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    Falling Rain Drop (Variable Mass)

    Yes it does thanks. Interesting for some reason I had it in my mind that they had to be polynomials. $$\frac{dv}{dt}+\frac{k}{m}v=g $$ $$\ln P(t)=\int \frac{k}{m_0+kt}dt=\ln{\left (m_0+kt\right )}+C $$ $$P(t)=e^C(m_0+kt)$$ $$\frac{d}{dt} ( v(m_0+kt)e^C ) = (m_0+kt)e^Cg$$ $$v(m_0+kt)-v_0m_0 =...
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    Falling Rain Drop (Variable Mass)

    $$mg = \frac{d}{dt}(mv)$$ $$d(mv) = mg\cdot dt$$ $$\int_{t=0}^{t=t} d(mv) = g\int_0^t(m_0+kt)dt$$ $$mv-m_0v_0 = g(m_0t+kt^2/2)$$ $$v = \frac{m_0v_0+g(m_0t+kt^2/2)}{m_0+kt}$$ Their expression for velocity is so simplified in the paper I can't even tell if this is correct. I have basically no...
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    Falling Rain Drop (Variable Mass)

    I was looking through my posts when I stumbled upon this one and I can't understand how they're solving the differential equation in the paper that was linked in response to this post. The author states that when ##\frac{dm}{dt}## is independent of velocity then the accretion equation can be...
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    I Stuck on the probability of rolling 'p' with 'n' s-sided dice

    I'm still finding this quite confusing. I understand what you're saying here that these are the boundaries for ##l## using ##k##. Then it seems like the last summation can be written as $$ \sum_{l=p-n}^{p-n-sn}\binom{p-sk-1}{p-sk-n}x^{p-sk-n} $$ but since ##l## has been replaced we also need...
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    I Stuck on the probability of rolling 'p' with 'n' s-sided dice

    Hi, I've been following the derivation of wolfram mathworld for this problem and I'm running into some trouble regarding the summation indices. Currently I am at the step where we have found that (it's pretty much just binomial expansion and taylor series to get to this point) $$ f(x) =...
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    Find B and H everywhere for a magnetized infinite cylinder

    Thanks! This pieces everything together.
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    Find B and H everywhere for a magnetized infinite cylinder

    Yes I agree that geometry doesn't make much sense to me either I was just copying what they had done. Setting up the problem as $$\vec{B}(2\pi s) = \mu_0(\int_s^a J_d\cdot 2\pi s\, ds - M_0\cdot 2\pi a) $$ results in the answer of $$\vec{B} = -\mu_0 M_0 (s/a)^2\hat{\phi}$$ which is just the...
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    Find B and H everywhere for a magnetized infinite cylinder

    Ok, the difference was in the book that their ##\vec{J_b}## was a constant so they could just multiply by the area to get the result of the integral however, my ##\vec{J_b}## is linear w.r.t. s although I am still running into trouble. Defining ##dA =l \,ds\Longrightarrow \int_s^a J_d\cdot dA...
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    Find B and H everywhere for a magnetized infinite cylinder

    Homework Statement An infinitely long cylinder of radius a has its axis along the z-direction. It has Magnetization ##M=M_0(s/a)^2\hat{\phi}## in cylindrical coordinates where ##M_0## is a constant and s is the perpendicular distance from the axis. Find the values of ##\vec{B}## and ##\vec{H}##...
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    Electric field inside a uniformly polarized cylinder

    Am I drawing it incorrectly? I still don't find the same result as you.
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    Electric field inside a uniformly polarized cylinder

    Homework Statement This is problem 4.13 from Griffiths. A long cylinder of radius a carries a uniform polarization P perpendicular to its axis. Find the electric field inside the cylinder. Homework Equations ##\int \vec{E}\cdot dA = q_{encl}/\varepsilon_0## The Attempt at a Solution [/B] We...
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    Drawing a Timing Diagram

    Hmm, now I'm confused because depending on whether or not I start at the top or bottom gate outputting 0 I will end up with either ##Q=1##, ##Q'=0## or ##Q=0##, ##Q'=1##
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    Drawing a Timing Diagram

    Homework Statement Draw the diagram for the following circuit given the following conditions: 1) X=Y=Z=1 2)X=Y=1, Z=0 3)X=Y=0, Z=1 4)X=1, Y=Z=0 Homework Equations The Attempt at a Solution [/B] ##W=XZ'+YZ##, ##V=Y'Z+XY## 1) W = 0 + 1 = 1 V = 0 + 1 = 1 and now I'm not sure how to get the...
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    Calculating the surface charge of a sphere and a conducting shell

