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
PLEASE help me!!! ASAP!! Ill mark you as brainliest!!!!
MakcuM [25]

Answer:

B

Explanation:

graph b shows a steady pace of movement for 20 minutes and then shows a plateau in the distance, showing that while time keeps moving (obviously), the distance doesn't change. then after 5 minutes, the student gets up and starts running again. hope this helped!

8 0
3 years ago
What force causes a rolling ball to slow down and stop
MAXImum [283]
Newtons first law of motion or friction
3 0
3 years ago
Read 2 more answers
What’s is the movement of one object around another
allsm [11]

Answer:

revolution

Explanation:

8 0
3 years ago
A 150 g egg is dropped from 3.0 meters. The egg is
Lynna [10]

<u><em>Answer:</em></u>

<u><em> </em></u>

<u><em>9.2 N, with significant figure rounding (2 s.f.) </em></u>

<u><em></em></u>

<u><em>Explanation:</em></u>

<u><em>This problem can be solved using momentum. The following equation relates momentum (mass & velocity) with force and time:</em></u>

<u><em></em></u>

<u><em>Note that  where v is the final velocity and v₀ is the initial velocity. Δv just means change in velocity.</em></u>

<u><em></em></u>

<u><em>Mass of the egg is 150 g, but we need to convert to kilograms if we want to use Newtons as a unit. 150 g is equal to 0.15 kg. since 1000 g = 1kg. </em></u>

<u><em>m = 0.15 kg</em></u>

<u><em></em></u>

<u><em>The dropped from 3.0 meters is irrelevant as the question tells us the initial velocity of the egg: 4.4 m/s before it hits the ground.</em></u>

<u><em>v₀ = 4.4 m/s [down]</em></u>

<u><em></em></u>

<u><em>When it comes to a stop, the egg will have a velocity of 0.</em></u>

<u><em>v = 0 m/s</em></u>

<u><em></em></u>

<u><em>The time it takes for the egg to stop is 0.072 seconds.</em></u>

<u><em>Δt = 0.072 s</em></u>

<u><em></em></u>

<u><em>Therefore, if down is positive, then</em></u>

<u><em></em></u>

<u><em>   </em></u>

<u><em></em></u>

<u><em>We round to two significant figures since every quantity has two sig. figs.</em></u>

<u><em>We only care about the magnitude, not direction. The answer is 9.2 N.</em></u>

<u><em>Unlimited, ad-free access to all of the questions with Brainly Plus</em></u>

<u><em>START 7 DAY FREE TRIAL</em></u>

<u><em>Click to let others know, how helpful is it</em></u>

<u><em>5.0</em></u>

5 0
3 years ago
Which part of the wave has the highest frequency?
trapecia [35]

Answer:

The last part on the right side of the diagram

Explanation:

Im on plato and just got it right :)

6 0
3 years ago
Read 2 more answers
Other questions:
  • ¿CUAL ES LA CAUSA DE UNA ONDA ESTACIONARIA?
    5·1 answer
  • Which example best illustrates the transfer of energy between two waves?
    5·2 answers
  • A cell containing cyanamide is least likely to carry on the process of what transport
    12·1 answer
  • Use the principle of superposition (Equation 1.3) to predict the electric field at the point 0.5 meters to the right and 1.5 met
    7·1 answer
  • 44. Belly-flop Bernie dives from atop a tall flagpole into a swimming pool below. His potential energy at the top is 10,000 J (r
    10·1 answer
  • Tarzan, whose mass is 94 kg, is hanging at rest from a tree limb. Then he lets go and falls to the ground. Just before he lets g
    15·1 answer
  • PLEASE HELP ME !!!!!
    9·1 answer
  • Can we detect em waves with our eyes?<br><br> tru or false
    14·2 answers
  • Deonte’s family sees a solar panel display and considers using solar power for their home. Deonte knows that solar energy is a n
    5·1 answer
  • Seeds are often found on which part of a gymnosperm?<br><br> branch<br> leaf<br> cone<br> stem
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!