THE PROBLEM  Consider a pendulum with an hollow spheric bob of negligible weight filled with 1.1 kg of fine sand and hanging from an inextensible wire of negligible mass and $l= 30 m$ long. At the point which is diametrically opposite to the point where it is hung to the wire, the sphere has a small hole from which the sand can come out with a constant flow of 1 g/s.  In the rest position the hole is at a distance $h = 1 m$ from the horizontal floor. At time $t = 0$ the sphere is placed at a distance $d = 30 cm$ from its rest position, the hole is opened and the sphere is let go (see the drawing for a better understanding).  Neglecting any friction and referring to a Cartesian axis that runs along the floor in the direction of the motion of the sphere and whose origin is at the point closest to the rest position of the sphere, compute:   the position where the sand falls as a function of the time $t$; the maximum distance at which the sand is deposited; the points where the greatest amount of sand is accumulated; the distribution of the mass of sand on the floor.   Assume that the sand remains exactly at the point where it touches the floor.  HINT Some useful trigonometric formulas: $$cos(arctan(x))=\frac{1}{\sqrt{1+x^2}}\qquad\qquad \sin(arctan(x))=\frac{x}{\sqrt{1+x^2}}$$    (WORKING ON) THE SOLUTION  I suppose that the first thing to notice is that the little-angle approximation can be used since the starting angle $\vartheta_{\text{MAX}}$ (at $t=0$) is easily computed as $$\vartheta_{\text{MAX}}=arcsin(d/l)=arcsin(0.01)\simeq 0.01000017$$  and it's very small. Hence $sin(\vartheta)$ can be approximated with $\vartheta$ and $cos(\vartheta)$ with $1$.  In this approximation, it is well known that the pendulum period is $$T = 2\pi\sqrt{\frac{l^*}{g}}$$ Now, strictly speaking, $l^*$ should be the distance between the point in which the wire is hung and the center of mass of the bob. At $t=0$ this means $l^* = l+R$, where $R$ is the radius of the hollow sphere. With the sand flowing away, the center of mass will shift downwards, increasing $l^*$ and hence increasing the time period only to be back at its initial position $l+R$ when there's no sand left inside the bob.  However, since the problem does not give the value for $R$, I assumed that a point-like approximation for the bob could be used, hence using $l^* = l$ regardless of how much sand is left inside the sphere.  I guessed this is what who wrote the problem was asking for since the period so computed is $T \simeq 11s$: a number matching too well the $1.1 g/s$ of the constant flow of sand to be a coincidence (giving the sphere emptied in $100$ periods).  Assuming all of the above is right, I went on to write the equations of motion for the flowing sand:   \begin{array}{ll}         x(t) &amp;= x_s + v_x t = l\,sin(\vartheta(t)) + l \dot{\vartheta(t)} t \simeq l\,\vartheta(t) + l \dot{\vartheta(t)} t\\         y(t) &amp;= y_s + v_y t - \frac{g}{2}t^2= h + l(1-cos(\vartheta)) + l \dot{\vartheta(t)} sin(\vartheta(t)) t - \frac{g}{2}t^2 \simeq h + l \dot{\vartheta(t)} \vartheta(t) t - \frac{g}{2}t^2         \end{array}    Then I'd like to set $y(t)=0$ in order to solve for $t$ and substitute inside the equation for $x(t)$...but I don't know how to proceed. Plus, how should the given HINT be useful?  Any help to solve this would be extremely appreciated!  EDIT (why this is not a duplicate)  This question is not a duplicate because I'm not asking of how the period of pendulum varies with the flowing of the sand but, assuming it does not vary (an approximation), I'm asking about the position where the sand falls as a function of the time $t$; the maximum distance at which the sand is deposited; the points where the greatest amount of sand is accumulated; the distribution of the mass of sand on the floor.
How do you find time period as a function of time for a simple pendulum that is in the form of a hollow sphere that is filled with mercury and there is a hole in the bottom through which the mercury is constantly falling at a fixed rate? I tried creating a function for the time period to check how it varies with the mass of mercury, but I had too many variables.   EDIT: This question was a concept based multiple choice question in my homework book. The option were time period becomes erratic, time period increases, time period decreases, time period first increases then decreases.
