Question

1. it is observed that the frequency of oscillation f of a guitar string depends upon the string mass m, length l, and tension force f. determine one possible equation for f based on m, l, and f. the equation must be dimensionally consistent.
2. ficks first law relates the diffusive flux to the gradient of the concentration. it postulates that the flux, j (kg/m²-s), the flux is dependent on the diffusion coefficient, d (m²/s), the density, ρ (kg/m³), and a displacement, x (m). deduce an equation for f that is dimensionally consistent.

Explanation:

Problem 9

Step1: Write the dimensions

The dimension of frequency \(f\) is \([T^{-1}]\). The dimension of mass \(m\) is \([M]\), length \(L\) is \([L]\), and tension force \(F\) (since \(F = ma\), dimension of \(F\) is \([MLT^{-2}]\))

Let \(f=k m^{a}L^{b}F^{c}\), where \(k\) is a dimensionless constant. Then \([T^{-1}]=[M]^{a}[L]^{b}[MLT^{-2}]^{c}=[M^{a + c}L^{b + c}T^{-2c}]\)

Step2: Equate the exponents

For \(M\): \(a + c=0\) (Equation 1)
For \(L\): \(b + c = 0\) (Equation 2)
For \(T\): \(-2c=-1\Rightarrow c=\frac{1}{2}\)

Substitute \(c = \frac{1}{2}\) into Equation 1: \(a=-\frac{1}{2}\)
Substitute \(c=\frac{1}{2}\) into Equation 2: \(b =-\frac{1}{2}\)

Step1: Write the dimensions

The dimension of flux \(J\) is \([ML^{-2}T^{-1}]\), diffusion coefficient \(D\) is \([L^{2}T^{-1}]\), density \(
ho\) is \([ML^{-3}]\), and displacement \(x\) is \([L]\)

Let \(J=kD^{a}
ho^{b}x^{c}\), where \(k\) is a dimensionless constant. Then \([ML^{-2}T^{-1}]=[L^{2}T^{-1}]^{a}[ML^{-3}]^{b}[L]^{c}=[M^{b}L^{2a-3b + c}T^{-a}]\)

Step2: Equate the exponents

For \(M\): \(b = 1\)
For \(T\): \(-a=-1\Rightarrow a = 1\)
For \(L\): \(2a-3b + c=-2\). Substitute \(a = 1\) and \(b = 1\) into it: \(2\times1-3\times1 + c=-2\Rightarrow c=-1\)

Answer:

\(f = k\frac{1}{L}\sqrt{\frac{F}{m}}\)

Problem 10