Polar Definition and 1000 Threads

I attempted to solve this problem by finding the angles of an intersection point by equalling both ##r=sin(\theta)## and ##r=\sqrt 3*cos(\theta)##. The angle of the first intersection point is pi/3. The second intersection point is, obviously, at the pole point (if theta=0 for the sine curve and...

$$-2\sin\theta=1\Leftrightarrow\theta=-\frac{\pi}{6},\,-\frac{5\pi}{6}\\ \begin{align*} \int_{-\frac{\pi}{6}}^{-\frac{5\pi}{6}}\frac 12\left(4\sin^2\theta-1\right)d\theta &=\int_{-\frac{\pi}{6}}^{-\frac{5\pi}{6}}\frac 12\left(1-2\cos2\theta\right)d\theta\\...

m a

The velocity of a particle below is expressed in polar coordinates, with bases e r and e theta. I know that the length of a vector expressed in i,j,k is the square of its components. But here er and e theta are not i,j,k. Plus they are changing as well. Can someone help convince me that the...

What does "F(r,θ) = 0" mean here?

If I have a physical problem, say, a particle which is constrained to move in the ##y## direction, which means that its ##x## coordinate remains fixed, does it make sense to write ##y## in terms of polar coordinates? That is, ##y = r \sin\theta##. Since now I have two parameters ##r,\theta##...

Hi, A well-known part of the formula for calculating the deflection stress is ##I_z=\int \int r^2 dA## Usually a moment of inertia is something related to how difficult is to move an object. In this case is understandable but i don't understand the meaning of the double integral. Using ##r^4##...

This is what I have so far, please need urgent help. I don't understand and know what to do. For the first part, I got a really long answer, for the second part I am trying in terms of mv^2/r = mg, or mg = m*(answer to first), but I am getting nowhere. PLease help

Ive found ##\delta x/\delta r## as ##sin\theta cos\phi## ##\delta r/\delta x## as ##csc\theta sec\phi## But unsure how to do the second part? Chain rule seems to give r/x not x/r?

When you graph something like ##θ=\frac{π}{4}## on a Polar Coordinate System: Why does the line go into the opposite quadrant as well? I can intuitively understand why it is in the first quadrant: ##θ = 45°## there and so all possible values of ##r## would apply there, giving you a straight line...

In spherical poler coordinates the volume integral over a sphere of radius R of $$\int^R_0\vec \nabla•\frac{\hat r}{r^2}dv=\int_{surface}\frac{\hat r}{r^2}•\vec ds$$ $$=4\pi=4\pi\int_{-\inf}^{inf}\delta(r)dr$$ How can it be extended to get $$\vec \nabla•\frac{\hat r}{r^2}=4\pi\delta^3(r)??$$

Hello there, I'm struggling in this problem because i think i can't find the right ##\theta## or ##r## Here's my work: ##\pi/4\leq\theta\leq\pi/2## and ##0\leq r\leq 2\sin\theta## So the integral would be: ##\int_{\pi/4}^{\pi/2}\int_{0}^{2\sin\theta}\sin\theta dr d\theta## Which is equal to...

Why first terms equal to zero ? ? ?

The area differential ##dA## in Cartesian coordinates is ##dxdy##. The area differential ##dA## in polar coordinates is ##r dr d\theta##. How do we get from one to the other and prove that ##dxdy## is indeed equal to ##r dr d\theta##? ##dxdy=r dr d\theta## The trigonometric functions are used...

I was playing around with a graphing program and sketching polar graphs involving tall power towers, when I noticed that ##sin(\theta) \uparrow \uparrow a## has an alternating appearance depending on whether ##a## is odd or even. I also noticed that the area enclosed by these alternating graphs...

1) the motion equations for ##m_2## are: $$T-m_2 g=0 \rightarrow T=m_2 g$$ ##m_1##: $$T=m_1\frac{v^2}{r_0} \rightarrow \vec {v_0}=\sqrt{\frac{r_0 g m_2}{m_1}}\hat{\theta}$$ 2) This is where I am stuck, first I wrote ##m_2## motion equation just like before, but in polar coordinates...

A solution of equations of motion for charged particle in a uniform magnetic field are well known (##r = const##, ## \dot{\phi} = const##). But if I tring to solve this equation using only mathematical background (without physical reasoning) I can't do this due to entaglements of variables...

