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1. ### Inverse Laplace Transformation

Homework Statement Solve the following: $$\mathscr{L}_s^{-1} \left\{ \frac{s}{s^2-s+\frac{17}{4}} \right\}$$ Homework Equations Table of Laplace Transforms. The Attempt at a Solution The solution is $$f(t) = (1/4 )e^{t/2} (\sin(2 t)+4 \cos(2 t))$$ I know I need to break up ##F(s)## into...
2. ### Is there an analytic solution to this system of equations?

Thank you for the in-depth explanation to a numerical approach, TheoMcCloskey! The technique you outlined is just beyond my current mathematical level of understanding, but it's educating to see a technique like that in action, and it's definitely brought up some points to research. Does your...
3. ### Is there an analytic solution to this system of equations?

I have the following system of equations with variables ##a,m##, and I'm wondering—can this system be solved symbolically/analytically? \begin{align} m &= 100 + \frac{ \left( 200 \frac{\ln{\frac{1}{2}}}{26.8} \right) }{\left(\dfrac{\ln{\frac{1}{2}}}{26.8} + a \right)} \\ \\ 50 &= me^{-a\left(...
4. ### What does ##\bar{x}_{\textrm{el}}## represent?

Thanks for your response, mathman. Here are some resources that utilize this notation: (page 1) http://www.sut.ac.th/engineering/Civil/CourseOnline/430201/pdf/05_review.pdf Vector Mechanics for Engineers:Statics and Dynamics by Ferdinand P. Beer & E. Russell Johnston J. Below is an example...
5. ### What does ##\bar{x}_{\textrm{el}}## represent?

In the context of centroids and moments, what do ##\bar{x}_{\textrm{el}}## and ##\bar{y}_{\textrm{el}}## represent? For example: $$\bar{x}L = \int \bar{x}_{\textrm{el}}dL$$
6. ### Reducing the function of a function to the ind. variable

Thanks for your enlightening responses, Andrew!
7. ### Least Squares Derivation—Simple Algebraic Simplification

Thank you, TitasB!
8. ### Least Squares Derivation—Simple Algebraic Simplification

Hey, PF I'm reading the following derivation of least squares, and I'm trying to figure out how the author went from the last step at the bottom of pg. 7 to the final equation (11) at the top of pg. 8. [http://isites.harvard.edu/fs/docs/icb.topic515975.files/OLSDerivation.pdf] More...
9. ### Reducing the function of a function to the ind. variable

Awesome explanation! That was exactly what I was trying to express, but you managed to say it more elegantly. I think I'm beginning to understand this problem at a deeper level now. By numerical techniques, I assume you mean we cannot apply symbolic techniques in those cases. I would also...
10. ### Reducing the function of a function to the ind. variable

Thanks for the example and clarification, Stephen! So I suppose I used the wrong notation to express my question. It still seems to me that ##A(t)## and ##B(t)## can be use to express either ##B## or ##A## as an implicit function of ##t##. This would be my ##A## as a function of ##B## or vice...
11. ### Reducing the function of a function to the ind. variable

Thanks for your responses, Fredrik and Andrew (I'm still trying to understand your post, Fredrik). I ran into function composition and decomposition while trying to understand this problem, and if I recall correctly, it seems like most function composition consists of substitution (which I...
12. ### Reducing the function of a function to the ind. variable

Hey, PF: If I have a function ##f(x)## where ##x## is itself a function of another variable (say time), is the following then true? $$f(x)=f(x(t))=f(t)$$ I ask this because if I have the following system of differential equations $$\frac{dA}{dt}=-Bb$$ $$\frac{dB}{dt}=-Aa$$ where litte ##a##...
13. ### Unfixed charges released from rest

Thanks, Merlin! I'm glad I was on the right track. I was able to get the correct answers to the rest of the problem thanks to your help. Best!
14. ### Unfixed charges released from rest

Homework Statement a.) [/B]A proton and an "positron" (identical to an electron, except positively charged) are brought ##5µm## apart and released from rest. What is the initial potential energy stored by this system? b.) In all of the previous problems on this homework, the system's...
15. ### How do we define "linear" for single and multivariable?

Thanks for the updated feedback, Simon. According to WolframAlpha, the DE is a first-order linear ordinary differential equation http://www.wolframalpha.com/input/?i=y%27%3Dxy Also, could you please define second order? Is the term in this given context synonymous with second degree?
16. ### How do we define "linear" for single and multivariable?

Thanks, Simon! The fact that the terms in the DE are not linear clarifies some things.
17. ### How do we define "linear" for single and multivariable?

