Polar coordinates Definition and 569 Threads

Homework Statement If (r, θ) are the polar coordinates of a point then describe the region defined by the restrictions -1 < r < 0, π/2 < θ < 3π/2 Homework Equations No clue The Attempt at a Solution I tried drawing the curve in a polar grid by starting at π/2 and finishing at 3π/2. I was...

Homework Statement The curve ##C## has polar equation ## r\theta =1 ## for ## 0<\theta<2\pi## Use the fact that ## \lim_{\theta \rightarrow 0}\frac{sin \theta }{\theta }=1## to show the line ## y=1## is an asymptote to ## C##.The Attempt at a Solution **Attempt** $$\ r\theta =1$$ $$\...

How do i get an idea, or a 'feel' of the components of the acceleration in polar coordinates which constitute the component in the eθ direction? from what i know, a= (r¨−rθ˙^2) er + (rθ¨+ 2r˙θ˙) eθ ; (where er and eθ are unit vectors in the radial direction and the direction of increase of the...

General question, how do you determine the limits of integration of a polar curve? Always found this somewhat confusing and can't seem to find a decent explanation on the internet.

Hi there, I've been trying to solve the following problem, which I found looks pretty basic, but actually got me really confused about the definition of angular momentum. Problem The trajectory of a point mass m is described by the following equations, in spherical coordinates: r(t) = r_0 +...

Homework Statement I am trying to calculate the laplacian in polar coordinates but I failed.Please see the attached Homework Equations The Attempt at a Solution My solution to this was uploaded in the attached.I was wondering what's wrong with the purple brackets since they shouldn't exist(...

I am learning about the polar coordinate system, and I have a few conceptual questions. I understand that in Cartesian coordinates there is exactly one set of coordinates for any given point. However, in polar coordinates there is an infinite number of coordinates for a given point. I see how...

Hello everyone, 1. Homework Statement Question : Find the volume of the region which remains inside the cyclinder x 2 + y 2 = 2y, and is bounded from above by the paraboloid surface x 2 + y 2 + z = 1 and from below by the plane z = 0 Homework Equations The Attempt at a Solution This looks...

Homework Statement Integrate by changing to polar coordinates: ## \int_{0}^6 \int_{0}^\sqrt{36-x^2} tan^{-1} \left( \frac y x \right) \, dy \, dx ## Homework Equations ## x = r \cos \left( \theta \right) ## ## y = r \sin \left( \theta \right) ## The Attempt at a Solution So this is a...

Homework Statement The functions are given: ##r(t)=pe^{kt}## ##\theta (t)=kt## ##v(r)=\sqrt2kr## ##a(t)=2k^2r## Find the radius of the curvature of the trajectory in the function of ##r## Homework Equations $$R=\frac{(\dot x^2 + \dot y^2)^{3/2}}{(\dot x\ddot y - \ddot x\dot y)}$$ There is also...

Homework Statement On the surface of a river at ##t=0## there is a boat 1 (point ##F_0##) at a distance ##r_0## from the point ##O## (the coordinate beginning) which is on the right side of the coast (picture uploaded below). A line ##OF_0## makes an angle ##θ_0=10°## with the ##x-axis## whose...

Homework Statement Transform given integral in Cartesian coordinates to one in polar coordinates and evaluate polar integral. ##\int_{0}^3 \int_{0}^x \frac {dydx}{\sqrt(x^2+y^2)}## Homework EquationsThe Attempt at a Solution I drew out the region in the ##xy## plane and I know that ##0 \leq...

Homework Statement $$ U_{tt}=\alpha^2\bigtriangledown^2U$$ in polar coordinates if solution depends only on R, t. Homework EquationsThe Attempt at a Solution So, the books solution is $$U_{tt}=\alpha^2[U_{rr}+\frac{1}{r}U_r]$$. I am getting stuck along the way can't figure out this last step I...

Hello, I have a question about polar coordinates. It is \vec r = \begin{pmatrix}r cos\phi \\ rsin\phi \\ z\end{pmatrix}=r\cdot \vec e_r + z\cdot \vec e_z and than is \ddot{\vec r} = (\ddot{r}-r\dot{\phi}^2)\vec e_r + (r\ddot{\phi} +2\dot{r}\dot{\phi})\vec e_{\phi} + \ddot{z}\vec e_z The...

Homework Statement [/B] Homework Equations The Attempt at a Solution [/B]

1. Homework Statement A particle of mass m is attached to the end of a light spring of equilibrium length a, whose other end is fixed, so that the spring is free to rotate in a horizontal plane. The tension in the spring is k times its extension. Initially the system is at rest and the...

Homework Statement As shown in image. 2. Homework Equations Moment of inertia of pulley = 1/2*M*R^2 Moment of inertia of rod (about end) = 1/3*M*L^2 Acceleration of end of rod in theta direction = L*α Acceleration of end of rod in radial direction = L*ω^2 The Attempt at a Solution...

