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
fiasKO [112]
3 years ago
6

Under the assumption that the beam is a rectangular cantilever beam that is free to vibrate, the theoretical first natural frequ

ency of the beam in terms of the length, L, width, b, thickness, h, density, p, and Young's Modulus, E, in Hertz is given by:
fn= h/2πL^2 √E/rho

The density of steel used in the beam is 7800 kg/m^3, and its Young's Modulus is 210 GPa. You measure the dimensions of the clamped steel beam with a set of calipers and gather the following data:

L = 233.5 mm, b = 24.9 mm, h = 3.3 mm

The calipers used to measure the dimensions of beam have a resolution of 0.01 mm and therefore have an uncertainty of +0.005 mm.

Required:
a. What is the uncertainty of the natural frequency (in Hz) due to the uncertainty of the length measurement?
b. What is the uncertainty of the natural frequency (in Hz) due to the uncertainty of the width measurement?
c. What is the uncertainty of the natural frequency (in Hz) due to the uncertainty of the thickness measurement?
d. What is the total uncertainty of the natural frequency due to the beam measurements (in Hz)?
e. What is the theoretical first natural frequency of the beam including the measurement uncertainty?
Physics
1 answer:
BartSMP [9]3 years ago
7 0

Answer:

a) Δf = 0.7 n , e)   f = (15.1 ± 0.7) 10³ Hz

Explanation:

This is an error about the uncertainty or error in the calculated quantities.

Let's work all the magnitudes is the SI system

The frequency of oscillation is

        f = n / 2π L² √( E /ρ)

where n is an integer

Let's calculate the magnitude of the oscillation

       f = n / 2π (0.2335)² √ (210 10⁹/7800)

       f = n /0.34257 √ (26.923 10⁶)

       f = n /0.34257    5.1887 10³

       f = 15.1464 10³ n

a) We are asked for the uncertainty of the frequency (Df)

       Δf = | df / dL | ΔL + df /dE ΔE + df /dρ Δρ

in this case no  error is indicated in Young's modulus and density, so we will consider them exact

       ΔE = Δρ = 0

       Δf = df /dL  ΔL

       df = n / 2π   √E /ρ   | -2 / L³ | ΔL

       df = n / 2π 5.1887 10³ | 2 / 0.2335³) 0.005 10⁻³

       df = n 0.649

Absolute deviations must be given with a single significant figure

        Δf = 0.7 n

b, c) The uncertainty with the width and thickness of the canteliver is associated with the density

 

In your expression there is no specific dependency so the uncertainty should be zero

The exact equation for the natural nodes is

          f = n / 2π L² √ (E e /ρA)

where A is the area of ​​the cantilever and its thickness,

In this case, they must perform the derivatives, calculate and approximate a significant figure

        Δf = | df / dL | ΔL + df /de  Δe + df /dA  ΔA

        Δf = 0.7 n + n 2π L² √(E/ρ A) | ½  1/√e | Δe

               + n / 2π L² √(Ee /ρ) | 3/2 1√A23  |

the area is

        A = b h

        A = 24.9  3.3  10⁻⁶

        A = 82.17 10⁻⁶ m²

        DA = dA /db ΔB + dA /dh Δh

        dA = h Δb + b Δh

        dA = 3.3 10⁻³ 0.005 10⁻³ + 24.9 10⁻³ 0.005 10⁻³

        dA = (3.3 + 24.9) 0.005 10⁻⁶

        dA = 1.4 10⁻⁷ m²

let's calculate each term

         A ’= n / 2π L² √a (E/ρ A) | ½ 1 /√ e | Δe

         A ’= n/ 2π L² √ (E /ρ)      | ½ 1 / (√e/√ A) |Δe

        A ’= 15.1464 10³ n ½ 1 / [√ (24.9 10⁻³)/ √ (81.17 10⁻⁶)] 0.005 10⁻³

        A '= 0.0266  n

        A ’= 2.66 10⁻² n

       A ’’ = n / 2π L² √ (E e /ρ) | 3/2  1 /√A³ |

       A ’’ = n / 2π L² √(E /ρ) √ e | 3/2  1 /√ A³ | ΔA

       A ’’ = n 15.1464 10³ 3/2 √ (24.9 10⁻³) /√ (82.17 10⁻⁶) 3 1.4 10⁻⁷

       A ’’ = n 15.1464 1.5 1.5779 / 744.85 1.4 10⁴

       A ’’ = 6,738 10²

we write the equation of uncertainty

     Δf = n (0.649 + 2.66 10⁻² + 6.738 10²)

The uncertainty due to thickness is

    Δf = 3 10⁻² n

The uncertainty regarding the area, note that this magnitude should be measured with much greater precision, specifically the height since the errors of the width are very small

     Δf = 7 10² n

 d)    Δf = 7 10² n

e) the natural frequency n = 1

       f = (15.1 ± 0.7) 10³ Hz

You might be interested in
The light waves have that particular interference from question #1 because they are emitted _____.
Elis [28]
I am attaching the rest of your question so it makes sense, 
<span>
Since lasers are made from stacking light waves that add together into a larger wave due to CONSTRUCTIVE INTERFERENCE.
</span>
Then, <span>light waves have that constructive interference (from question #1) because they are emitted IN PHASE with each other.

