Question

Part 1 - Basic Equations

The basic equation connecting wave speed v, wavelength ? and frequency f is

v = f?
The wave speed in a stretched string depends on the string's tension T and mass per unit length ? (also called the "linear mass density") with the equation

v =
sqrt1a.gif T/?
1. If the string's length is L, what is the fundamental wavelength ?1?

(I already got this answer, it is 2L)



2. Derive a formula for the speed of the wave in terms of the fundamental frequency f1 and the string length L.

v=



3. Derive a formula for linear mass density ? in terms of the wave speed v and string tension T.

?=


Assume you have the following intermediate (i.e., significant figures may be incorrectly stated) experimental results :

L = 0.812 ? 0.019 m
f1 = 29.02 ? 1.45 Hz
T = 5.08 N (exact)

4. What is the linear mass density of the string (without uncertainty, units requred)?

?=



5. Use the above results to calculate the fractional uncertainty in the wave speed ?v:



6. Use the above results to calculate the fractional uncertainty in the linear mass density ??:



7. Then calculate the absolute uncertainty in the linear mass density ??:



8. Finally, state the linear mass density with its uncertainty using correct significant figures (include units):

Answer 1

Answer:

1. λ = 2 L, 2.  v = 2L f₁ , 3.    v = √ T /μ², 4.   μ = 2,287 10⁻³ kg / m , 5.   Δv / v = 0.058 , 6.    Δμ /  μ = 0.12 , 7. Δ μ = 0.3  10⁻³ kg / m ,

8.  μ = (2.3 ±0.3)  10⁻³ kg / m

Explanation:

The speed of a wave is

            v = λ f                1

Where f is the frequency and λ the wavelength

     

The speed is given by the physical quantities of the system with the expression

            v = √ T /μ²                   2

1) The fundamental frequency of a string is when at the ends we have nodes and a maximum in the center, therefore this is

                 L = λ / 2

                 λ = 2 L

2) For this we substitute in equation 1

              v = 2L f₁

3) let's clear from equation 2

             

The speed of a wave is

            v = λ f₁

Where f is the frequency and Lam the wavelength

The speed is given by the physical quantities of the system with the expression

           v = √ T /μ²                            2

4) linear density is

           μ = T / (2 L f₁)²

           μ = 5.08 / (2 0.812 29.02)²

           μ = 2,287 10⁻³ kg / m

We maintain three significant length figures, so the result is reduced to

           μ = 2.29 10⁻³ kg / m

5) the speed of the wave is

            v = 2 L f₁

The fractional uncertainty is

         Δv / v = ΔL / L + Δf₁ / F₁

         Δv / v = 0.02 / 0.812 + 1 / 29.02

         Δv / v = 0.024 + 0.034

         Δv / v = 0.058

6) the equation for linear density is

              μ = T / (2 L f₁)²

             Δμ / μ = 2 ΔL / L + 2Δf₁ / f₁

The tension is an exact value therefore its uncertainty is zero ΔT = 0

            Δμ / μ = 2 0.02 / 0.812 + 2 1 / 29.02

             Δμ /  μ = 0.12

7) absolute uncertainty

           Δ μ = $e_{r}$   μ

           Δ μ = 0.12 2.29 10⁻³ kg / m

           Δ μ = 0.3  10⁻³ kg / m

8)

           μ = (2.3 ±0.3)  10⁻³ kg / m