angle is significant for two reasons: it impacts the efficiency of the bolt at converting rotational motion into linear motion, and it influences the frictional forces at play due to a change in the 'effective' normal force as the lead angle changes.

When calculating torque for a bolt, the lead angle is used to adjust the frictional resistance encountered. As the lead angle increases, the frictional force component aligned with the bolt axis becomes larger, requiring more torque to overcome it. Therefore, calculating the lead angle accurately is vital for determining the minimum torque needed to loosen the bolt.

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Most popular questions from this chapter

A man attempts to support a stack of books horizontally by applying a compressive force of \(F=120 \mathrm{~N}\) to the ends of the stack with his hands. If each book has a mass of \(0.95 \mathrm{~kg}\), determine the greatest number of books that can be supported in the stack. The coefficient of static friction between his hands and a book is \(\left(\mu_{s}\right)_{h}=0.6\) and between any two books \(\left(\mu_{s}\right)_{b}=0.4\).

The tractor has a weight of \(16000 \mathrm{lb}\) and the coefficient of rolling resistance is \(a=2\) in. Determine the force \(\mathbf{P}\) needed to overcome rolling resistance at all four wheels and push it forward.

A worker walks up the sloped roof that is defined by the curve \(y=\left(5 e^{0.01 x}\right) \mathrm{ft}\), where \(x\) is in feet. Determine how high \(h\) he can go without slipping. The coefficient of static friction is \(\mu_{s}=0.6\).

A cylinder having a mass of \(250 \mathrm{~kg}\) is to be supported by the cord that wraps over the pipe. Determine the largest vertical force \(\mathbf{F}\) that can be applied to the cord without moving the cylinder. The cord passes (a) once over the pipe, \(\beta=180^{\circ}\), and (b) two times over the pipe, \(\beta=540^{\circ}\). Take \(\mu_{s}=0.2\).

Determine the angle \(\phi\) at which the applied force \(\mathbf{P}\) should act on the pipe so that the magnitude of \(\mathbf{P}\) is as small as possible for pulling the pipe up the incline. What is the corresponding value of \(P ?\) The pipe weighs \(W\) and the slope \(\alpha\) is known. Express the answer in terms of the angle of kinetic friction, \(\theta=\tan ^{-1} \mu_{k}\)
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