In my multivariable calculus class, we briefly went over Taylor polynomial approximations for functions of two variables. My professor said that the second degree terms include any of the following: $$x^2, y^2, xy$$ What surprised me was the fact that xy was listed as a nonlinear term. In...
18. ### Point Charge Given Field Maps

Thank you for your replies! You were right; each field line represents an equal fraction of the original charge. The answer came out to around $$-7.1 \mathrm{C}$$.
19. ### Point Charge Given Field Maps

Homework Statement The electric field map is shown below. (Take it to be a self-consistent scale drawing at a particular moment in time.) where \mathrm{q}_4 = +10 C, \mathrm{q}_6 = +10 C What is the charge of \mathrm{q}_5 ? Homework Equations The Attempt at a Solution I've tried the...
20. ### Electric Field of Point Charges in Equilateral Triangle

Thank you for your feedback, Simon! The answer I posted was indeed correct.
21. ### Where does friction come from?

Have you tried: http://en.wikipedia.org/wiki/Friction ? "Friction is not itself a fundamental force but arises from interatomic and intermolecular forces between the two contacting surfaces. "
22. ### Electric Field of Point Charges in Equilateral Triangle

Homework Statement A point charge q =+6.4 \mu \mathrm{C} is placed at each corner of an equilateral triangle with sides 0.29 \mathrm{m} in length. What is the magnitude of the electric field at the midpoint of any of the three sides of the triangle? Hint: if you are careful to examine the...
23. ### Partial Fraction Decomposition—Multiple Variables

What's the best approach to solving the partial-fraction decomposition of the following expression? $$\frac{1}{(a-y)(b-y)}$$ The expression is not of the following forms: But I know the solution is $$= \frac{1}{(a-b)(y-a)}-\frac{1}{(a-b)(y-b)}$$
24. ### Net Ionic Equation of Four Solutions

The problem statement The solutions of ##NaC_2H_3O_3 + BaCl_2 +NH_4NO_3 + K_2CrO_4## are mixed together. 1. Write the Net Ionic Equation. 2. What is the driving force of the reaction? The attempt I'm not sure how to solve for the net ionic equation without a balanced equation. My...
25. ### Mechanics: Acceleration in Pulley System

1. The problem A 3.0-kg block sits on top of a 5.0-kg block which is on a horizontal surface. The 5.0-kg block is pulled to the right with a force ##\overrightarrow{F}## shown in the figure. The coefficient of static friction between all surfaces is 0.65 and the kinetic coefficient is...
26. ### Tension between two sliding blocks

Homework Statement The problem: Two blocks made of different materials connected together by a thin cord, slide down a plane ramp inclined at an angle θ to the horizontal as shown in the figure (block B is above block A). The masses of the blocks are mA and mB and the coefficients of...

29. ### General Chem—Telluric Acid

Thanks, Jim. So would ##H_2TeO_4\cdot2H_2O## then be considered a hydrate due to the presence of water molecules within it? In addition, how are we supposed to know that telluric acid is a hydrate? (i.e., that it has a ##nH_2O##)
30. ### Tension on a Tyrolean Traverse

Homework Statement Christian is making a Tyrolean traverse as shown in the figure. That is, he traverses a chasm by stringing a rope between a tree on one side of the chasm and a tree on the opposite side, 25m away. The rope must sag sufficiently so it won't break. Assume the rope can...
31. ### Integral with e^x

Step 1. Try to simplify the problem (i.e., take any constants out of the integral). Step 2. You may want to try u-substitution.
32. ### General Chem—Telluric Acid

According to Wikipedia, the proper chemical formula for Telluric Acid is Te(OH)_6. From what I've learned in chemistry thus far, acids made from polyatomic ions within the same group follow a similar pattern. For example, Selenic Acid H_2SeO_4 is analogous to Sulphuric acid H_2SO_4. If this...
33. ### Vectors: Average Velocity, Acceleration, Speed

Thank you, ehild. You were correct in stating that A and C cannot be solved, and I was able to compute the average acceleration by breaking the vectors into their components.
34. ### Vectors: Average Velocity, Acceleration, Speed

Homework Statement A car is moving with speed ##17.4m/s## due south at one moment and ##28.3m/s## due east ##7.50s## later. A. Determine the magnitude and direction of its average velocity over this time interval. B. Determine the magnitude and direction of its average acceleration over...
35. ### Standard Deviation Conceptual [intro. Stats]

Ah, bounded was the word I was looking for! Thank you, Stephen. Would the example then be considered a Truncated normal distribution ? If this is the case, what would 2.5 or 3 standard deviations imply in relation to the three-sigma rule when the values simply cannot extend beyond the...
36. ### Standard Deviation Conceptual [intro. Stats]

Hello, PF! [My question pertains to a non-rigorous, undergraduate introductory Probability and Statistics course. I'm no math major, so please correct me if I've mishandled any terms or concepts as I try to express myself. I'm always eager to learn!] * * * In a discussion of the...