In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. The radial coordinate is often denoted by r or ρ, and the angular coordinate by φ, θ, or t. Angles in polar notation are generally expressed in either degrees or radians (2π rad being equal to 360°).
Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term polar coordinates has been attributed to Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of circular and orbital motion.
Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates.
The polar coordinate system is extended to three dimensions in two ways: the cylindrical and spherical coordinate systems.
In the following%3A%20https://pubs.rsc.org/en/content/articlehtml/2013/sm/c3sm00140g?casa_token=3O_jwMdswQQAAAAA%3AaSRtvg3XUHSnUwFKEDo01etmudxmMm8lcU4dIUSkJ52Hzitv2c_RSQJYsoHE1Bm2ubZ3sdt6mq5S-w'] paper, the surface velocity for a moving, spherical particle is given as (eq 1)...
The 2D Fourier transform is given by: \hat{f}(k,l)=\int_{\mathbb{R}^{2}}f(x,y)e^{-ikx-ily}dxdy
In terms of polar co-ordinates: \hat{f}(\rho,\phi)=\int_{0}^{\infty}\int_{-\pi}^{\pi}rf(r,\theta)e^{-ir\rho\cos(\theta-\phi)}drd\theta
For Fourier transforms in cartesian co-ordinates, relating the...
I made this exercise up to acquire more skill with polar coordinates. The idea is you're given the acceleration vector and have to find the position vector corresponding to it, working in reverse of the image.
My attempts are the following, I proceed using 3 "independent" methods just as you...
I have a function in polar coordinates:
t (rho, phi) = H^2 / (H^2 + rho^2) (1)
I have moved the center to the right and want to get the new formulae.
I use cartesian coordinates to simplify the transformation (L =...
Hi,
I am trying to find open-form solutions to the integrals attached below. Lambda and Eta are positive, known constants, smaller than 10 (if it helps). I would appreciate any help! Thank you!
I had an equation. $$T=\frac{1}{2}m[\dot{x}^2+(r\dot{\theta})^2]$$ Then, they wrote that $$\mathrm dr=\hat r \mathrm dr + r \hat \theta \mathrm d \theta + \hat k \mathrm dz$$ I was thinking how they had derived it. The equation is looking like, they had differentiate "something". Is it just an...
>10. Let a family of curves be integral curves of a differential equation ##y^{\prime}=f(x, y) .## Let a second family have the property that at each point ##P=(x, y)## the angle from the curve of the first family through ##P## to the curve of the second family through ##P## is ##\alpha .## Show...
I am trying to derive the tangential acceleration of a particle. We have tangential velocity, radius and angular velocity. $$v_{tangential}= \omega r$$ then by multiplication rule, $$\dot v_{tangential} = a_{tangential} = \dot \omega r + \omega \dot r$$ and $$a_{tangential} = \ddot \theta r +...
I got a polar function.
$$ \psi = P(\theta )R(r) $$
When I calculate the Laplacian:
$$ \ \vec \nabla^2 \psi = P(\theta)R^{\prime\prime}(r) + \frac{P(\theta)R^{\prime}(r)}{r} + \frac{R(r)P^{\prime\prime}(\theta)}{r^{2}}
$$
Now I need to convert this one into cartesian coordinates and...
The equations of motion are:
\ddot{r}-r{\dot{\theta}} ^{2} = -\frac{1}{r^{2}}
for the radial acceleration and
r\ddot{\theta} + 2\dot{r}\dot{\theta}= 0
for the transverse acceleration
When I integrate these equations I get only circles. The energy of the system is constant and the angular...
To begin with, I posted this thread ahead of time simply because I thought it may provide me some insight on how to solve for another problem that I have previously posted here: https://www.physicsforums.com/threads/special-relativity-test-particle-inside-suns-gravitational-field.983171/unread...
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 have a right triangle: one of the angles is ##60°## (that's ##\theta##), one of the sides is ##40 m## long, and the hypotenuse is equal to the radius. Now I can find an expression for ##r## and that expression is ##r=\frac{height}{sin \theta}##. If I differentiate it, I'll get ##\dot r## and...
Well, what I've done so far is calculating the magnitude of velocity and acceleration replacing ##t=2## in ##\theta (t)## and ##r(t)## so I could get the expressions for ##\dot r##, ##\dot \theta##, ##\ddot r## and ##\ddot \theta##. But that's not my problem... my problem is related to the...
Hello to all good people of physics forums. I just wanted to ask, whether the angular and linear (orbital) speed in perihelion of eliptical orbit are related the same way as in circular orbit (v = rw). If we take a look at the angular momentum (in polar coordinates) of reduced body moving in...
Summary: I can't figure out how the solver carries out the conversions from cartesian to cylindrical coordinates and vice-versa.
