I have a question regarding the concept of &quot;work done&quot; in Classical Mechanics, inspired by my time in real life situation of me the gym. &quot;Work done&quot; depends on the direction of the force and the displacement.  If the force is aligned with the displacement, then the work done is positive.  If the force is anti-aligned with the displacement, then the work dine is negative. This means that, if the force I applied is anti-aligned with the displacement, then energy is transferred to me.  However, in weight lifting, I still need to put force and (therefore) effort when I lower my dumbbell: in essence, I was still using up chemical energy when lowering my dumbbell.  This means that chemical energy is used up, even when the dumbbell transfers energy to me: isn't this a violation of conservation of energy?
I read the definition of  as  $$W ~=~ \vec{F} \cdot \vec{d}$$  $$\text{ Work  =  (Force) $\cdot$ (Distance)}.$$ If a book is there on the table, no work is done as no distance is covered. If I hold up a book in my hand and my arm is stretched, if no work is being done, where is my energy going?
Getting an error that a flow "was" activate and cannot be overwritten A twice differentiable function $f$ satisfies $f′′(x)+f(x)=−xg(x)f′(x)$, $\forall x\ge 0$. Consider $f\in C^2$ so that $$f''(x)+f(x)=-x\,g(x)f'(x), \ \forall x\ge0 $$ where $g(x)\ge 0$. Then ($\forall x\ge 0$)         (A) $f(x)^2+f'(x)^2$ is non-increasing,   (B) $f(x)^2&lt;3f(0)^2+(2f'(0))^2$,   (C) $|f(x)|\le\alpha$, where $\alpha$ is a fixed real constant.   (D) $\lim_{x\to\infty} f(x)\sin\left(\dfrac{1}{x}\right)$ exists.        Answer Given: (A),(B),(C),(D)  What I have tried : I differentiated option (A) and then with the help of the given equation in the question I was able to show that  it is less than zero . Hence I was able to infer that option (A) is correct .   Now coming to option (B).  I assumed the function in option (A) to be $h(x)$.   $$ h(x) = f(x)^2 + f'(x)^2 $$  Since $h'(x)&lt;0$ ,therefore $h(x)$ is a decreasing function   hence for all $x\ge 0$  $h(x)&lt; h(0)$  which implies $(f(x))^2 + (f'(x))^2&lt;(f(0))^2 + (f'(0))^2$  From the above expression we can get the second option .   Is my method correct ?. If not , please show the right method .  I am unable to get the the 3rd and 4th option . What HTML tags are allowed on Stack Exchange sites? Why didn't Star Trek TOS have better medical equipment for Vulcans? In  when Ambassador Sarek became critically ill, why didn't they send him home or send for a Vulcan medical expert? Since they were hosting many prominent political figures from many races, it seems strange that they did not have much medical support for a race so common as Vulcans (who are part of the Federation). Prove that the normalisation of $A=k[X,Y]/(Y^2-X^2-X^3)$ is $k[t]$ where $t=Y/X$ (Reid, Exercise 4.5) This is a problem about finding the normalisation of a quotient polynomial ring. So I have to find the integral closure of the ring in its field of fractions. The problem statement is as follows:     Let $A=k[X,Y]/(Y^2-X^2-X^3)$. Prove that the normalisation of $A$ is $k[t]$ where $t=Y/X$.   Can I do this by showing that the field of fractions $\text{Frac} A$ of $A$ is equal to $k[t]$, and subsequently showing that the field of fractions is normal? (This could be done by showing that $k[t]$ is a UFD?)  I am lost at calculating/determining $\text{Frac}A$, or similarly proving that $k[t]=\text{Frac}A$. Also, how do I show that it is normal?  I hope you can help! Why is the empty set considered an interval? What is the definition of an interval and why is the empty set an interval by that definition? The discrimination between short and long option is only `-`? On Unix system, options begin with - is short option, while options begin with -- is long option. Is my understanding right? hot network questions have inconsistent CSS for visited/not visited depending on the site you're on This is a HNQ excerpt from Stack Overflow. Visited questions are objectively darker.    This is a HNQ excerpt from Super User. Visited questions are objectively lighter.    This is a HNQ excerpt from Seasoned Advice. Visited questions are objectively darker.    This is a HNQ excerpt from Puzzling, a beta site. Visited questions are objectively darker, although the difference is hard to spot.    This is a HNQ excerpt from Travel. Visited questions actually have a completely different color.    These are 5 different sites, with 5 different formats for how the HNQ sidebar displays links. Some make visited links darker. Some make them lighter. Some make them a little big darker. some give visited links a completely different color. This is very confusing: If someone visits one of these HNQ questions and wants to continue browsing the HNQ from there, they likely expect that the links in that HNQ segment have consistent layout compared to the HNQ segment from the website they came from.  