First, I assumed that the wine glass is a paraboloid in shape - this is a parabola of the form y = ax2 that has been rotated around the y-axis (I don't really see the point in the more general format of rotating y2 = 2ax about the z-axis myself, and the horizontal orientation would make the situation far less convincing in any case) to form a, well, 'wine glass shape'! Then, I took a 3-D plane of the form y = xtan(θ) and analysed the intersections it makes with the paraboloid - this plane effectively models the surface of the liquid.
The development of this short project is seen below. The part I am most proud of is that I have learned how to integrate under a surface within a circular region using polar coordinates, which is much simpler than the traditional rectangular coordinate method, requiring multiple trigonometric substitutions to achieve the same result. Anyway, enjoy reading. For the first time I know someone is reading this, so I hope it is all correct and logical to follow...
The gameplan for my method might be useful:
Now, the bulk of the maths:
The implications of these formulae are quite powerful really, but it is quite difficult to draw using Desmos - I have not yet worked out how to parametrically add the z-axis into the 2-D plotter. However, I have done a couple of graphs on the online graphing calculator: http://tinyurl.com/zgsvzc8. The angle is changed with the slider 'T' between -π/2 and π/2, and the volume of liquid and shape of the glass are also altered with their respective sliders.
Seeing the changing shape of the surface of the liquid (red) and the cross-sectional profile of the glass at the same time as the angle of tilt changes is very revealing - the red ellipse disappears at the point where the surface no longer straddles the minimum point of the glass (the bottom, at the origin): although real wine glasses actually have a slight curve back in towards the top, a quadratic curve has no such 'lip'. Therefore as soon as the liquid is all on one wall of the glass, it is poured out!
Please bear in mind that I have caused the surface of the liquid to rotate instead of the glass itself because it is much simpler to do. Therefore the graph is from the perspective of the glass instead of the Earth's gravitational field!
The development of this short project is seen below. The part I am most proud of is that I have learned how to integrate under a surface within a circular region using polar coordinates, which is much simpler than the traditional rectangular coordinate method, requiring multiple trigonometric substitutions to achieve the same result. Anyway, enjoy reading. For the first time I know someone is reading this, so I hope it is all correct and logical to follow...
The gameplan for my method might be useful:
Now, the bulk of the maths:
The implications of these formulae are quite powerful really, but it is quite difficult to draw using Desmos - I have not yet worked out how to parametrically add the z-axis into the 2-D plotter. However, I have done a couple of graphs on the online graphing calculator: http://tinyurl.com/zgsvzc8. The angle is changed with the slider 'T' between -π/2 and π/2, and the volume of liquid and shape of the glass are also altered with their respective sliders.
Seeing the changing shape of the surface of the liquid (red) and the cross-sectional profile of the glass at the same time as the angle of tilt changes is very revealing - the red ellipse disappears at the point where the surface no longer straddles the minimum point of the glass (the bottom, at the origin): although real wine glasses actually have a slight curve back in towards the top, a quadratic curve has no such 'lip'. Therefore as soon as the liquid is all on one wall of the glass, it is poured out!
Please bear in mind that I have caused the surface of the liquid to rotate instead of the glass itself because it is much simpler to do. Therefore the graph is from the perspective of the glass instead of the Earth's gravitational field!