This means that they arrive at the same point of space with the same characteristics and their effects do not cancel each other, but the opposite, their intensity increases.</span>

7 0
3 years ago
Read 2 more answers
What is average speed?what is the formula for average speed?what does it mean?
Rama09 [41]
Average speed = total distance traveled/total time taken
it is the total distance traveled in a total time the total distance is attained
4 0
3 years ago
Assuming that each of the following objects is a typical example of its class, rank them by increasing density.
inysia [295]

molecular cloud <interstellar cloud <1 Msun protostar <1 Msun star <intercloud gas

Explanation:

<u>Molecular cloud-</u> They are a variety of interstellar cloud in which molecular hydrogen can sustain themselves. They have a very low temperature ranging from -440 to -370 degrees Fahrenheit or between<u> 10 to 50 Kelvin. </u>Owing to their extremely low temperature, they appear mostly dark when viewed through telescopes.

<u>Interstellar cloud-</u> They are a congregation of a large number of interstellar gases, dust and plasma in any galaxy or universe. They have varying temperature depending on their proximity to a star. E.g. Neutral hydrogen atom clouds have a temperature of around <u>just 100 Kelvin</u> while those in the near vicinity of a star have temperatures as high as 10,000 Kelvin.

<u>1 Msun star-</u> These stars have temperature anywhere between <u>5300 and 6000 Kelvin</u>. The main source of such high surface temperature is nuclear fusion process where elemental hydrogen molecules are fused to form helium molecules.  

<u>1 Msun protostar-</u> protostar is rather a young star which is still in formation phase (i.e. gathering mass from the parent molecular cloud). They have temperature anywhere between <u>2000-3000</u> kelvin and are accompanied by dust usually.

<u>Intercloud gas- </u>These are the remainder gases that are spread throughout the interstellar space. This Intercloud gas is divided into warm intercloud medium and extremely hot coronal gas with temperatures comparing to Sun’s corona. Warm intercloud forms the dominant part of intercloud gas with a temperature around <u>8000 Kelvin</u>.

8 0
3 years ago
A child rides her bicycle in her level driveway . what form of energy does she possess
Elden [556K]
B. Kinetic

Kinetic energy depends on motion and mass
7 0
2 years ago
What evidence did astronomers use to prove jets travel in opposite directions?
mr Goodwill [35]

The evidence that astronomers use to prove jets travel in opposite direction is Spectral lines from a very fast moving ionized gases.

<h3>Who are the astronomers?</h3>

Astronomer is a person who study astronomy which is a scientific study that has to do with space, space bodies, comets, planets, stars and so on.

Therefore, The evidence that astronomers use to prove jets travel in opposite direction is Spectral lines from a very fast moving ionized gases.

Learn more about astronomers below.

brainly.com/question/10826203

#SPJ11

7 0
2 years ago
Read 2 more answers
Other questions:
  • Give one advantage and one disadvantage of using nuclear power stations rather than
    15·1 answer
  • Why do you think the area swept out by a planet in a given period of time remains constant, even as the planet speeds up and slo
    15·2 answers
  • Astronomers have discovered a planetary system orbiting a star, which is at a distance of 3.9 × 1014 m from the earth. One plane
    6·1 answer
  • A bowler lifts his bowling ball a distance of 2 meters using 50 joules of energy. If, on Earth, a 1.0 kilograms mass weighs 9.8
    9·2 answers
  • Fission is the process of creating energy by
    15·1 answer
  • Two copper rods are separated by a small gap at B. Rod AB has a diameter of 200mm and rod BC has a diameter of 150mm. Find the f
    12·1 answer
  • A CIRCULAR COIL OF 20 TURNS OF RADIUS 10.0 cm IS PERPENDICULAR TO A 0.30 tesla MAGNETIC FIELD. HOW MUCH INDUCED emf IS CREATED W
    6·1 answer
  • The unit of the electric resistance is ???​
    6·1 answer
  • the gravitational force acts on all objects in proportion to their mass. neglecting air resistance, why don’t heavy objects fall
    5·1 answer
  • sports photographers often use large aperture, long focal length lenses. what limitations do these lenses impose on the photogra
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!