Is this something worth looking into? I realize that each site has their own look and feel, but an element that appears on every site should be consistent IMHO. People have expectations about how links work, and one of those is that if you click on a link within the same domain (which the HNQ segment arguably is), they behave similary on that page as they do on the page you come from. I have two passports/nationalities. How do I use them when I travel? I am a citizen of two different countries, and have two passports.  How should I use my passports when traveling? how many distinct values does it have? How can I remove the qed symbols from all lemma proofs? the following code will only make the qed not appear in the second proof. However, it it possible to make the qed symbol never appear at the end of the proofs without having to specify it locally every time?  \documentclass{article} \usepackage{amsthm} \begin{document} \begin{proof} This has the QED symbol. \end{proof} \begin{proof}\renewcommand{\qedsymbol}{} This hasn't. \end{proof} \begin{proof} And this has it again. \end{proof} \end{document} Whenever the Doppler effect is mentioned, it's typically in the context of sound waves or electromagnetic radiation. On the cosmological scale, red-shifting is also important because of the enormous speed of receding galaxies, thanks to the expansion of the universe.   Yet, red-shift is always discussed as the red-shifting of electromagnetic waves. Can gravitational waves be red-shifted? If so, could observations of them be used like red-shifted electromagnetic waves from distant sources are; that is, to figure out how fast an object is receding? Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$ What happens, if a rocket is filled with a vacuum instead of high pressured air? Suppose you put wheels under a compressed air tank    so that it can move horizontally to the right and to the left. Suppose there is a nozzle on the right hand side of the tank (in the picture on the backside, if you like).  If the tank is filled with high pressured air and you open the nozzle, the tank will move to the left. This is how rockets work and one can easily imagine that.   What happens when the tank is filled with a vacuum and you open the nozzle?  My first impression was, that the tank is moving to the right but someone told me, that it would be stand still. Is this true? What is an intuitive explanation for that? Non-finite clause complementation of complex transitive verbs This question has been bothering me for a while. It came up when I was reading Chapter 16 of "A Comprehensive Grammar of the English Language."   How to explain the grammatical structure of the following four sentences?    "We knew him to be a spy." "I saw her leave the room." "I heard someone shouting." "I got the watch repaired."    What category do the four non-finite clauses in boldface fall into? They do not seem to fall into any of relative, nominal, comparative and adverbial clauses. How can you make an armor that has infinite protection armor? When I was using the new attribute command, I saw an attribute called &quot;armor.&quot; I don't know what it actually does, but I think it has to do with protection. I was wondering if I could make an armor that has infinite protection so I tried to make a command: give @s netherite_helmet{AttributeModifiers:[{AttributeName:&quot;generic.armor&quot;,Name:&quot;generic.armor&quot;,Amount:1.79769313486232E+308d,Operation:0,UUID:[I;439,684,724,99]},Slot:&quot;head&quot;}]}  I thought it would give me infinite protection, but it only gives me a helmet with the empty &quot;protection&quot; attribute. It works fine on older versions, but not on the 1.17 snapshots. I have the following problem:  Let $a, b \in\mathbb{Z}$. Show that $\,\{ ax + by\ :\ x, y \in \mathbb{Z}\} = \{ n \gcd(a,b)\ :\ n\in \mathbb{Z} \}$   I understand that the Bezout's lemma says that $gcd(a,b) = ax +by$, so Im not really how you would go about proving the above, it doesn't really make sense to me. Any help is appreciated1 How can I tell if an ETF is expensive? For stocks, I can easily look at the price/earnings ratio, and, according to The Intelligent Investor, stand clear of those with a ratio higher than 20.  How can I tell if the current price of an ETF is a good price to buy it? Are there such simple metrics? How to make a circular array using offset object as center? We know that the there is an Object Offset option for Array modifier, but when trying to make a circular array with that offset object as the center of the array circle, it always doesn't behave quite like what's expected. So how to make a nice and controllable circular array?  P.S.: There is an old  about moving the object's origin. I wonder if there is any way to do this without changing the location of the origin. :) How to handle dynamic changing ID's In XPath?