Continuous Images of $GL_n(\mathbb{R})$ Which of the following is a continuous image of $GL_n(\mathbb{R})$?  The real line $\mathbb{R}$  The subspace $\{(x,\frac1x):x\in \mathbb{R},x \ne 0\}$ of $\mathbb{R}^2$  $\mathbb{R}^2 \setminus \{(x,\frac1x):x\in \mathbb{R},x \ne 0\}$  $\{(x_1,x_2,...,x_n):x_1^2+x_2^2+...+x_n^2 \le 1\}$ of $\mathbb{R}^n$   How do I proceed? $GL_n(\mathbb{R})$ is not compact, not connected, so I cannot rule out some options using topological properties. Colorize symbols with lstdefinelanguage in latex Getting Acquaintances to become Friends Induction Proof: Fibonacci Numbers Identity with Sum of Two Squares Using induction, how can I show the following identity about the fibonacci numbers? I'm having trouble with simplification when doing the induction step.  Identity: $$f_n^2 + f_{n+1}^2 = f_{2n+1}$$  I get to:  $$f_{n+1}^2 + f_{n+2}^2$$  Should I replace $f_{n+2}$ using the recursion? When I do that, I end up with the product of terms, and that just doesn't seem right. Any guidance on how to get manipulate during the induction step?  Thanks! This is a follow-up question to the discussion about the finite axiom of choice .  Suppose we have a countable collection of non-empty sets $\{A_1, A_2, A_3,\cdots\}$ Reasoning as indicated in that discussion, we can prove the existence of a choice function $c_n$ such that $c_n(i)$ belongs to $A_i$ for $i = 1,\cdots,n$  Again reasoning as suggested and using induction, we can prove the existence of a function $C$ defined over $\{1, 2, \cdots\}$ such that $C(n)$ is a choice function defined for $i=1,\cdots,n$ with $C(n)(i)$ an element of A_i. We can require in addition that $C(n)(i)$ and $C(n-1)(i)$ are equal for $i=1,\cdots,n-1$.  Now it would it seem that we can prove the countable axiom of choice by taking the union of $C(1)$, $C(2)$, $\cdots$  The question is as follows: why is that proof wrong? I suspect the answer may have something to do with the axiom of union. DOS commands for Epson dot matrix not working Our Epson dot matrix printer is not printing on the following DOS command.:  Type text.txt&gt;prn   But it does print perfectly on:  open TEXT.txt&gt;File&gt;Print    What is the issue here? Elsevier actions following US sanctions on Iran Deputy badge after a reputation of 3000 Resistor in series needed with LED string? What is the status icon that looks like a house? There is a strange icon that shows up in my status bar only when I'm at a certain restaurant. It looks like a cross between a house and wifi.   I suspect it might be a free wifi hotspot or something related to a nearby phone store, but I can't find any information about it in manual for my phone.  See the following screenshot: Which is the preferred way to convert an Enum to a String in .NET 3.5?   Enum.GetName Enum.Format ToString   Why should I prefer one of these over the others? Does one perform better? $f,\overline f$ are both analytic in a domain $\Omega$ then $f$ is constant? is it true if $f,\overline f$ are both analytic in a domain $\Omega$ then $f$ is constant? I am not able to  find out what property of holomorphic map I need to apply. please help.Thank you.  $f(z)=u(x,y)+iv(x,y)$, $\bar{f}(z)=u(x,y)-iv(x,y)$ Difference between confidence intervals and prediction intervals For a prediction interval in linear regression you still use $\hat{E}[Y|x] = \hat{\beta_0}+\hat{\beta}_{1}x$ to generate the interval. You also use this to generate a confidence interval of $E[Y|x_0]$. What's the difference between the two? Let $$G_\beta(w) = e^{\beta w^2}$$  Now I get the process of applying a fourier transform (or inverse) to get a new  gaussian:  $$G_\beta(x) = G_\beta(0) e^{\frac{-x^2}{4\beta}}$$  but in doing the derivation the constant $G_\beta(0)$ never arose. I know how to calculate the constant, but where did it come from in the derivation?  edit: Fourier form used: $$\int_{-\infty}^{\infty} f(w) e^{-iwx} dw$$ How do I delete a file whose name begins with "-" (hyphen a.k.a. dash or minus)? Can Reduce *really* not solve for x here? Can you change your email for Pokemon Go? How do I format a String in an email so Outlook will print the line breaks? I'm trying to send an email in Java but when I read the body of the email in Outlook, it's gotten rid of all my linebreaks. I'm putting \n at the ends of the lines but is there something special I need to do other than that? The receivers are always going to be using Outlook.   I found a page on microsoft.com that says there's a 'Remove line breaks' "feature" in Outlook so does this mean there's no solution to get around that other than un-checking that setting?  Thanks For any prime $p > 3$, why is $p^2-1$ always divisible by 24? Does python optimize modules when they are imported multiple times? If a large module is loaded by some submodule of your code, is there any benefit to referencing the module from that namespace instead of importing it again?  For example: I have a module MyLib, which makes extensive use of ReallyBigLib. If I have code that imports MyLib, should I dig the module out like so  import MyLib ReallyBigLib = MyLib.SomeModule.ReallyBigLib   or just  import MyLib import ReallyBigLib