1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Svetradugi [14.3K]
2 years ago
13

Find the mass and the center of mass of a wire loop in the shape of a helix (measured in cm: x = t, y = 4 cos(t), z = 4 sin(t) f

or 0 ≤ t ≤ 2π), if the density (in grams/cm) of the wire at any point is equal to the square of the distance from the origin to the point
Mathematics
1 answer:
Sholpan [36]2 years ago
3 0

Answer:

<u>Mass</u>

\sqrt{17}(\displaystyle\frac{8\pi^3}{3}+32\pi)

<u>Center of mass</u>

<em>Coordinate x</em>

\displaystyle\frac{(\displaystyle\frac{(2\pi)^4}{4}+32\pi)}{(\displaystyle\frac{8\pi^3}{3}+32\pi)}

<em>Coordinate y</em>

\displaystyle\frac{16\pi}{(\displaystyle\frac{8\pi^3}{3}+32\pi)}

<em>Coordinate z</em>

\displaystyle\frac{-16\pi}{(\displaystyle\frac{8\pi^3}{3}+32\pi)}

Step-by-step explanation:

Let W be the wire. We can consider W=(x(t),y(t),z(t)) as a path given by the parametric functions

x(t) = t

y(t) = 4 cos(t)

z(t) = 4 sin(t)  

for 0 ≤ t ≤ 2π

If D(x,y,z) is the density of W at a given point (x,y,z), the mass  m would be the curve integral along the path W

m=\displaystyle\int_{W}D(x,y,z)=\displaystyle\int_{0}^{2\pi}D(x(t),y(t),z(t))||W'(t)||dt

The density D(x,y,z) is given by

D(x,y,z)=x^2+y^2+z^2=t^2+16cos^2(t)+16sin^2(t)=t^2+16

on the other hand

||W'(t)||=\sqrt{1^2+(-4sin(t))^2+(4cos(t))^2}=\sqrt{1+16}=\sqrt{17}

and we have

m=\displaystyle\int_{W}D(x,y,z)=\displaystyle\int_{0}^{2\pi}D(x(t),y(t),z(t))||W'(t)||dt=\\\\\sqrt{17}\displaystyle\int_{0}^{2\pi}(t^2+16)dt=\sqrt{17}(\displaystyle\frac{8\pi^3}{3}+32\pi)

The center of mass is the point (\bar x,\bar y,\bar z)

where

\bar x=\displaystyle\frac{1}{m}\displaystyle\int_{W}xD(x,y,z)\\\\\bar y=\displaystyle\frac{1}{m}\displaystyle\int_{W}yD(x,y,z)\\\\\bar z=\displaystyle\frac{1}{m}\displaystyle\int_{W}zD(x,y,z)

We have

\displaystyle\int_{W}xD(x,y,z)=\sqrt{17}\displaystyle\int_{0}^{2\pi}t(t^2+16)dt=\\\\=\sqrt{17}(\displaystyle\frac{(2\pi)^4}{4}+32\pi)

so

\bar x=\displaystyle\frac{\sqrt{17}(\displaystyle\frac{(2\pi)^4}{4}+32\pi)}{\sqrt{17}(\displaystyle\frac{8\pi^3}{3}+32\pi)}=\displaystyle\frac{(\displaystyle\frac{(2\pi)^4}{4}+32\pi)}{(\displaystyle\frac{8\pi^3}{3}+32\pi)}

\displaystyle\int_{W}yD(x,y,z)=\sqrt{17}\displaystyle\int_{0}^{2\pi}4cos(t)(t^2+16)dt=\\\\=16\sqrt{17}\pi

\bar y=\displaystyle\frac{16\sqrt{17}\pi}{\sqrt{17}(\displaystyle\frac{8\pi^3}{3}+32\pi)}=\displaystyle\frac{16\pi}{(\displaystyle\frac{8\pi^3}{3}+32\pi)}

\displaystyle\int_{W}zD(x,y,z)=4\sqrt{17}\displaystyle\int_{0}^{2\pi}sin(t)(t^2+16)dt=\\\\=-16\sqrt{17}\pi

\bar z=\displaystyle\frac{-16\sqrt{17}\pi}{\sqrt{17}(\displaystyle\frac{8\pi^3}{3}+32\pi)}=\displaystyle\frac{-16\pi}{(\displaystyle\frac{8\pi^3}{3}+32\pi)}

You might be interested in
I need the answer for the screenshot
Lesechka [4]
Second option
5/8>1/2
3 0
1 year ago
Read 2 more answers
The width of a large, rectangular-shaped tree farm is 1.9 kilometers. The length of the farm is 4.4 kilometers. What are the bes
Vinvika [58]

Answer:

9 square kilometers

Step-by-step explanation:

Let's round the width to 2.0 kilometers, and round the length to 4.5 kilometers.

We know that area is length times width, so

A = lw

A = 2.0*4.5

<u>A = 9.0 square kilometers</u>

If we do this on the calculator, it's around 9.36 square kilometers, so our estimate was good.

6 0
3 years ago
In a typical stoichiometric problem, the given quantity is first converted to______. Then the ____ ratio from the balanced equat
ryzh [129]

Answer:

The correct answer is

moles

mole

moles

Step-by-step explanation:

In a typical stoichiometric problem, the given quantity is first converted to moles. Then the mole ratio from the balanced equation is used to calculate the moles of the wanted substance. Finally, the moles are converted to any other unit of measurement related to the unit mole.

4 0
3 years ago
Are squares always rectangles?
zaharov [31]


Yes, squares are generally known as rectangles.

Hope it helps

4 0
3 years ago
Idk i just chose something and b was the correct answer
Sonja [21]

Answer:

what happened

Step-by-step explanation:

8 0
3 years ago
Other questions:
  • What is the value of x^2+4 for x=5?<br>A)14<br>B)21<br>C)29<br>D)41<br>​
    6·2 answers
  • Problem solving math
    9·1 answer
  • A video game maker claims that number of glitches in code for their
    11·1 answer
  • Find the remainder when f(x)=2x3-5x2+8x+4 is divided by 2x-1 and use it to determine if 2x-1 is a factor
    12·1 answer
  • Suppose you want to get a loan for $ 5 , 000 at an interest rate of 5.3 % for 2 years to be paid back in 24 monthly installments
    7·1 answer
  • The price of an item has dropped to$63 today.Yesterday it was $140.find the percentage decrease
    8·1 answer
  • The picture is my question ( 2 math/algebra questions)
    9·2 answers
  • The sum of two numbers is 58. The difference between the same two numbers is 6. Find the two numbers, and then find the PRODUCT
    13·1 answer
  • You take a math quiz that has 25 regular points and 5 bonus points. You get a score of 20, which includes 2 bonus points. Which
    12·1 answer
  • What is the slope of the line
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!