Homework Statement Consider the 'ice cream cone' bounded by z = 14 − x2 − y2 and z = x2 + y2 .(a) Find the equation of the intersection of the two surfaces in terms of x and y. (b) Set up the integral in polar coordinates. Homework EquationsThe Attempt at a Solution I got part a without...

Homework Statement Let ##x##, ##y##, and ##z## be the usual cartesian coordinates in ##\mathbb{R}^{3}## and let ##u^{1} = r##, ##u^{2} = \theta## (colatitude), and ##u^{3} = \phi## be spherical coordinates. Compute the metric tensor components for the spherical coordinates...

Note: All bold and underlined variables in this post are base vectors I was reading the book 'Introduction To Mechanics' by Kleppner and Kolenkow and came across an example I don't quite understand. The example is this: a bead is moving along the spoke of a wheel at constant speed u m/s. The...

Homework Statement What is the magnitude of the velocity vector if ##\vec{v} = 4 \hat{r} + 6 \hat{\theta}## Homework EquationsThe Attempt at a Solution I know how do do this in Cartesian coordinates (use the Pythagorean theorem), but not so sure how to do it in polar coordinates.

I have read that in polar coordinates, we can form the position vector, velocity, and acceleration, just as with Cartesian coordinates. The position vector in Cartesian coordinates is ##\vec{r} = r_x \hat{i} + r_y \hat{j}##. And any choice of ##r_x## and ##r_y## maps the vector to a position in...

1. The question The position of a particle is given by r(t) = acos(wt) i + bsin(wt) j. Assume a and b are both positive and a > b. The plane polar coordinates of a particle at a time t equal to 1/8 of the time period T will be given by _ Homework Equations r(t) = acos(wt) i + bsin(wt) j. The...

I am trying to find and solve the geodesics equation for polar coordinates. If I start by the definition of Christoffel symbols with the following expressions : $$de_{i}=w_{i}^{j}\,de_{j}=\Gamma_{ik}^{j}du^{k}\,de_{j}$$ with $$u^{k}$$ is the k-th component of polar coordinates ($$1$$ is for...

According to this video the length of basis is r. It grows as we further from the origin . Why?

Homework Statement I need to find the work done by the force field: $$\vec{F}=(5x-8y\sqrt{x^2+y^2})\vec{i}+(4x+10y\sqrt{x^2+y^2})\vec{j}+z\vec{k}$$ moving a particle from a to b along a path given by: $$\vec{r}=\frac{1}{2}\cos(t)\vec{i}+\frac{1}{2}\sin(t)\vec{j}+4\arctan(t)\vec{k}$$ The Attempt...

Homework Statement "Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates." ##7.##...

I am trying to evaluate \int\int xy dxdy over the region R that is defined by r=sin(2theta), from 0<theta<pi/2. I am struggling on where to begin with this. I have tried converting to polar coordinates but am not really getting anywhere. Any guidance would be really appreciated (Crying)

We know, that the infinitesimal area element in Cartesian coordinate system is ##dy~dx## and in Polar coordinate system, it is ##r~dr~d\theta##. This inifinitesimal area element is calculated by measuring the area of the region bounded by the lines ##x,~x+dx, ~y,~y+dy## (for polar coordinate...

Homework Statement Evaluate ∫∫D (3x + 4y 2 ) dA, where D = {(x, y) : y ≥ 0, 1 ≤ x 2 + y 2 ≤ 4} with the use of polar coordinates. Homework Equations The Attempt at a Solution I made a sketch of the circle. It's radius is = 1 and it's lowest point is at (0,0), highest at (0,2), leftmost point...

Homework Statement Using the cylindrical polar co ordinates ##(ℝ,θ,z)## calculate the gradient of ##f=ℝ sin θ + z^2## the textbook says that the scale factors are ## h1=1, h2=ℝ & h3=1## how did they arrive at this?[/B]Homework EquationsThe Attempt at a Solution ##h1=|∂f/∂ℝ|= sin θ...

H! I wonder how to solve: I=\int_{-\infty}^{\infty}e^{-u^2}\frac{1}{1+Cu} du I have solved: \int_{-\infty}^{\infty}e^{-u^2}du which equals \sqrt{\pi} and I solved it with polar coordinates and variable substitution. Thankful for help! Edison

I don't understand why the conservation of angular momentum can imply an acceleration, in absence of a force. Consider for istance planetary motion. The angular momentum \vec{L} of the planets is conserved and that means \mid \vec{L} \mid=mr^2 \dot{\theta}=mrv_{\theta} is conserved too...

Homework Statement This is a physics olympiad problem; and I am still struggling with it. I will quote it here: " A particle moves along a horizontal track following the trajectory $$r=r_{0}e^{-k\theta}$$, where $$\theta$$ is the angle made by the position vector with the horizontal. Recall...

