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
Fudgin [204]
2 years ago
11

A homogeneous rectangular lamina has constant area density ρ. Find the moment of inertia of the lamina about one corner

Mathematics
1 answer:
frozen [14]2 years ago
7 0

Answer:

I_{corner} =\frac{\rho _{ab}}{3}(a^2+b^2)

Step-by-step explanation:

By applying the concept of calculus;

the moment of inertia of the lamina about one corner I_{corner} is:

I_{corner} = \int\limits \int\limits_R (x^2+y^2)  \rho d A \\ \\ I_{corner} = \int\limits^a_0\int\limits^b_0 \rho(x^2+y^2) dy dx

where :

(a and b are the length and the breath of the rectangle respectively )

I_{corner} =  \rho \int\limits^a_0 {x^2y}+ \frac{y^3}{3} |^ {^ b}_{_0} \, dx

I_{corner} =  \rho \int\limits^a_0 (bx^2 + \frac{b^3}{3})dx

I_{corner} =  \rho [\frac{bx^3}{3}+ \frac{b^3x}{3}]^ {^ a} _{_0}

I_{corner} =  \rho [\frac{a^3b}{3}+ \frac{ab^3}{3}]

I_{corner} =\frac{\rho _{ab}}{3}(a^2+b^2)

Thus; the moment of inertia of the lamina about one corner is I_{corner} =\frac{\rho _{ab}}{3}(a^2+b^2)

You might be interested in
6. Solve using any method.<br> x+6y=7<br> 5x + 8y = 13
Norma-Jean [14]

Answer:

<h2>x = 1, y = 1 → (1, 1)</h2>

Step-by-step explanation:

\bold{ELIMINATION\ METHOD}\\\\\left\{\begin{array}{ccc}x+6y=7&\text{Multiply both sides by (-5)}\\5x+8y=13\end{array}\right\\\\\underline{+\left\{\begin{array}{ccc}-5x-30y=-35\\5x+8y=13\end{array}\right}\qquad\text{add both sides of the equations}\\.\qquad-22y=-22\qquad\text{divide both sides by (-22)}\\.\qquad\boxed{y=1}\\\\\text{Put it to the first equation}\\\\x+6(1)=7\\x+6=7\qquad\text{subtract 6 from both sides}\\\boxed{x=1}

\bold{SUBSTITUTION\ METHOD}\\\\\left\{\begin{array}{ccc}x+6y=7&\text{subtract}\ 6y\ \text{from both sides}\\5x+8y=13\end{array}\right\\\\\left\{\begin{array}{ccc}x=7-6y&(1)\\5x+8y=13&(2)\end{array}\right\\\\\text{Substitute (1) to (2):}\\\\5(7-6y)+8y=13\qquad\text{use the distributive property}\\(5)(7)+(5)(-6y)+8y=13\\35-30y+8y=13\qquad\text{subtract 35 from both sides}\\-30y+8y=13-35\qquad\text{combine like terms}\\-22y=-22\qquad\text{divide both sides by (-22)}\\\boxed{y=1}

\text{Put it to (1):}\\\\x=7-6(1)\\x=7-6\\\boxed{x=1}

4 0
3 years ago
Read 2 more answers
The graph shows a vertical translation of y = x.
iris [78.8K]

Answer:

D is the answer

Step-by-step explanation:

3 0
3 years ago
A Blimp is 400 feet in the air. There is an angle of depression of 40 degrees to a football stadium. How far is the Blimp from t
VMariaS [17]

Answer:

Step-by-step explanation:

The diagram of the stadium and the blimp is shown in the attached photo. A right angle triangle, ABC is formed. Angle A is alternate to the angle of depression. This mean that they are equal. So angle A =40 degrees.

The distance of the the Blimp from the stadium is x feet which is the hypotenuse of the triangle.

Applying trigonometric ratio,

Sin 40 = 400/x

xsin40 = 400

x = 400/sin40 = 400 / 0.6428

x = 622.3 feet

5 0
3 years ago
Find the center-radius form of the equation of the circle described.
Kitty [74]

Answer:

Correct answer:  (x-√2)² + (y-√5)² = 3

Step-by-step explanation:

Given data: Center (x,y) = (√2,√5) and r = √3

The canonical or cartesian form of the equation of the circle is:

( x-p )² + ( y-q )² = r²  

Where p is the x coordinate of the center, q is the y coordinate of the center and r is the radius of the circle.

God is with you!!!

8 0
3 years ago
Help 6th grade math please help i will give brainliest
lilavasa [31]

Answer:

multiply 12 times 9

Step-by-step explanation:

I'm pretty sure this is right, I'm sorry if it's wrong

3 0
2 years ago
Other questions:
  • PLEASE HELP BIG BIG POINTS 20 POINTS!!!!
    12·2 answers
  • 9+5divided5+2 value 2
    15·2 answers
  • What is the greatest common factor 21,35,49
    5·2 answers
  • ............... help :)????????????????
    9·1 answer
  • How to solve if y=14 when x=8, find y when x=12
    11·1 answer
  • The price of a video game system dropped by 18%. What is the current price if it used to be $140?
    15·1 answer
  • Plz help meee <br> (exponents)
    6·1 answer
  • Pls help 10 points pls!
    7·1 answer
  • The figure shows Triangle ABC and some of its transformed images on a coordinate grid:
    9·2 answers
  • The length of a driveway is 18 yards what is it in feet?​
    13·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!