I have a set of points of a finite element mesh which when inputted into a solver (ansys) gives the displacement of each node. I can get the displacement values of...
Homework Statement
A particle of a mass ##m## is embedded in a circular rail, (radius: ##R##), without any friction. In a given moment, the particle finds itselfs without velocity at point C, and a force is applied on the rail, which starts moving with an ## \vec A ## constant acceleration. Use...
Could anyone please explain how can I know mathematically whether the logarithmic spiral curve spirals inward or outward? In which sense does the outward spiral spirals?
Thank you very much for your help
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...
My question is why isn't the radial component e→r of acceleration in cylindrical coords simply r'' ?
If r'' is the rate at which the rate of change of position is changing in the radial direction, wouldn't that make it the radial acceleration? I.e, the acceleration of the radius is the...
Homework Statement
Question attached in attachments
Homework Equations
Area enclosed by polar graph is ∫0.5r^2
where r is the radius as a function of angle theta
The Attempt at a Solution
I attempted to use the formula above and I subtracted the area of the inside from the outside but it...
Homework Statement
-infinity < r > +infinity
Which of the following are equations for the line y=m*x for m<0:
a. theta = -arctan(m)
b. theta = arctan(m)
c. theta = arctan(-m)
d. theta = arctan(m) + pi
e. theta = arctan(m) - pi
f. r = 1/(sin(theta - arctan(m)))
2. The attempt at a solution...
Homework Statement
I tried to answer the following questions is about the curve surface z= f (x, y) = x^2 + y^2 in the xyz space.
And the three questions related to each other
A.)
Find the tangent plane equation at the point (a, b, a^2+ b^2) in curved surface z .
The equation of the...
Homework Statement
13.43 A child slides down the helical water slide AB. The description of motion in cylindrical coordinates is
##R=4m##, ##θ=ω^2t^2## and ##z=h[1-(\frac {ω^2t^2} {π})]##, where h=3m and ω=0.75rad/s.
Compute the magnitudes of the velocity vector and acceleration vector when...
Homework Statement
13.34 The curved portion of the cloverleaf highway interchange is defined by
##R^2=b^2sin2θ##, 0<=θ<=90deg. If a car travels along the curve at the constant speed v0,
determine its acceleration at A
Homework Equations
The Attempt at a Solution
[/B]
Is this wrong?
Homework Statement
13.30 The colar B slides along a guide rod that has the shape of the spiral R = bθ.
A pin on the collar slides in the slotted arm OC. If the speed of the collar is constant at v0,
determine the angular speed ##\dot θ## of the arm OC in terms of v0, b, and θ.
Homework...
Homework Statement
13.29 The colar B slides along a guide rod that has the shape of the spiral R = bθ.
A pin on the collar slides in the slotted arm OC. If OC is rotating at the constant angular
speed ##\dot θ = ω##, determine the magnitude of the acceleration of the collar when
it is a A...
Homework Statement
13.25 The rod OB rotates counterclockwise about O at the constant angular speed
of 45 rev/min while the collar A slides toward B with the constant speed 0.6 m/s,
measured relative to the rod. When collar A is in the position R = 0.24m, θ = 0, calculate
(a) its velocity...
Homework Statement
13.24 A particle travels along a plane curve. At a certain instant, the polar
components of the velocity and acceleration are vR=90mm/s, vθ=60mm/s,
aR=-50mm/s2, and aθ=20mm/s2. Determine the component of acceleration that is tangent to the path of the particle at this...
So when finding the Area from a double integral; or Volume from a triple integral: If the curve/surface has a negative region: (for areas, under the x axis), (for volumes, below z = 0 where z is negative)
What circumstances allow the negative regions to be taken into account as positive when...
Homework Statement
We've got an object/person in the center of a cicrcle that spins around with an angular velocity ω. That same object is moving at a constant speed k with the direction of the radius, that is from the center to the outside of the sphere. That object describes then a spiral...
Homework Statement
Find the divergence of the function ##\vec{v} = (rcos\theta)\hat{r}+(rsin\theta)\hat{\theta}+(rsin\theta cos\phi)\hat{\phi}##
Homework Equations
##\nabla\cdot\vec{v}=\frac{1}{r^2}\frac{\partial}{\partial r}(r^2v_r)+\frac{1}{r sin\theta}\frac{\partial}{\partial...
Homework Statement
question :
find the value of
\iint_D \frac{x}{(x^2 + y^2)}dxdy
domain : 0≤x≤1,x2≤y≤x
Homework Equations
The Attempt at a Solution
so here, i tried to draw it first and i got that the domain is region in first quadrant bounded by y=x2 and y=x
and i decided to...
devinaxxx
Thread
calculus
derivation
double integral
polarcoordinates
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.
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...
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...
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...
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...
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...