/*! 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 39 Find the work done by a person w... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 39

Find the work done by a person weighing 150 pounds walking exactly one revolution up a circular helical staircase of radius 3 feet if the person rises 10 feet.

Short Answer

Expert verified
The work done by the person is 4226.172 Joules.

Step by step solution

01

Convert the weight to Newtons

As the standard unit of force in the International System of Units (SI) is Newtons, the weight needs to be converted from pounds to Newtons. The conversion factor can be found in a physics textbook or online. Given, 1 pound is equal to 4.44822 Newtons. Hence the weight in Newtons = 150 pounds * 4.44822 N/lb = 667.233 N.
02

Calculate the distance travelled (displacement)

The person travels in a helical path. The linear distance travelled is the hypotenuse of the formed right triangle, with the height of the staircase and the perimeter of the staircase's base as the other sides. The length of the base is calculated by multiplying the radius with \(2\pi\), as it's a circular path. This gives \(2\pi*3 feet = 18.849 feet\). The total distance (hypotenuse) is calculated using Pythagoras鈥� Theorem, \(a^2 + b^2 = c^2\), which gives \(\sqrt{(18.849 feet)^2 + (10 feet)^2} = 20.794 feet\). Convert 20.794 feet to meters (1 meter = 3.28084 feet) to get 6.336 meters.
03

Calculate work done

The formula to calculate work done is force * displacement. Therefore, work done = 667.233 N * 6.336 m = 4226.172 J.

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Most popular questions from this chapter

In Exercises 19 and \(20,\) find the total mass of two turns of a spring with density \(\rho\) in the shape of the circular helix \(\mathbf{r}(t)=3 \cos t \mathbf{i}+3 \sin t \mathbf{j}+2 t \mathbf{k}\) \(\rho(x, y, z)=\frac{1}{2}\left(x^{2}+y^{2}+z^{2}\right)\)

Find the divergence of the vector field \(\mathrm{F}\). \(\mathbf{F}(x, y, z)=x e^{x} \mathbf{i}+y e^{y} \mathbf{j}\)

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Work Consider a particle that moves through the force field \(\mathbf{F}(x, y)=(y-x) \mathbf{i}+x y \mathbf{j}\) from the point (0,0) to the point (0,1) along the curve \(x=k t(1-t), y=t .\) Find the value of \(k\) such that the work done by the force field is 1

In Exercises 17 and 18 , evaluate \(\int_{C}\left(2 x+y^{2}-z\right) d s\) along the given path. \(C:\) line segments from (0,0,0) to (1,0,0) to (1,0,1) to (1,1,1)

A particle moves along the path \(y=x^{2}\) from the point (0,0) to the point (1,1) . The force field \(\mathbf{F}\) is measured at five points along the path and the results are shown in the table. Use Simpson's Rule or a graphing utility to approximate the work done by the force field. $$ \begin{array}{|l|c|c|c|c|c|} \hline(x, y) & (0,0) & \left(\frac{1}{4}, \frac{1}{16}\right) & \left(\frac{1}{2}, \frac{1}{4}\right) & \left(\frac{3}{4}, \frac{9}{16}\right) & (1,1) \\ \hline \mathbf{F}(x, y) & \langle 5,0\rangle & \langle 3.5,1\rangle & \langle 2,2\rangle & \langle 1.5,3\rangle & \langle 1,5\rangle \\ \hline \end{array} $$
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