Studying the acceleration expressed in polar coordinates I came up with this doubt: is this frame to be considered inertial or non inertial? (\ddot r - r\dot{\varphi}^2)\hat{\mathbf r} + (2\dot r \dot\varphi+r\ddot{\varphi}) \hat{\boldsymbol{\varphi}} (1) I do not understand what is the...

Homework Statement B⃗ = -2.0ι^ + 3.0 j^. Find the polar coordinates r and theta. Homework Equations n/a The Attempt at a Solution r=sqrt((-2.0)^2+(3.0^2)) r = 3.6 theta = tan^-1(3/-2) = -56 degrees The answers seem to be wrong, can I get any guidance on this question please?

Homework Statement Consider Minkowski space in the usual Cartesian coordinates ##x^{\mu}=(t,x,y,z)##. The line element is ##ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}=-dt^{2}+dx^{2}+dy^{2}+dz^{2}## in these coordinates. Consider a new coordinate system ##x^{\mu'}## which differs from these...

Show that, in polar coordinates, the system is given by r′ = r(r^2 − 4) θ′ = 1x′1 = x1 − x2 − x1^3 x′2 = x1 + x2 − x2^3

Homework Statement Find the scalar product of diracs delta function ##\delta(\bar{x})## and the bessel function ##J_0## in polar coordinates. I need to do this since I want the orthogonal projection of some function onto the Bessel function and this is a key step towards that solution. I only...

I know that an arc length in polar coordinates can be computed by integrating $$\int ds$$ using the formula ##ds=\sqrt{\rho^2 + \frac{dr}{d\theta}^2}d\theta##. But, seeing that ##s=\rho\theta## and ##ds = \rho d\theta##, why is it wrong to calculate arc lengths with this expression for ##ds##?

Homework Statement Homework EquationsThe Attempt at a Solution I have stared at this for hours and don't know where to start. I think I need to get r in terms of t but I don't really know how with the information given. I just need a good hint to get started.

Homework Statement Homework Equations Vθ = r x θdot Vr = rdot Accelθ = r ⋅ θdot^2 + 2(r dot)(theta dot) Accelr = r double dot - (r ⋅(θdot)^2) The Attempt at a Solution I know that the answer is supposed to be But I'm not quite sure on how I'm supposed to get to an answer like that. I'm...

Hi this isn't my homework, but it is taken from a worksheet for a Maths course(trying to refresh my rusty math), so I hope it fits in here. 1. Homework Statement two cylindrical polar vectors with same origin: P(2,55°,3); Q(4,25°,6) units in m Homework Equations a) Express in cartesian...

I am looking at this derivation of velocity in spherical polar coordinates and I am confused by the definition of r, theta and phi. http://www.usna.edu/Users/math/rmm/SphericalCoordinates.pdf I thought phi was the co latitude in the r,θ,∅ system and not the latitude. Of course the two are...

I am looking to understand more about ##a=(\ddot{r}-r(\ddot{\theta})^2)\hat{r}+(r\ddot{\theta}+2\dot{r}\dot{\theta})\hat{\theta}## I understand the terms ##\ddot{r}## and ##r\ddot{\theta}## ,but why ##-r(\ddot{\theta})^2## has opposite direction to ##\hat{r}## and why ##2\dot{r}\dot{\theta}##...

Homework Statement Find the volume of the solid lying inside both the sphere x^2 + y^2 + z^2 = 4a^2 and the cylinder x^2 + y^2 = 2ay above the xy plane. Homework Equations Polar coordinates: r^2 = x^2 + y^2 x = r\cos(\theta) y = r\sin(\theta) The Attempt at a Solution So I tried this...

Homework Statement Express the quantity ∂2/∂x2+∂2/∂y2 in polar coordinates. Homework Equations x=ρcosφ y=ρsinφ ρ=sqrt(x2+y2) The Attempt at a Solution This is my first post, so I apologize for any weird looking equations, etc. I know that this is not a difficult problem, but I just cannot...

Homework Statement Change the Cartesian integral into an equivalent polar integral and then evaluate. Homework Equations x=rcosθ y=rsinθ I have: ∫∫r2cosθ dr dθ The bounds for theta would be from π/4 to π/2, but what would the bounds for r be? I only need help figuring out the bounds, not...

Homework Statement I am currently trying to calculate the moment and products of inertia of a ring rotating about the x-axis at the moment the ring lies in the xy plane. The problem is that the notations I have from textbook are denoted for Cartesian coordinates. i.e. Ixx=∑i mi(yi2+zi2), and...

Hi all, I'm having some problems in grasping/properly understanding the usage of the del operator ( ##\nabla## ) in spherical co-ordinates, and I was wondering if someone could point me to some good resources on the subject, or take a bit of time to try to explain it to me. It just doesn't seem...