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lesya [120]
3 years ago
12

A flat solid disk of radius R and mass M is around vertical axle going through a center of mass of a disk. Suddenly a hoop of ma

ss M and radius R is falling coaxially on the disk. Hoop did not rotate initially. If rotational energy of the system was initially RE, what is rotational energy of the system after hoop has landed?
Physics
1 answer:
Softa [21]3 years ago
4 0

Answer:

Kf = 2/9 RE

Explanation:

The initial mechanical energy of the system is the sum of the kinetic and potential energy of the two bodies as the height does not change we can take the zero in the position of the disk, the ring is still so it has no energy and the energy disk is energy rotation kinetics

      K = ½ m₁ w₁²

In the final position the disc and the ring rotate together, so calluses contribute energy

      Kf = K₁ + K₂ = ½ (m₁ + m₂) w₂²

Where m1 is the mass of the disk, m2 the mass of the ring and w is the initial and final angular velocity

To find the final angular velocity, we treat the case as an inelastic shock, where the kinetic moment (L) is preserved, the system is formed by the two bodies.

       L₀ = Lf

       L₀ = I₁ w₁ + 0

       Lf = (I₁ + I₂) w₂

       I₁ w₁ = (I₁ + I₂) w ₂

       w₂ = I₁ / (I₁ + I₂) w₁

We take the kinetic moments of the bodies

Disk     I₁ = ½ m₁ R₁²

Hoop   I₂ = m₂ R₂²

Let's calculate the final angular velocity

     w₂ = ½ m₁ R² / (1/2 m₁ R² + m₂ R²) w₁

     w₂ = ½ m₁ / (m₁/2 + m₂) w₁

With this value we can substitute and calculate the final kinetic energy

     Kf = ½ (m₁ + m₂) [½ m₁ / (m₁ /2 + m₂) w₁]²  

     Kf = 1/8 [(m₁ + m₂) m₁² / (m₁/2 + m₂)²] w₁²

Let's substitute the values ​​that the mass and radius of the disc and ring give us are the same (M, R)

    Kf = 1/8 [2M M² / (3M/2)²] w₁² = ¼ M³ / (9M² /4) w₁²

    Kf = 1/9 M w₁²

This is the final kinetic energy, let's say it based on the initial (RE)

    Ko = RE = ½ M w₁²

    Kf / Ko = (1/9 M w₁²) / (1/2 M w₁²)

    Kf / RE = 2/9

    Kf = 2/9 RE

This loss of kinetic energy is transformed into internal energy during the crash

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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

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F is the force

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