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onent can shorten her backswing yet still hit hard by using your pace making a safe yet effective return hit hard and flat to the open court off a high short ball but otherwise try hitting with more top spin defensive options just as an attacking ball need not be a fast ball a defensive ball need not be a slow ball rather a defensive ball is one where our aim is to return to an approximately 50 50 chance of winning the point if we are off balance or are facing a challenging ball we cannot go for too much or we risk hitting out and immediately losing the point reducing the risk of hitting out can be achieved in various ways do not swing as fast or with excessive spin take the ball later after the bounce giving it more time to slow down block the ball back perhaps taking it on the rise to a safe region well away from the lines focus on being balanced even if it means hitting with less power importantly everyone is different and you should learn what your preferred defensive shots are for example while flat balls inherently have a lower margin for error nonetheless some people may find they are more accurate hitting flat than hitting with top spin simply due to their biomechanical structure and past practice placement of a defensive shot is crucial because speed spin and or time have been sacrificed for greater safety it is quite likely that if the opponent has time to get into a comfortable position they can punish your ball this is yet another example of where small differences can have large consequences if the opponent can step into the ball you might be in trouble yet hitting just a metre deeper or a metre further to the side might prevent this moreover never forget that placement is relative to where the opponent currently is hitting deep to the backhand is normally a good shot unless the opponent is already there changes in depth can be very effective but they must be relative to where the opponent is if an opponent is attacking she might have moved to being on the baseline looking to take the next ball early hitting a deep top spin shot not necessarily very hard is very effective in this scenario because the opponent is forced to move backwards and thus cannot generate as much power a skilled opponent can still hit hard but even a 10 km h reduction in ball speed makes a large difference technical considerations watching the slow motion clips on youtube of professional players hitting groundstrokes shows that the same player has many different ways of hitting her forehand and backhand at the point of contact you can look to see where her feet are which way her hips are pointing which way her shoulders are pointing the angle of her wrist and the contact point of the ball relative to the body and as she hits you can look to see which parts of the body are moving and which have momentarily become static while an upcoming article may consider such technical aspects in greater detail here it is simply noted that while you should be able to hit every kind of shot from any reasonable contact point each has advantages and disadvantages and since tennis is a game of probabilities it comes as no surprise that the top players instinctively know not just what type of shot to hit but also how technically to hit it in the best possible way for them at that particular moment if they have time and want to hit a heavy top spin they will probably choose a contact point further away from their body and step into the ball while if they must return a very fast ball they may instead use a more open stance and hit the ball more in line with their eyes well out in front next time you are on the court experiment with how changing the contact point even over a relatively small range of 10 20 cm can change how hard you can hit the ball how accurately you can hit the ball and how much top spin you can generate and do not forget this may change depending on the type of incoming ball for example generating pace off a slow ball is better done using a different technique than returning a fast incoming ball failing to recognise this may mean you feel you are not hitting the ball well when in fact you are just not using the best technique for the particular type of shot you want to hit the other side of the coin is recognising that sometimes it is necessary to play a ball using a non optimal technique due to a funny bounce or lack of time or inherent laziness in this situation it is important to adjust the type of shot you hit if you are forced to return a hard hit ball at full stretch you will lose accuracy so do not go anywhere near the lines if you are jammed you are not going to be able to hit as heavy a ball as you may wish so you may change to hit flatter opting to take time away from the opponent of course changing from heavy to flat generally means changing where you want your shot to land a shorter top spin that bounces above shoulder height is generally good whereas a shorter flat shot is generally bad for example categories uncategorized does the voltage across an inductor immediately reverse if the inductor is suddenly disconnected august 3 2018 jmanton leave a comment consider current flowing from a battery through an inductor then a resistor before returning to the battery again what happens if the battery is suddenly removed from the circuit online browsing suggests that the voltage across the inductor reverses to maintain current flow but the explanations for this are either by incomplete analogy or by emphatic assertion moreover one could argue for the opposite conclusion if an inductor maintains current flow then since the direction of current determines the direction of the voltage drop the direction of the voltage drop should remain the same not change to understand precisely what happens it is important to think in terms of actual electrons when the battery is connected there is a stream of electrons being pushed out of the negative terminal of the battery being pushed through the resistor being pushed through the inductor then being pulled back into the battery through its positive terminal the question is what happens if the inductor is ripped from the circuit thereby disconnecting its ends from the circuit the explanation of what happens does not change in any substantial way whether it is the battery or the inductor that is removed the analogy of an inductor is a heavy water wheel the inductor stores energy in a magnetic field while a water wheel stores energy as rotational kinetic energy but if we switch off the water supply to a water wheel and the water wheel keeps turning what happens nothing much and if we disconnect an inductor so there is no circuit for current to flow in what can happen one trick is to think not of a water wheel but of a heavy fan inside a section of pipe ripping the inductor out of the circuit corresponds to cutting the piping on either side of the fan and immediately capping the ends of the pipes this capping mimics the fact that electrons cannot flow past the ends of wires not taking sparks into consideration crucially then when we disconnect the fan there is still piping on either side of the fan and still water left in these pipes consider the water pressure in the capped pipe segments on both sides of the fan assume prior to cutting out the fan water had been flowing from right to left through the fan indeed when the pump is first switched on it will cause a pressure difference to build up across the fan this pressure difference is what causes the fan to start to spin as the fan spins faster this pressure difference gets less and ideally goes to zero in the limit initially then there is a higher pressure on the right side of the fan the fan keeps turning powered partly by the pressure difference but mainly by its stored rotational kinetic energy think of its blades as being very heavy therefore not wanting to slow down so water gets sucked from the pipe on the right and pushed into the pipe on the left these pipes are capped therefore the pressure on the right decreases while the pressure on the left increases voltage drop is a difference in pressure therefore the voltage drop across the inductor is changing there is no discontinuous change in pressure the claim that the voltage across an inductor will immediately reverse direction is false that said the pressure difference is changing and there will come a time when the left pipe will have a higher pressure than the right pipe now there are two competing forces the stored kinetic energy in the fan wants to keep pumping water from right to left while the larger pressure on the left wants to force water from left to right the fan will start to slow down and eventually stop albeit instantaneously at the very moment the fan stops spinning there is a much larger pressure on the left than on the right therefore this pressure difference will force the fan to start spinning in the opposite direction under ideal conditions then the voltage across the inductor will oscillate why should we believe this analogy though returning to the electrons the story goes as follows assume an inductor in a circuit has a current flowing through it from left to right therefore electrons are flowing through the inductor from right to left because benjamin franklin had 50 chance of getting the convention of current flow correct if the inductor is ripped out of the circuit the magnetic field that had been built up will still push electrons through the inductor in an attempt to maintain the same current flow the density of electrons on the right side of the inductor will therefore decrease while the density on the left side will therefore increase electrons repel each other so it becomes harder and harder for the inductor to keep pushing electrons from right to left because every electron wants its own space and it is getting more and more crowded on the left side of the inductor eventually the magnetic field has used up all its energy trying to cram as many electrons as possible into the left side of the inductor the electrons on the left are wanting to get away from each other and are therefore pushing each other over to the right side of the inductor this force induces a voltage drop across the inductor as electrons want to flow from left to right we say the left side of the inductor is more negative than the right side the voltage drop has therefore reversed but it did not occur immediately nor will it last forever because the system will oscillate as the electrons on the left move to the right they cause a magnetic field to build up in the inductor and the process repeats ad infinitum adding to the explanation we can recognise a build up of charge as a capacitor there is always parasitic capacitance because charge can always accumulate in a section of wire therefore there is no such thing as a perfect inductor for if there were we could not disconnect it rather an actual inductor can be modelled by an ideal inductor in parallel with an ideal capacitor technically there should also be a resistor in series to model the inevitable loss in ordinary inductors an inductor and capacitor in parallel form what is known as a resonant lc circuit which as the name suggests resonates categories electronics informal classroom notes tags basic electronic principles disconnecting an inductor inductor intuition behind caratheodory s criterion think sharp knife and shrink wrap august 24 2017 jmanton 3 comments despite many online attempts at providing intuition behind caratheodory s criterion i have yet to find an answer to why testing all sets should work https www thestudentroom co uk showthread php t 4284694 https mathoverflow net questions 34007 demystifying the caratheodory approach to measurability therefore i have taken the liberty of proffering my own intuitive explanation for the impatient here is the gist justification and background material are given later the set u is non measurable the set a is measurable we will think of our sets as rocks as explained later a rock is not measurable if its boundary is too jagged in the above image the rock is not measurable because it has a very jagged boundary on the other hand the rock has a very nice boundary and hence we would like to consider to be measurable for to be measurable according to caratheodory s criterion the requirement is for to hold for all sets while other references generally make it clear that this is a sensible requirement if is measurable and we will go over this below what makes caratheodory s criterion unintuitive is the requirement that must hold for all sets including non measurable sets however we argue that a simple change in perspective makes it intuitively clear why we can demand holds for all an outer measure uses shrink wrap to approximate the boundary if a cuts the rock u cleanly then the same errors are made when approximating the boundaries of the three objects in the picture hence caratheodory s criterion holds the two ingredients of our explanation are that the boundary of serves as a knife and that an outer measure calculates volume by using shrink wrap for example lebesgue outer measure approximates a set by a finite union of cubes that fully cover the set and uses the volume of the cubes as an estimate of the volume of the set precisely the infimum over all such approximations is used in the above image the red curve represents the shrink wrap the outer measure is given by the area enclosed by the red curve for rocks which are three dimensional we would use the volume encased by the shrink wrap when we take and we think of using part of the boundary of the part that intersects to cut if is a rectangle or cube then we produce a very clean straight cut of the set or rock consider shrink wrapping and individually as shown by the red curves in the above image two observations are critical if the cut made by is clean then the shrink wrap fits the cut perfectly no error is made for this part of the boundaries of and for all other parts of the boundaries of and i e the jagged parts the errors made by the shrink wrap for and precisely equal the errors made by the shrink wrap for because it is the same jagged boundary being fitted it follows that holds whenever has a nice boundary regardless of whether is measurable or not caratheodory s criterion is blind to the boundary of and instead is only testing how smooth the boundary of is different sets will test different parts of the boundary of from the beginning first it should be appreciated that caratheodory s criterion is not magical it cannot always produce the exact sigma algebra that you are thinking of so it suffices to understand why in certain situations it is sensible in particular this means we can assume that the outer measure we are given works by considering approximations of a given set by larger but nicer sets such as finite collections of open cubes whose measure we know how to compute taking the infimum of the measures of these appr...