    So in an insulator the electrons can't flow freely therefore they won't be able to redistribute across the surface? Yes, is it just a conductor because it's metal?
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    Calculating the surface charge of a sphere and a conducting shell

    I thought that charge only entirely resided on the surface of conductors otherwise why would they mention this as a property of conductors and not just in general? After looking around it seems like the charge will always distribute across the surface of anything in order to minimize the...
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    Calculating the surface charge of a sphere and a conducting shell

    Homework Statement (Problem 2.38 From Griffth's Electrodynamics): A metal sphere of radius R, carrying charge q, is surrounded by a thick concentric metal shell (inner radius a, outer radius b). The shell carries no net charge. Find the surface charge density ##\sigma## at R, a, and b...
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    I Why can we WLOG derive Simpson's rule over interval -1 to 1

    On the Simpson's Rule wikipedia page they mention in their derivation that the calculation can be simplified if one notices that there is no loss in generality in setting ##a=-1## and ##b=1## for the integral ##\int_{a}^{b}P(x)\cdot dx## as a result of scaling. I'm not entirely sure what...
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    Determining if a card is in the same pile as another card

    While the two statements are equivalent only one will be stored as a fact. A pile could be expressed as a bunch of above(x,y) facts, under(x,y) facts, or a combination of both so it seems necessary to include both.
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    Determining if a card is in the same pile as another card

    Homework Statement Write a predicate to determine if two cards are in the same pile. The placement of the cards is given as facts above(x,y), x is above y, or below(x,y), x is below y. I'm supposed to do this using Prolog which is a first-order logic language. Homework Equations The Attempt...
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    Optimal Path from source to multiple destinations

    Ok makes sense. What if instead of the optimal path we are interested in a reasonably accurate approximation?
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    Optimal Path from source to multiple destinations

    Homework Statement You are placed in a 2-dimensional labyrinth at a starting location ##s## and must travel to n goal locations ##g_1, g_2, ..., g_n##. Determine the optimal path using the A* algorithm. g = goal cell, s = start cell example input: 3 11 //row length, column length # # # # # # #...
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    For a round-robin of n teams find the number of different outcomes

    Oh, I believe I need to rewrite the proof anywayssince it relied on ##a_i < a_{i+1}##
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    For a round-robin of n teams find the number of different outcomes

    Yes I suppose that doesn't make sense after how you've defined it.
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    For a round-robin of n teams find the number of different outcomes

    Im a bit confused as to why we've defined ##a_i < a_{i+1}## instead of ##a_i \leq a_{i+1}## since this doesn't have to hold for tournaments in general. Proof: Consider the series ##A=a_1+a_2+\cdots +a_n## with ##0\leq a_1<a_2<\cdots < a_n## then ##\sum a_i=\frac12n(n-1)=0+1+\cdots+(n-1)## then...
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    For a round-robin of n teams find the number of different outcomes

    Whoops forgot about that. For some reason I thought I would be able to avoid the constraints when coding an algorithm, clearly this is not the case. I took this code for the unique partitions of an integer and implemented some basic checks to see if the current partition could be a potential...
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    For a round-robin of n teams find the number of different outcomes

    Yikes. Just to make sure I'm understanding this correctly, we are solving for the cases in which one team beats all the others or one team loses to all the others in order to remove sums involving a 0 as one of the digits which then allows us to sum that with the partitions of n(n-1)/2 that...
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    For a round-robin of n teams find the number of different outcomes

    Yep there are no draws. We only care about the overall result and not which team beat which. E.g. Team A:(1-0) and Team B:(0-1) is the same as Team A:(0-1) and Team B:(1-0). Yes the way Orodruin explained it is correct. Hopefully my reply to Haruspex clarifies any confusion about order.
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    For a round-robin of n teams find the number of different outcomes

    Homework Statement In a round-robin tournament each team plays every other team once, find the number of different outcomes possible for ##n## teams. e.g. for 4 teams the possible outcomes are: |3-0 | 3-0 | 2-1 | 2-1 |...
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    Showing equivalent potential expressions for a Transverse String

    Hmmm, I can see that this gives the same result as the answer but I don't understand it. It's written in the form ##V_1=\frac{1}{2}(k)d^2## where you subbed in the effective spring constant however I can't understand why we are allowed to write the displacement of the string as merely the...
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