A thin cylindrical rod of length 1.5m has a mass of 2.4 kg, the density of the material is 2700 kg m-3 and the Young’s modulus of the rod is 69 GPa. What is the characteristic impedance of the rod to longitudinal waves?

Question

BlackTom AI · Accepted Answer

We are asked for the characteristic impedance of the rod to longitudinal waves. For longitudinal waves in a solid rod, the characteristic impedance Z is given by Z = ρ c, where ρ is the mass density (kg/m^3) and c is the wave speed in the rod. The wave speed for longitudinal waves in a solid is c = sqrt(E/ρ), with E being Young's modulus (Pa).

Step 1: Compute the wave speed c.
- E = 69 GPa = 69 × 10^9 Pa
- ρ = 2700 kg/m^3
- c = sqrt(E/ρ) = sqrt(69×10^9 / 2700) = sqrt(2.555...×10^7) ≈ 5055 m/s

Step 2: Compute the impedance Z = ρ c.
- Z = ρ c = (2700 kg/m^3) × (5055 m/s) ≈ 13.63×10^6 kg/(m^2·s) ≈ 1.36×10^7 kg/(m^2·s)

Notes on the provided options:
- The resulting numeric value, about 1.36×10^7 kg/(m^2·s), is vastly larger than all the given choices.
- Option a: 503 kg m-4 s-1 would correspond to 5.03×10^2 kg/(m^4 s), which does not match the dimensional form we expect (the standard impedance has units of kg/(m^2·s)). This indicates a mismatch in units, or perhaps a misprint in the provided unit notation.
- Option b: 8.1×10^3 kg m-4 s-1 (≈ 8.1×10^3 kg/(m^4·s)) is still off both in magnitude and in the expected unit structure for Z.
- Option c: 4.1×10^3 kg m-4 s-1 is similarly inconsistent with the expected kg/(m^2·s) form and far from the computed value.

If one were to check dimensional consistency: Z should have units of kg/(m^2·s). The given options all present kg/(m^4·s), which suggests a probable misprinting or a missing factor of area (A) in the formulation. If an area factor were intended and Z were given per unit area, the numeric scale could be altered by a factor of A, but with the data provided (no radius or area explicitly given beyond the density and modulus), the clean derivation yields Z ≈ 1.36×10^7 kg/(m^2·s).

In summary, following the standard theory for longitudinal waves in solids, the characteristic impedance should be Z = sqrt(ρ E) or equivalently ρ sqrt(E/ρ), which numerically gives about 1.36×10^7 kg/(m^2·s). None of the offered options match this value, and their units are inconsistent with the conventional impedance units.

A thin cylindrical rod of length 1.5m has a mass of 2.4 kg, the density of the material is 2700 kg m-3 and the Young’s modulus of the rod is 69 GPa. What is the characteristic impedance of the rod to longitudinal waves?单项选择题

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