studiot

I hope they can explain themselves more fully than you have done. Did you have a question?

Try a PM to a mod. dave is online right now.

OK so there are two approaches to integration. The anti derivative is linked to the area approach by what is known as the Fundamental Theorem of Calculus. This theorem is difficult and therefore not normally studied until advanced courses in calculus. Here is the simple route to area. Note this is 'intuition' only, not a formal proof since this is a very quick answer. Consider any general curve (function) as drawn. Divide it into strips by drawing verticals. The strips are almost rectangular. It is obvious from elementary geometry that the area of any strip is greater than the area generated by multiplying the low side by dx and less than the areas generated by multiplying the high side by dx. An approximation can be obtained by using the average value of y1 and y2 (low side and high side) times dx. Equally the total area between any two x values can be obtained bu adding up (summing) the areas of all the strips. So if we let the strip beocme very thin (dx tends to zero) and the number of them very large we obtain an infinite sum. We can write this in limit form for a formal proof, but it demonstrates the principle of why the Area = Integral ydx Does this help?

How were you introduced to the definite integral, or for that matter any integral? I need to start from what you know, and proceed to what you want to know.

What purpose does this serve? My objective is calculation not drafting perfection. How do I draw a velocity to scale? This is an instantaneous picture. The velocities are linked by formulae. I have modelled my human as a vertical straight line, since this line will meet the rain before any other part of the human. I have no idea what you mean by draw the rain behind. Again I have no idea what you mean. Raindrops falling from the perimeter ? All the raindrops have reached terminal velocity and are falling at the same constant speed. These were the explicit conditions stated by the OP. Water splashes, water from another source, etc are all outside the scope of this model. But then so is the slope of the ground, the partial evaporation of rain drops and other phenomena present in the real world.

Here is my analysis of the situation. The notation provides a basis for futher discussion. Here is some explanation. OP is the projected length of the umbrella on the horizontal. This blocks all rain from passing. As the front edge, B, moves forwards it covers rain that has already fallen below the level of the umbrella. This rain continues to fall vertically, impacting the ground between C and D. C is the point where rain falling when the umbrella edge was at A strikes the ground. These points create a diagonal line BC, separating the area under the umbrella into two zones. Quad OBCP where is it totally dry and triangle, BCD where it is still raining. Thus any walker behind line AC cannot get wet under any circumstances Further, although there is rain in triangle CED, none of this is accessible to the walker, since it will have fallen to the ground by the time the walker arrives. Thus only the rain in triangle BEC could ever hit the walker. Now since BD is constant the longer AB is the smaller the slope of CB and therefore the smaller the triangle BCE. But the condition for making AB as long as possible is for it to be horizontal. This answers the original question, but an interesting further exercise is to superimpose the area within BCE that the walker is going fast enough to enter. This is the point of line A'B'E'C' and the velocities. When the rain is vertical and the umbrella horizontal the they meet at 90 degrees. If the umbrella is now tilted at x from the horizontal, the rain and umbrella are now at (90-x) to each other.

Sorry I can't agree with this. No rain can fall directly down through the horizontal projection of the umbrella, wherever it is. The umbrella creates a 'dry zone' immediately in front of the walker. The walker is always walking into this zone. Since the horizontal position maximizes the shadow it maximizes the size of this dry zone. The umbrella travels with the walker. So yes, the walker is walking into the rain, but the only rain that she can encounter has already fallen below the level of the umbrella, before the umbrella has arrived. So it becomes a race between the speed of the walker and the speed of falling drops. Yes the vertical projection of the umbrella will push the drops in the upper level out of the way. But this projection must be small to keep off the drops still falling vertically above the walker. The walker will still catch any drops that have already fallen below this level on the lower part of their body, shoes etc, if he walks fast enough. The risk of that is minimised by maximizing the dry zone and therefore the horizontal projection. This is unlike the situation of an open topped sports car, which is going much faster, so a (near) vertical windscreen will push the drops in the upper half away and the car's speed will carry you past the vertically falling drops before more can fall. Further the driver is protected at low level by the bodywork.

