/*! 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 42 An object with weight \(W\) is d... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 42

An object with weight \(W\) is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle \(\theta\) with a plane, then the magnitude of the force is $$F = \frac { \mu W } { \mu \sin \theta + \cos \theta }$$ where \(\mu\) is a constant called the coefficient of friction. For what value of \(\theta\) is \(F\) smallest?

Short Answer

Expert verified
\(\theta = \tan^{-1}(-\mu)\) minimizes the force \(F\).

Step by step solution

01

Understanding the Problem

We need to find the angle \( \theta \) at which the force \( F \) needed to drag an object on a horizontal plane is minimized. \( F \) is given by the expression \( F = \frac{ \mu W }{ \mu \sin \theta + \cos \theta } \). Our task is to determine the value of \( \theta \) that results in the smallest possible value for \( F \).
02

Expressing the Objective Function

The objective function to minimize is \( F(\theta) = \frac{ \mu W }{ \mu \sin \theta + \cos \theta } \). We are looking for the angle \( \theta \) that makes this expression as small as possible.
03

Finding the Derivative

To find the minimum value of \( F \), we first differentiate \( F \) with respect to \( \theta \). Using the quotient rule, the derivative is:\[F'(\theta) = \frac{d}{d\theta} \left( \frac{\mu W}{\mu \sin \theta + \cos \theta} \right) = \frac{-(\mu^2 W \cos \theta + \mu W \sin \theta)}{(\mu \sin \theta + \cos \theta)^2}\].
04

Setting the Derivative to Zero

To find critical points, we set \( F'(\theta) = 0 \):\[-(\mu^2 W \cos \theta + \mu W \sin \theta) = 0\].This simplifies to \( \mu^2 \cos \theta + \mu \sin \theta = 0 \).
05

Solving the Trigonometric Equation

Simplify \( \mu^2 \cos \theta + \mu \sin \theta = 0 \) to: \( \mu \sin \theta = -\mu^2 \cos \theta \). This simplifies to \( \tan \theta = -\mu \) (since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)).
06

Conclusion

The angle \( \theta \) which minimizes the force \( F \) is given by \( \theta = \tan^{-1}(-\mu) \). This value of \( \theta \) results in the smallest force needed to drag the object along the horizontal plane.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Coefficient of Friction
The concept of the coefficient of friction is crucial when studying the force needed to move objects. It is a dimensionless constant denoted by \( \mu \). This constant represents the frictional properties between two surfaces. In simpler terms, it tells us how much force is required to slide one object over another.
  • If \( \mu \) is high, it means more force is required to move the object because the surfaces are rougher.
  • A lower \( \mu \) value indicates smoother surfaces, which need less force to move an object.
When pulling an object with a rope, the coefficient of friction helps calculate the minimum force needed. It's a critical variable in the equation provided: \[F = \frac{ \mu W }{ \mu \sin \theta + \cos \theta }\]Understanding \( \mu \) can help predict the effort required to maintain movement, especially when angles come into play by affecting the overall force dynamics.
Derivative Application
Derivatives are fundamental to finding optimal solutions like minimizing force in our exercise. The derivative of a function gives us the rate at which the function is changing. In practical terms, it's like checking how steep or flat a curve is at any given point.
  • For our exercise, we are interested in how the force \( F \) changes as the angle \( \theta \) changes.
  • By calculating the derivative of \( F \) with respect to \( \theta \), we can identify tendencies towards increase or decrease in the force.
We use the quotient rule to find this derivative, which handles the division of two functions. The derivative of our problem appears as:\[F'(\theta) = \frac{-(\mu^2 W \cos \theta + \mu W \sin \theta)}{(\mu \sin \theta + \cos \theta)^2}\]This function allows us to explore where the force might be minimized by analyzing its critical points.
Critical Points
Identifying critical points in a function like \( F(\theta) \) is essential for optimization problems such as finding the minimum force. Critical points occur where the derivative of a function equals zero or is undefined.To find these points:
  • First, set the derivative \( F'(\theta) \) to zero: \[-(\mu^2 W \cos \theta + \mu W \sin \theta) = 0\]
  • This equation simplifies to \( \mu^2 \cos \theta + \mu \sin \theta = 0 \), further leading to \( \tan \theta = -\mu \).
The angle \( \theta = \tan^{-1}(-\mu) \) marks the critical point where the condition of force minimization is met. When we identify such a point, it helps us ensure that we are applying the least necessary amount of force to drag the object horizontally.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The first appearance in print of I'Hospital's Rule was in the book Analyse des Infiniment Petits published by the Marquis de I'Hospital in \(1696 .\) This was the first calculus textbook ever published and the example that the Marquis used in that book to illustrate his rule was to find the limit of the function $$y=\frac{\sqrt{2 a^{3} x-x ^ {4}}-a \sqrt[3]{\operatorname{aax}}}{a-\sqrt[4]{a x^{3}}}$$ as \(x\) approaches a, where a \( > 0 .\) (At that time it was common to write aa instead of a'.) Solve this problem.

(a) If \(C ( x )\) is the cost of producing \(x\) units of a commodity, then the average cost per unit is \(c ( x ) = C ( x ) / x\) . Show that if the average cost is a minimum, then the marginal cost equals the average cost. (b) If \(C ( x ) = 16,000 + 200 x + 4 x ^ { 3 / 2 } ,\) in dollars, find \(( \mathrm { i } )\) the cost, average cost, and marginal cost at a production level of 1000 units; (ii) the production level that will minimize the average cost; and (iii) the minimum aver- age cost.

What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve \(y = 3 / x\) at some point?

Produce graphs of \(f\) that reveal all the important aspects of the curve. In particular, you should use graphs of \(f^{\prime}\) and \(f "\) to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. \(f(x)=x^{6}-15 x^{5}+75 x^{4}-125 x^{3}-x\)

Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than over land because air generally rises over land and falls over water during the day. A bird with these tendencies is released from an island that is 5\(\mathrm { km }\) from the nearest point \(B\) on a straight shoreline, flies to a point \(C\) on the shoreline, and then flies along the shoreline to its nesting area \(D .\) Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points \(B\) and \(D\) are 13\(\mathrm { km }\) apart. (a) In general, if it takes 1.4 times as much energy to fly over water as it does over land, to what point \(C\) should the bird fly in order to minimize the total energy expended in returning to its nesting area? (b) Let \(W\) and \(L\) denote the energy (in joules) per kilometer flown over water and land, respectively. What would a large value of the ratio \(W / L\) mean in terms of the bird's flight? What would a small value mean? Determine the ratio \(W / L\) corresponding to the minimum expenditure of energy. (c) What should the value of \(W / L\) be in order for the bird to fly directly to its nesting area \(D ?\) What should the value of \(W / L\) be for the bird to fly to \(B\) and then along the shore to \(D ?\) (d) If the omithologists observe that birds of a certain species reach the shore at a point 4\(\mathrm { km }\) from \(B\) , how many times more energy does it take a bird to fly over water than over land?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.