I have a little question about converting Velocity formula that is derived as, ##\vec{V}=\frac{d\vec{r}}{dt}=\frac{dx}{dt}\hat{x}+\frac{dy}{dt}\hat{y}+\frac{dz}{dt}\hat{z}## in Cartesian Coordinate Systems ##(x, y, z)##. I want to convert this into Polar Coordinate System ##(r, \theta)##...

I considered the work done by the frictional force in an infinitesimal angular displacement: $$dW = Frd\theta = (kr\omega) rd\theta = kr^{2} \frac{d\theta}{dt} d\theta$$I now tried to integrate this quantity from pi/2 to 0, however couldn't figure out how to do this$$W =...

Well, I drew the polar and standard axis centered in the particle and wrote which angles were equal to 60° so I could decompose the velocity. The problem says "moves towards it (the radar) with velocity v=5 m/s, so that's one of the components. But I realized that the velocity "cuts" the angle...

Which HPLC column can separate high polar compounds?

I am modeling a cylindrical source in MCNP6 and would like to use the FMESH tally in cylindrical coordinates. I am looking for the dose to water from the source as a function of radial distance as well as polar angle running from 0 to 180 degrees in the YZ plane not around Z. Is there a way to...

What is the advantage of using a polar coordinate system with rotating unit vectors? Kleppner's and Kolenkow's An Introduction to Mechanics states that base vectors ##\mathbf{ \hat{r}}## and ##\mathbf{\hat{\theta}}## have a variable direction, such that for a Cartesian coordinates system's base...

the graph of x= √4-y^2 is a semicircle or radius 2 encompassing the right half of the xy plane (containing points (0,2); (2,0); (0-2)) the graph of x=y is a straight line of slope 1 The intersection of these two graphs is (√2,√2) y ranges from √2 to 2. Therefore, the area over which we...

##{dx}^2+{dy}^2=3^2+3^2=18## ##{dr}^2+r^2{d\theta}^2=0^2+3^2*(\theta/2)^2\neq18## I have a feeling that what I'm doing wrong is just plugging numbers into the polar coordinate formula instead of treating it as a curve. For example, I naively plugged in 3 for r even though I know the radius...

Does direct radiation increase as the zenith increases? I know the diffuse radiation increases as zenith does, but is it the same for direct Also is it right to say that when diffuse radiation increase, the direct should decrease and vice versa? edit: My book says diffuse increases as we...

I am labelling this as undergraduate because I got it from an undergraduate physics book (Tipler and Mosca). The uniform semicircle has radius R and mass M. I am getting the wrong answer but I can't see where I am going wrong. Any help would be appreciated. My solution: The centre of mass...

A) Find all values on [0,2pie) such that (thita0) produces the tip of a petal (maximum magnitude of r) all values for which r=0, and sketch a graph? a) r = 5 sin 2 (thita0) a) r = 5 sin 3 (thita0) a) r = 5 sin 4 (thita0) B) considering what you can observe in the previous graphs, what are...

There is a simple geometric derivation of the area element ## r dr d\theta## in polar coordinates such as in the following link: http://citadel.sjfc.edu/faculty/kgreen/vector/Block3/jacob/node4.html Is there an algebraic derivation as well beginning with Cartesian coordinates and using ##...

Determine the polar coordinates of the two points at which the polar curves r=5sin(theta) and r=5cos(theta) intersect. Restrict your answers to r >= 0 and 0 <= theta < 2pi.

Say "I have grid in polar coordinates (r, theta). How do I plot it in tecplot. Tecplot plots it in cartesian coordinates."

Let $S^{n-1} = \left\{ x \in R^2 : \left| x \right| = 1 \right\}$ and for any Borel set $E \in S^{n-1}$ set $E* = \left\{ r \theta : 0 < r < 1, \theta \in E \right\}$. Define the measure $\sigma$ on $S^{n-1}$ by $\sigma(E) = n \left| E* \right|$. With this definition the surface area...

Express -3-3i in polar form. I know that r=3√2. And I understand that now we take tan^-1(b/a) which I did. tan^-1(-3/-3) = π/4. So I put my answer as z = 3√2 [cos(π/4) + isin(π/4)]. However the answer manual told me this was incorrect I am unsure of where I went wrong...

Homework Statement Use Desmos to graph the spiral ##r=\theta## on the interval ##0\leq\theta\leq4\pi##, and then determine the exact length of the curve and a four decimal approximation. Hint: ##\int \sec^3(x)dx=\frac{1}{2}\sec(x)tan(x)+\frac{1}{2}\ln\left|\sec(x)+\tan(x)\right|+C## Homework...

