ENGINEERINGAn incompressible fluid oscillates harmonically ($V=V_{0}\sin \omega t$, where V is the velocity) with a frequency of 10 rad/s in a 4 -in.-diameter pipe. $A\frac{1}{4}$ scale model is to be used to determine the pressure difference per unit length, $\Delta p_{\ell}$ (at any instant) along the pipe. Assume that:
$$
\Delta p_{\ell}=f\left(D, V_{0}, \omega, t, \mu, \rho\right)
$$
where D is the pipe diameter, $\omega$ the frequency, t the time, $\mu$ the fluid viscosity, and $\rho$ the fluid density. $\textbf{(a)}$ Determine the similarity requirements for the model and the prediction equation for $\Delta p_{f}$. $\textbf{(b)}$ If the same fluid is used in the model and the prototype, at what frequency should the model operate? ENGINEERINGCalculate ∫_C F(r)*dr for the given data. If F is a force, this gives the work done by the force in the displacement along C. Show the details. F=[x-y,y-z,z-x], C:r=[2 cos t, t, 2 sin t] from (2, 0, 0) to (2, 2π, 0)