/*! 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 15 An object moves along the line s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 15

An object moves along the line segment from (1,1) to \((2,5),\) subject to the force \(\boldsymbol{F}=\) \(\left\langle x /\left(x^{2}+y^{2}\right), y /\left(x^{2}+y^{2}\right)\right\rangle .\) Find the work done.

Short Answer

Expert verified
The work done by the force field is \(W = 0.\)

Step by step solution

01

Understanding Work Done

The work done by a force on an object moving along a path is calculated as the line integral of the force vector along the path. For a vector field \(\boldsymbol{F}(x, y) = \langle P(x, y), Q(x, y) \rangle\), the work done is given by \(W = \int_{C} \boldsymbol{F} \cdot d\boldsymbol{r}\), where \(C\) is the path from the starting point to the endpoint.
02

Parameterize the Path

Parameterize the line segment from \((1, 1)\) to \((2, 5)\) using a parameter \(t\). A suitable parameterization is \(\boldsymbol{r}(t) = (1 + t, 1 + 4t)\) where \(t\) ranges from 0 to 1.
03

Compute the Derivative of \(\boldsymbol{r}(t)\)

The derivative of \(\boldsymbol{r}(t)\) with respect to \(t\) is \(\boldsymbol{r}'(t) = \frac{d}{dt}[(1 + t, 1 + 4t)] = \langle 1, 4 \rangle\). This represents the differential element \(d\boldsymbol{r}\).
04

Express the Force Field in Terms of the Parameter \(t\)

Substitute the parameterized path \((x(t), y(t)) = (1 + t, 1 + 4t)\) into the force field. Thus, \(\boldsymbol{F}(1+t, 1+4t) = \left\langle \frac{1+t}{(1+t)^2 + (1+4t)^2}, \frac{1+4t}{(1+t)^2 + (1+4t)^2} \right\rangle\).
05

Compute the Dot Product \(\boldsymbol{F}(t) \cdot \boldsymbol{r}'(t)\)

The dot product of \(\boldsymbol{F}(1+t, 1+4t)\) and \(\boldsymbol{r}'(t) = \langle 1, 4 \rangle\) is \(\frac{1+t}{(1+t)^2 + (1+4t)^2} \cdot 1 + \frac{1+4t}{(1+t)^2 + (1+4t)^2} \cdot 4\). Thus, \(\boldsymbol{F}(t) \cdot \boldsymbol{r}'(t) = \frac{1+t + 4(1+4t)}{(1+t)^2 + (1+4t)^2}\).
06

Evaluate the Integral

Set up the integral to find the work done: \(W = \int_{0}^{1} \frac{1 + t + 4(1 + 4t)}{(1+t)^2 + (1+4t)^2} \, dt\). Simplify and solve this integral to find the work done by the force field on the object.

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.

Work Done
In physics, work done refers to the amount of energy transferred by a force moving an object along a path. The concept of work done can be explored through line integrals in mathematics. When a force is represented by a vector field, and an object moves along a path, the work done by this force is the line integral of the force along the path.

  • For example, if a force \( \boldsymbol{F}(x, y) = \langle P(x, y), Q(x, y) \rangle \) acts on an object as it moves from a point \( A \) to a point \( B \), the work \( W \) done is given by the line integral \( W = \int_{C} \boldsymbol{F} \cdot d\boldsymbol{r} \), where \( C \) is the path of movement.
  • This technique captures not only the magnitude of the force applied but also considers how the force is applied along the trajectory.
Understanding this principle helps you apply the concept of work done to various physical situations beyond basic math problems involving force and movement.
Vector Fields
A vector field assigns a vector to every point in space. In this context, a force like \( \boldsymbol{F}(x, y) = \left\langle \frac{x}{x^2 + y^2}, \frac{y}{x^2 + y^2} \right\rangle \) depicts how the forces vary at each point on a plane.

  • The first component \( \frac{x}{x^2 + y^2} \) suggests how much force acts in the x-direction, while the second component \( \frac{y}{x^2 + y^2} \) indicates the force in the y-direction.
  • In vector fields, each vector can represent multiple physical quantities such as velocity, gravitational fields, or in our case, a force field.
Visualizing vector fields helps understand the dynamics of forces interacting within a defined space, especially when imagining how the force behaves around various points.
Parameterization
Parameterization is a method used to represent a curve or line using one or more parameters. For curves in a plane, this usually involves representing x, y coordinates with a parameter \( t \).

  • For instance, the line from \((1, 1)\) to \((2, 5)\) is parameterized as \( \boldsymbol{r}(t) = (1 + t, 1 + 4t) \), where \( t \) changes from 0 to 1.
  • It provides a convenient way to handle the path of an object, transforming a range into linear equations that describe a continuous movement along the curve.
  • The derivative of these functions gives the direction and the rate of movement, expressed as \( \boldsymbol{r}'(t) \), which is crucial for calculating things like the work done when an object moves along this path.
Parameterization simplifies complex trajectories into manageable linear relationships, aiding calculations in integral evaluations over curves.

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

Find an \(f\) so that \(\nabla f=\left\langle x^{2} y^{3}, x y^{4}\right\rangle,\) or explain why there is no such \(f\).

Evaluate \(\iint_{D}\langle x, y, 3\rangle \cdot N d S,\) where \(D\) is given by \(z=3 x-5 y, 1 \leq x \leq 2,0 \leq y \leq 2\), oriented up.

Let $$ \boldsymbol{F}=\left\langle\frac{-x}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}, \frac{-y}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}, \frac{-z}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}}\right\rangle $$ Find the work done by this force field on an object that moves from (1,1,1) to (4,5,6) .

Describe or sketch the surface with the given vector function. (a) \(\boldsymbol{r}(u, v)=\langle 6 v-u, 3+v, 1-4 v\rangle\) (b) \(\boldsymbol{r}(u, v)=\langle 2 \cos u, 5 \cos u, v\rangle\) (c) \(\boldsymbol{r}(s, t)=\left\langle u+v, 2 u+v, u^{2}+v^{2}\right\rangle\) (d) \(\boldsymbol{r}(s, t)=\langle\sin u+\cos u, u, v\rangle\)

Investigate the vector field \(\langle-x,-y\rangle .\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Geometry

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.