I don't fully understand how to work out the impedance from the given equation (5j-5)x(11j-11)/(5j-5)+(11j-11). Any help would be greatly appreciated. Thanks. The answer needs to be in rectangular and polar form.

Hello, I get that both polar unit vectors, ##\hat{r}## and ##\hat{\theta}##, are unit vectors whose directions varies from point to point in the plane. In polar coordinates, the location of an arbitrary point ##P## on the plane is solely given in terms of one of the unit vector, the vector...

Homework Statement Write ##5-3i## in the polar form ##re^\left(i\theta\right)##. Homework Equations $$ |z|=\sqrt {a^2+b^2} $$ The Attempt at a Solution First I've found the absolute value of ##z##: $$ |z|=\sqrt {5^2+3^2}=\sqrt {34} $$. Next, I've found $$ \sin(\theta) = \frac {-3} {\sqrt...

Homework Statement Find the area of the region that lies inside the first curve and outside the second curve. ##r=6## ##r=6-6sin(\theta)## Homework Equations ##A=\frac {1} {2}r^2\theta## The Attempt at a Solution \[/B] If I'm correct, the area should just be ##\frac {1} {2}\int_{0}^{2\pi} 6^2...

The equation is: r= 1/1+sin(theta) I know the answer is supposed to be: x^2+y^2=(1-y)^2 I can't figure out the steps to get to the answer.

My professor gave us a study guide with the solutions: The equation is: sec(theta)=2 I am supposed to convert it to a rectangular equation. I know the answer is going to be y^2-3(x)^2=0 I don't know how to get to the answer he gave us.

Homework Statement [/B] A particle is moving along a curve described by ##p(t) = Re^{\omega t}## and ##\varphi (t) = \omega t##. What is the particles transverse acceleration? Homework Equations [/B] None The Attempt at a Solution [/B] The position vector is ##Re^{\omega t} \vec{e_p}##...

Homework Statement Integrate from 0 to 1 (outside) and y to sqrt(2-y^2) for the function 8(x+y) dx dy. I am having difficulty finding the bounds for theta and r. Homework Equations I understand that somewhere here, I should be changing to x = r cost y = r sin t I understand that I can solve...

Homework Statement Find the eigenfunctions and eigenvalues of the isotropic bidimensional harmonic oscillator in polar coordinates. Homework Equations $$H=-\frac{\hbar}{2m}(\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial...

Hi, I have a general question. How do I show that an operator expressed in spherical coordinates is self adjoint ? e.g. suppose i have the operator i ∂/∂ϕ. If the operator was a function of x I know exactly what to do, just check <ψ|Qψ>=<Qψ|ψ> But what about dr, dphi and d theta

Hello, I am in need of some clarification on tangential velocity in polar coordinates. As far as I know, the tangential velocity vector is ##\vec{v} = v\vec{e_t}##, where ##\vec{e_t} = \frac{\vec{v}}{v}##. This gives us the ##\vec{e_r}## and ##\vec{e_\varphi}## coordinates of the tangential...

Homework Statement Hello everyone, I have an assignment (Spivak's Calculus) to show that the polar equation of a hyperbola with the right focus in the origin is ##r=\frac {±\Lambda} {1+εcos(\theta)}##, but the equation I reached was slightly yet somewhat disturbingly different, and I'm not sure...

x^2+y^2=4 I have so far: (r^2)cos^(theta)+(r^2)sin(theta)=4 Idk what I'm supposed to do from here

Convert the equation to polar form 8x=8y I thought it would be 8*r*cos(theta)=8*r*sin(theta) Said it was incorrect then I thought I needed to divide by 8 to remove it, giving me: r*cos(theta)=r*sin(theta) But that was also incorrect and now I am stuck

Homework Statement Replace the polar equation with an equivalent Cartesian equation. ##r^2 = 26r cos θ - 6r sin θ - 9## a)##(x - 13)^2 + (y + 3)^2 = 9## b)##(x + 26)^2 + (y - 6)^2 = 9## c)##26x - 6y = 9## d)##(x - 13)^2 + (y + 3)^2 = 169## Homework Equations ##x= r cos \theta## ##y= r sin...

Homework Statement Graph the set of points whose polar coordinates satisfy the given equation or inequality. 0 ≤ θ ≤ , 0 ≤ r ≤ 4 Homework Equations - The Attempt at a Solution Is it correct ?