Fundamentals of Fluid Mechanics Questions and Answers  
Question by AME536A Student
I am getting stuck on problem #4 on the homework (the canoe question). I am able to do part A and it gives me the correct answer for part c, but I am having trouble with the integral on part B. The task is to integrate the forces instead of using Archimedes principle. The basic premise is pressure multiplied by area, so starting off, we can say the area integral for the submerged part of the boat is $$ \int_{\theta_0}^{\theta_1} \int_{0}^x rd \theta dx $$ (I couldn't get Latex to work with subscripts in the integral bounds. It should be $\theta^{}_{0}$ and $\theta^{}_{1}$). Next we can use Pascal's law to find the pressure: $$ P = P^{}_{atm} + \rho^{}_{w}g(H-y) $$ Here is where I am getting stuck. Originally I had the integral set up using Y and integrating from 0 to H multiplied by the area to get force: $$ \int_{\theta}^\theta \int_{0}^x \int_{0}^H r(P^{}_{atm} + \rho^{}_{w}g(H-y)) \sin(\theta) d \theta dx dy $$ This doesn't seem like the right way to integrate to find the force. There isn't really a y direction, it's just backward along the radius. So, my thought was to find Y as a function of $\theta$: $$ y = R(1-\sin(\theta)) $$ So the integral becomes $$ \int_{\theta}^\theta \int_{0}^x r(P^{}_{atm} + \rho^{}_{w}g(H-R(1-\sin(\theta))) \sin(\theta) d \theta dx $$ Neither of the ways I've done the problem thus far are resulting in the correct answer. I'm unsure of what is going wrong. Are either of the two ways I mentioned the correct way to find the force applied on the bottom of the canoe?
It's difficult to say where the problem is because you don't define $y$ and $\theta_0$ and $\theta_1$. Is $y$ pointing up or down? What is $\theta_0$? This should be made clear through text or figure. You seem to be on the right track thus. There's probably a small mistake in the setup of the integral. Have you tried integrating only half of the canoe and then multiplying by 2? This will get rid of one potential error related to the angles.. PS. I fixed your problem in your post. To see how I did it, right click on the equation and choose “Show Math As” -> “TEX Commands”.
Question by AME536A Student
I have a quick question about Problem 6, based on the figure provided, do we need to take into account of mercury surface tension on the side walls of the tank ?
If there is enough information to take this into account, then yes you have to. If not you have to mention this in the assumptions.
Question by AME536A Student
In question #2 for Assignment #3, can we assume that the duct that the fluid is traveling through is square?
No, you shouldn't make this assumption. Your answer should work for any type of duct as long as the intake is aligned with the $y$ axis.
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