The rain is falling vertically so there is no wind. In these circumstances you should hold the umbrella vertically. The umbrella cast a 'dry shadow' over the holder which is maximal at the vertical umbrella position, whether the umbrella is moving or stationary. At any other angle the shadow is oblique and equal to the projection of the umbrella area onto the horizontal. This is very much like the question, "Is the normal reaction at an angle when on object moves over its support?"

Ask a mod to move this topic to speculations. Get a surveying textbook and find out about deflection angles. Surveyors often have to measure or set out curves, only part of which, are accessible and the centre is too far away to be accessible. There are other methods than deflection angles .

Did I not mention high rate of discharge? The outflow is controlled solely by the outflow geometry and configuration. Once the siphonic action is initiated the inflow exerts no control over the outflow, rather like the toilet flush I referred to in post2. Do you understand this? The inflow in not going to be constant, which you originally stated. Of course perhaps this might be the only absolutely constant, naturally occurring flow, in existence. As a matter of interest have you heard of Turloghs in the Burren in County Claire in Eire? You should look these up.

The condition under which my diagram will act as a siphon or simply an overflow depend upon the inlet and outlet conditions. I thought I had already explained that, did you not understand it? The diagram was meant to be able to reproduce this stated behaviour, which it could do in suitable circumstances. Here is a modified drawing, exaggerated to demonstrate the point. Water enters the initially empty chamber and fills it to level CC a total of volume 3 There is no outflow at this stage. Further inflow fills the volume marked 2 plus the left hand pipe of the siphon. When the water reaches level AA it starts to trickle out over what you have dubbed a weir. So long as the inflow remains low it will trickle out over this weir. However inflow is variable and a sudden influx of water equal only the the small additional volume 1 and taking the level to BB will initiate the siphonic action. The siphon will now discharge through the outflow until the level falls to CC. That is the whole of volumes 1 and 2 will discharge continuously in a sudden rush. There will then be a dry period where volume 2 recharges, the duration will depend upon the size of volume 2 and the inflow rate. The siphon is then reset ready for action. Note that unlike the cup and straw siphon in the video the intake for the outlet is not at the bottom (though it could be) so volume 3 is never discharged once filled. The straw emptied the cup completely since one open end went to the bottom. Note also that this mechanism is capable of providing a much larger temporary sudden discharge, plus a lower base rate flow, depending upon the geometry. All it requires is volume 1 to be much smaller than volume 2.

Thank you so much for the link, I got sidetracked by the pop pop boats. Ideal for some younger relatives! The siphon is full immersed in the cup so completely empties it. If the outlet is to one side and does not reach the bottom then it will not completely empty. I should have noted that an anticline is tension dominated, whereas a syncline is compression dominated. This is because the outer top is stretched in an anticline, leading to cracking and weakness in an otherwise impervious layer, allowing a route for water entry.

#First let me apologise for the poor quality of my hasty sketch. This may have led you astray. Start with the chamber or porous zone empty. Water percolates in and the chamber collects until the level reaches B Percolation continues, but the water still cannot escape until its level is such that it is enough to initiate the siphonic action, which I have shown as A. In fact there will be a small discharge as soon as the level reaches the lower part of the outlet, you have shown as C. Once the water between my A and B has been discharged the level will be too low to sustain siphonic action so only this water will exit. The water below B will remain. Of course other factors which affect the discharge are the rate of influx and the available rate of eflux. A heavy rainshower, for instance, will rapidly charge up a finite chamber, but the outlet may be restricted so the eflux will occur in sudden heavy bursts. As I said the most likely situation is that the collection is not in an open chamber but in porous material that is bent around an anticline (upfold). Does this make it any clearer?

OOps You seem to have contrdicted yourself within the first few lines. Two things for the universe so where did charge come from - That's three things or is it four by my counting.

