To find the value of [tex]\( C \)[/tex] from the given data, we need to use the provided equation:
[tex]\[
\frac{V}{t} = \frac{\pi P r^4}{8 C l}
\][/tex]
Given the following measurements:
- [tex]\(\frac{V}{t} = 1.20 \times 10^{-6} \, \text{m}^3/\text{s}\)[/tex]
- [tex]\(P = 2.50 \times 10^{3} \, \text{N/m}^2\)[/tex]
- [tex]\(r = 0.75 \, \text{mm} = 0.75 \times 10^{-3} \, \text{m}\)[/tex]
- [tex]\(l = 0.250 \, \text{m}\)[/tex]
We first rearrange the equation to solve for [tex]\( C \)[/tex]:
[tex]\[
C = \frac{\pi P r^4}{8 \left( \frac{V}{t} \right) l}
\][/tex]
Next, let's substitute the given values into the equation:
1. [tex]\(\pi \approx 3.14159\)[/tex]
2. [tex]\(P = 2.50 \times 10^{3} \, \text{N/m}^2\)[/tex]
3. [tex]\(r = 0.75 \times 10^{-3} \, \text{m}\)[/tex]
4. [tex]\(\left( \frac{V}{t} \right) = 1.20 \times 10^{-6} \, \text{m}^3/\text{s}\)[/tex]
5. [tex]\(l = 0.250 \, \text{m}\)[/tex]
Performing the substitution:
[tex]\[
C = \frac{\pi \times 2.50 \times 10^{3} \times (0.75 \times 10^{-3})^4}{8 \times 1.20 \times 10^{-6} \times 0.250}
\][/tex]
Simplifying inside the parenthesis and exponentials:
[tex]\[
= \frac{3.14159 \times 2.50 \times 10^{3} \times (0.75)^4 \times 10^{-12}}{8 \times 1.20 \times 10^{-6} \times 0.250}
\][/tex]
Continuing the calculation step by step:
1. Calculate [tex]\( (0.75)^4 \)[/tex]:
[tex]\[
(0.75)^4 = 0.31640625
\][/tex]
2. Multiply powers of ten:
[tex]\[
0.31640625 \times 10^{-12} = 0.31640625 \times 10^{-12}
\][/tex]
3. Multiply [tex]\( \pi \times 2.50 \times 10^3 \times 0.31640625 \)[/tex]:
[tex]\[
3.14159 \times 2.50 \times 10^3 \times 0.31640625 = 2.480502625 \times 10^3
\][/tex]
4. Combine constants:
[tex]\[
= \frac{2.480502625 \times 10^3 \times 10^{-12}}{8 \times 1.20 \times 10^{-6} \times 0.250}
\][/tex]
This expression simplifies to:
[tex]\[
C = \frac{2.480502625 \times 10^{-9}}{2.4 \times 10^{-6}} = \frac{2.480502625}{2.4} \times 10^{-3} = 1.0335428 \times 10^{-3} \approx 0.001035 \, \text{Ns/m}^2
\][/tex]
Thus, the value of [tex]\(C\)[/tex] is approximately:
[tex]\[
C = 0.001035 \, \text{Ns/m}^2
\][/tex]
