The discussion revolves around deriving a height-time equation for water flowing out of a cylinder through a pinhole. The primary equation presented is $$\dot{W}=\rho g C_d A \sqrt{2gh}$$, where $$\dot{W}$$ is the weight flow rate, $$\rho$$ is the water density, $$g$$ is gravitational acceleration, $$C_d$$ is the discharge coefficient, $$A$$ is the orifice area, and $$h$$ is the water height. Participants explore the relationship between height and time, leading to the equation $$\frac{dh}{dt}=-\frac{A}{A_x}C_D\sqrt{2gh}$$ and discuss integration techniques to express height as a function of time. The conversation concludes with a focus on the accuracy of the derived discharge coefficient and the importance of initial conditions in solving the equations.
PREREQUISITESThis discussion is beneficial for students and educators in physics and engineering, particularly those interested in fluid mechanics, as well as hobbyists conducting experiments related to fluid flow and calculus.
Now that you know the value of Cd, you can plot this equation up and see what you get. Plot a graph of this equation so that we can compare it with the Logger Pro graph for dW/dt.Jonathan Densil said:
You calculate the derivative from the difference in the times, but you plot the derivative at the average of the times. It is just a coincidence that the difference and the average are about the same value for this particular case.Jonathan Densil said:
Excellent. This is almost the exact negative of dW/dt calculated by Logger Pro from the raw data.Jonathan Densil said:
I don't know how to do this. Maybe, if you submit it to the mathematics forums (specifying very precisely what is involved mathematically), you can get some help there. Mention that you have an analytic expression for the rate of weight loss vs time, and experimental data on weight as a function of time.Jonathan Densil said: