Imagine you are biking, and you come across a ledge. I want to figure out what minimum velocity do you need to bike at to overcome a ledge, of some height $H$.   Roughly, it looks something like this:   The wheel is essentially a thin ring, with mass $M$, radius $R &gt; H$, velocity (moving rightward) $v$, and angular velocity $\omega$. We know that the wheel is purely rolling so $v=\omega r$, as long as it is on the rough surface.   I know that energy at the bottom has to be more than the gravitational potential energy of the wheel at the top (the limiting case is when the wheel has no velocity when it mounts the edge). Therefore, $$\frac{1}{2}I\omega ^2 + \frac{1}{2} Mv^2 = MgH$$ where $I$ is the moment of inertia.   I do not think this all you need to know, though. As in you could be going really fast, but a bike won't be able to climb the Empire State Building. Can we conserve angular momentum about the point of contact between the ledge and the wheel? Does the normal force do any work on the wheel?  How do you go about finding the minimum velocity required to climb the ledge?
A sphere of radius R is rolling without slipping with a velocity v and collides inelastically with a step of height h &lt; R. What is the minimum velocity for which the sphere will be over the step?    Relevant equations  Total kinetic energy (maybe): $\frac{1}{2}I\omega^{2}+\frac{1}{2}mv^2$  Gravitational potential energy: $mgh$   Moment of inertia about a point a distance d from the CM from de center of mass: $I=I_{cm}+md^{2}$ Rolling without slliping  $v=\omega R$  A discussion of the problem This problem is confusing to me: I know the collision is inelastic so there is no conservation of kinetic energy, there is torque when the sphere pivots the step so the angular momentum is not conserved either, the linear momentum doesn't give a lot information and even though I tried to see what condition needs the torque about the pivot point (to give a net torque in the direction of rotation and thus passing the step), I'm not able to see which torque can overcome the one from gravity which doesn't let the sphere slide up:    The sphere has an angular velocity, but I'm missing something conceptual because I can't figure out how to formally state that the angular momentum changed the right amount so that the sphere made it past the step.  The attempt at a solution  My (almost certainly wrong) attempt was to say that the sphere lost kinetic energy because of the change on potential energy:  $\frac{1}{2}I\omega^{2}+\frac{1}{2}mv^2 +mgR = mg(R+h)\\ v=\sqrt{\frac{10}{7}gh} $  I hope you can help me :)
Where is "Regular Solids" option? Following an online tutorial that uses an object named "icosahedron" from Add > Mesh > Regular Solids > Archimedean > Icosahedron. This object is not available in my version of Blender (2.74). Presenter asks me to include an add-on named "Add Mesh: Regular Solids", which again is not listed in my User Preferences dialog. What can I do about it? OpenLayers 3 WMS GetFeatureInfo popup I have a map a at the link below:    I added one of my GeoServer WMS layers to the map, now I try to add the attributes window pop up on click. I found to examples on the OpenLayers 3 homepage, but I cannot put together a working code...      Is there any chance to have a working sample somewhere? I'm trying to build a datapack that lets you craft xp bottles from glass bottles at the expense of 7 xp per bottle (because they drop 3 to 11 xp so the average is 7). I have almost everything working except for a check for if they have too little xp. If they have 7 or less xp, I would like to remove one bottle of enchanting from their inventory and give them one glass bottle, because the function for removing xp is called after they've crafted one and a scoreboard has counted it. My remove function currently looks like this:  #create dummy scoreboard to multiply bottles crafted by seven to get xp to be removed scoreboard objectives add 7xp dummy scoreboard players set @s 7xp -7 scoreboard players operation @s xpCrafted *= @s 7xp  #how do I get the value in xpCrafted into the command to remove xp? xp add @s -7 points  #cleanup scoreboard players reset xpCrafted scoreboard players reset 7xp scoreboard players set @e xpCrafted 0 scoreboard objectives remove 7xp   And my tick function like this:  execute as @e[scores={xpCrafted=1..}] at @s run function craft_xp:removexp   As the inline comment says, how do I get that value in xpCrafted into my xp removing command, and also how should I check to make sure the player has enough xp for the operation in the tick command (or elsewhere)? Presumably I would run a different command to remove the extra bottle if they didn't have enough xp, but I don't know how because there doesn't seem to be any type of if/else structure in commands. Finally, should a question like this be in two different questions or is one ok on this forum? If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$ If $f,g$ are continuous at $a$, show that $h(x)=\max\{f(x),g(x)\}$ and $k(x)=\min\{f(x),g(x)\}$ are also continuous at $a$. Here is my attempt at a proof. It feels very elaborate and I am not sure if it is correct. Can someone please point out any mistakes or places where I may improve. Thanks!  By the definition of continuity at $a$ of $f,g$ we have that $\lim\limits_{x\to a}f(x)=f(a)$ and $\lim\limits_{x\to a}g(x)=g(a)$. Suppose that $f(a)&gt;g(a)$. Then there exists a $\delta&gt;0$ such that $f&gt;g$ for all $x$ satisfying $|x-a|&lt;\delta$. Then for $x$ satisfying $|x-a|&lt;\delta$ we have $h(x)=f(x)$ and $k(x)=g(x)$ and thus $h$ and $k$ are continuous at $a$ because $f$ and $g$ are continuous at $a$. Now if $f(a)&lt;g(a)$ we simply relabel $f=\tilde{g}$ and $g=\tilde{f}$, so that $\tilde{f}(a)&gt;\tilde{g}(a)$ and we apply the previous result to show that again $h$ and $k$ are continuous at $a$.\  Now if $f(a)=g(a)$ then we can distinguish three cases. $f\geq g$ in a small neighborhood about $a$, $f\leq g$  in a small neighborhood about $a$ or $f\geq g$ on one side of $a$ and $f\leq g$ on the other side. In the first case we assume that $f\geq g$ in some small neighborhood about $a$. Then, since $f(a)=g(a)$ we have $h(x) = f(x)$ in this neighborhood and $k(x) = g(x)$ and again we see that $h$ and $k$ are continuous at $a$ due to the continuity of $f$ and $g$ at $a$. Similarly if $f\leq g$ in a neighborhood around $a$ then $h(x)=g(x)$ and $k(x)=f(x)$ in this neighborhood and $h$ and $k$ are continuous at $a$.   Suppose that just to the left of $a$ we have $f\geq g$ and just to the right of $a$ we have $f\leq g$. Then $\lim\limits_{x\to a^-}h(x)=\lim\limits_{x\to a^-}f(x)=f(a)=h(a)=g(a)=\lim\limits_{x\to a^+}g(x)=\lim\limits_{x\to a^+}h(x)$. Similarly $\lim\limits_{x\to a^-}k(x)=\lim\limits_{x\to a^-}g(x)=g(a)=k(a)=f(a)=\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to a^+}k(x)$. So $\lim\limits_{x\to a}h(x)=h(a)$ and $\lim\limits_{x\to a}k(x)=k(a)$ which means that $h$ and $k$ are continuous at $a$. Lastly if $f\leq g$ just to the left of $a$ and $f\geq g$ just to the right of $a$ we can relabel $f=\tilde{g}$ and $g=\tilde{f}$ and apply the last result to show $h$ and $k$ are continuous at $a$.   Again, please point out any mistakes I may have made. Thanks!! if $f^{(n)}(x_0),g^{(n)}(x_0)$ exist and $\lim_{x\to x_0} \dfrac{f(x)-g(x)}{(x-x_0)^n}=0$ then $f^{(r)}(x_0)=g^{(r)}(x_0 )$ for all $0\leq r\leq n.$ Show:   if $f^{(n)}(x_0),g^{(n)}(x_0)$ exist and   $$\lim_{x\to x_0} \dfrac{f(x)-g(x)}{(x-x_0)^n}=0$$   then $f^{(r)}(x_0)=g^{(r)}(x_0 )$ for all $0\leq r\leq n.$   My attempt:  let n = 1,  since $f'(x_0),g'(x_0)$ exists then The taylor polynomial exists: $$T_{f,1}(x)=f(x_0)+f'(x_0)(x-x_0), \lim_{x\to x_0} \dfrac{f(x)-T_{f,1}(x)}{(x-x_0)}=0$$  $$T_{g,1}(x)=g(x_0)+g'(x_0)(x-x_0), \lim_{x\to x_0} \dfrac{g(x)-T_{g,1}(x)}{(x-x_0)}=0$$  Then, \begin{align} \lim_{x\to x_0} \dfrac{f(x)-g(x)}{(x-x_0)^n} &amp; =\lim_{x\to x_0} \dfrac{f(x)-g(x)+T_{f,1}(x)-T_{f,1}(x)+T_{g,1}(x)-T_{g,1}(x)}{(x-x_0)^n}\\ &amp;=\lim_{x\to x_0} \dfrac{f(x)-g(x)+T_{f,1}(x)-T_{f,1}(x)+T_{g,1}(x)-T_{g,1}(x)}{(x-x_0)^n}\\ &amp;=\lim_{x\to x_0} \dfrac{f(x)-T_{f,1}(x)}{x-x_0}-\lim_{x\to x_0} \dfrac{g(x)-T_{g,1}(x)}{x-x_0}+\lim_{x\to x_0} \dfrac{T_{f,1}(x)-T_{g,1}(x)}{x-x_0}\\ &amp;=\lim_{x\to x_0}\dfrac{T_{f,1}(x)-T_{g,1}(x)}{x-x_0}\\ &amp;=0 \end{align}  Then, \begin{align} \lim_{x\to x_0}\dfrac{T_{f,1}(x)-T_{g,1}(x)}{x-x_0} &amp;=\lim_{x\to x_0}\dfrac{f(x_0)+f'(x_0)(x-x_0)-g(x_0)-g'(x_0)(x-x_0)}{x-x_0}\\ &amp;=\left(f(x_0)-g(x_0)\right)\lim_{x\to x_0}\dfrac{1}{x-x_0}+f'(x_0)-g'(x_0) \\&amp;=0 \end{align} I am confused because  $\lim_{x\to x_0}\dfrac{1}{x-x_0}$ does not exist. What's the difference between anonymous methods (C# 2.0) and lambda expressions (C# 3.0)? jQuery attribute selector variable The coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)...(1+x^{100})$.  I tried the following concept, how to sum 9 using 1-9 only without repetition.  1:9, 2:1+8,3:7+2, 4: 6+3, 5: 5+4,  6:1+2+6 ,7:1+3+5, 8:2+3+4  The answer is 8.  How will it be solved using Binomial theorem. Why can we distinguish different pitches in a chord but not different hues of light? In music, when two or more pitches are played together at the same time, they form a chord. If each pitch has a corresponding wave frequency (a pure, or fundamental, tone), the pitches played together make a , which is obtained by simple addition. This wave is no longer a pure sinusoidal wave.  For example, when you play a low note and a high note on a piano, the resulting sound has a wave that is the mathematical sum of the waves of each note. The same is true for light: when you shine a 500nm wavelength (green light) and a 700nm wavelength (red light) at the same spot on a white surface, the reflection will be a superposition waveform that is the sum of green and red.  My question is about our perception of these combinations. When we hear a chord on a piano, we’re able to discern the pitches that comprise that chord. We’re able to “pick out” that there are two (or three, etc) notes in the chord, and some of us who are musically inclined are even able to sing back each note, and even name it. It could be said that we’re able to decompose a Fourier Series of sound.  But it seems we cannot do this with light. When you shine green and red light together, the reflection appears to be yellow, a “pure hue” of 600nm, rather than an overlay of red and green. We can’t “pick out” the individual colors that were combined. Why is this?  Why can’t we see two hues of light in the same way we’re able to hear two pitches of sound? Is this a characteristic of human psychology? Animal physiology? Or is this due to a fundamental characteristic of electromagnetism? After the death of Lord Voldemort, Harry was taken to the Dursleys and became known as "The Boy Who Lived".  We know that the Wizarding World found out  that Voldemort was dead and undoubtedly it was Dumbledore that advised people that Harry had somehow survived the attack and caused Voldemort's death. According to Snape (in HP:HBP) there were even persistent rumours that Harry was a powerful Dark Wizard in the making until he turned up at Hogwarts a decade later.  My question is : Where did the Wizarding World think Harry was for the intervening 10 years? How to perform a true GIS clip of polygons layer using a polygon layer in R? I would like to do a true GIS Clip in R of soils polygons using a series of single boundary polygons, but I cannot find an R function to properly do it. It should work just like the clip function in ESRI's ArcMap. I've tried the over method in sp package but it doesn't seem to work for polys over polys.   