You have asked a very good question, that is worthy of more than a few moments thought. Do you know the difference between intensive properties and extensive properties? Extensive properties are additive. They depend on the quantity of matter so each tiny element of the body contributes and they all add up. Examples are mass and volume. Both of these can be measured pretty accurately. Intensive properties, however, do not depend upon the quantity of matter. Examples are temperature, density and pressure. Normally we want one single value to represent the whole body concerned. By asking this we are effectively want the whole to be homogenous and isotropic. Chemists try to can achieve this by stirring for instance. When the body is not homogenous, ie the property varies from point to point within it we can indeed take an average, summed over the whole body (or part of it) An example would be the average surface temperature of the earth. Intensive properties are therefore, by nature likely to be less accurately availabale than extensive ones. Does this help?

The essence of Chaos mathematics is that it is non linear, so why would you expect linear algebra to be applicable? Since Linear maths is much much easier than non linear maths most disciplines try to use it for models and to 'linearize' whenever practicable. I am no expert in astro stuff but I expect they are no different. Incidentally, you have referred to chaos theory as another astro science, please elaborate as this is the first I have heard of this connection.

What's so difficult? This sketch is a very simple way for a self discharging siphon to work. I would have to know considerably more about the strata to work out anything better. I have shown pipe like voids, but is more likely to be be faults / fractures or other features since it is natural, although natural clay filled 'pipes' do occur. and the 'void' may be a pocket of pervious rock, rather than an actual chamber. Of course, the siphon may simply be formed from the local folding of a thin pervious layer, sandwiched between two impervious ones, that gets recharged when it rains, until there is enough water in it to drive the siphon over the top of the anticline, thus starting the siphon until the aquifer is drained. It may then issue forth as a spring on the other side of the anticline at a local fault. The water may be entering not directly but along the interface between to layers ot strata. There are just oodles of possibilities. go well

With regard to the blog, which led me here, I thoroughly enjoyed the thread on Bohemian Gravity. http://www.scienceforums.net/topic/78785-bohemian-gravity/?hl=%2Bbohemian+%2Bgravity However I don't see any essential difference between this piece of pure entertainment and any other non interactive presentation.

It's been much better today.

But you have a mechanical valve in the cistern. If that failed would ther not be overflow? Have you never seen the situation where the flush does not terminate but continues at the reduced rate of the inflow?

Have you looked at your toilet lately? Does it not also have the same extended low rate inflow plus intermittent high outflow characteristics?

Such a broad ranging question, it's difficult to answer or narrow down. Try reading Cats' Paws and Catapaults By Steven Vogel Penguin. It is a small book that compares how nature does things with how man's technology achieves results and examines what you need to know to understand the workings of the world and apply it to your own needs. From what you say you should have no trouble understanding it and it may inspire you to narrow down you interest to some specific area. The book is probably available very cheaply second hand.

Not quite. Most chemical reactions require the breaking of bonds, before new ones can be made (if there are new ones). But breaking of bonds alone is only the first part of the reaction. The second part may release energy. The terms exothermic and endothermic refer to the entire reaction. It is the net direction of heat energy that decides. So if more heat energy is evolved than needs to be input at the beginning the reaction is exothermic. Likewise if you need to put in more heat than you get back the reaction is endothermic. Pleae note that the terms refer to specifically to heat energy. Does this help?

Multidimensions have application when you come to advanced mechanics. You should look up "generalised co-ordinates" https://www.google.co.uk/#q=generalised+coordinates+in+mechanics I once wrote a paper entitled "The use of the fifth quadrant".

I agree with the answers in the first parts. Whilst the two boxes are moving together they act as a single object as far as the rope is concerned and the friction between them counts as an internal force, so your calculatuion does not show any frictional retarding force on the lower box. As soon as they start to slide past each other the upper box exerts a retarding force on the lower box. thus you need to calculate this and do a horizontal force balance to find the actual accelerating force on the lower box. Does this help?