One suggestion was to use the gIntersection in rgeos package as a clip using the following code:  #------------------------------------ library(rgeos) library(maptools)  #Read layers as SpatialPolygonsDataFrame (both the same Albers projection) Soils_poly = readShapePoly("Soils_polygons")  #Note - Has 400 polygons clipper_poly = readShapePoly("clipper_polygon")  #Note - Has 1 polygon  #Try gintersection as clip  Clipped_polys = gIntersection(Clipper_Tile_poly, Soils_poly)  #-----------------------------------   This takes 5 minutes to run (way too slow) and errors with this:     Error in RGEOSBinTopoFunc(spgeom1, spgeom2, byid, id, drop_not_poly, "rgeos_intersection") :      TopologyException: no outgoing dirEdge found at -721459.77681285271 2009506.5980877089   I also tried this code to check for overlap:  gIntersects(Clipper_Tile_poly, Soils_poly)   and the result was TRUE. clip function in ESRI ArcMap works fine for this data.   Anyone knows of an R function to properly do a clip on spatial polygons using spatial polygons? `--no-install-recommends` only specified packages I want to install xfce4-goodies but I don't want Midori Browser. With --no-install-recommends I don't know how to specify only Midori. I want the rest of packages. How to do that? Thanks! How do you take off body armor in GTA Online? It's kind of hard to shop for clothes with a big bullet proof vest on.  Is there a way to take off the armor after missions, or at least hide it? Deriving input based on other parameters in Python script tool? I have a folder and it contains two feature classes, e.g., a.shp and b.shp. The script tool created has two parameters. The first one is the model location, and the second one needs a feature class (a.shp), as shown in the below picture.  What I really want is every time when I set up the model location for the first input, the second input will automatically find the a.shp as the input rather than the user needs to choose a.shp from that folder. I noticed that the dependency is probably what I really need, but there is no code example. Any suggestions?   I am using ArcGIS 10.3.1 Advanced License. Use chown to set the ownership of all a folder's subfolders and files? Can you create diagrams of chemical structures in LaTeX?   Example(s) of "chemical structure diagrams": I have been attending a number of academic / professional conferences recently and one thing that really awed me was the flawless and natural way in which the speakers presented on their topics.  Sure, they had the occasional mispronunciations or awkward pauses but they spoke with authority and confidence.   Some of them used powerpoint slides but were not reading from the slides; they were just talking on the points and they had so much to say on each point that I felt they could not do it without notes, but they were not even looking at their notes most of the time.  (The advantages of not looking at their notes too much was that they could maintain eye contact and could use very interesting slides that were not crammed with the dot points of their speeches.)  I hate to think that they had memorised their speech notes but some of the presentations went for more than an hour.  I know rehearsal is important but I wonder how could anyone remember so much in a nervous situation. (Its one thing to know everything on your topic; its another thing to present that 'everything'!)  Question:  What background / foreground things do presenters do that make their presentations flawless and natural? How to make a grid-to-sphere animation like this? Let $a_n=\sqrt[n]n-1$. Does $\sum_{n=1}^\infty a_n$ converge? Are there tips and tricks to overcome a fear of flying? Are there tips and tricks for people with fear of flying that can makes them have a more pleasant experience when flying? I know drugs can do magic, and . I'm ideally after official advice and recommendations, but other tips and tricks would be nice as well.