Circle Definition and 1000 Threads

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.
Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a disc.
A circle may also be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations.

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  1. C

    Rubber Band Spinning in midair

    Homework Statement A uniform thin circular rubber band of mass M and spring constant k has an original radius R. Now it is tossed into the air. Assume it remains circular when stabilized in air and rotates at angular speed ω about its center uniformly. Derive an expression for the new radius...
  2. P

    General Relativity and the Circumference of a Circle

    Homework Statement Calculate the circumference of the circle θ = θ0 (a constant) in the spatial geometry \begin{eqnarray*} dS^2 = a^2(d\theta^2 + sin^2\theta cos^2\theta d\phi^2) \end{eqnarray*} Hence, (by finding R(z)) sketch the cross section of the surface embedded in three dimensions via...
  3. Biker

    Induced EMF in a circle: Faraday's law

    Homework Statement Explain what happens in the following situation: You have a loop in the form of a circle and there is a varying magnetic field inside that loop as the following picture illustrates: Homework Equations Faraday's law lorentez force The Attempt at a Solution I first thought...
  4. F

    Understanding Mohr's Circle Homework Equations

    Homework Statement http://web.mit.edu/course/16/16.unified/www/FALL/materials/documents/HO-M-6(Mohrs)(08).pdf I found this on the Internet . At there , the author defined positive as counterclockwise rotation . and also “ Positive shear would cause a clockwise rotation of the infinitesimal...
  5. F

    How Can θp and θs Have the Same Sign in Mohr's Circle?

    Homework Statement In this question , how could the $$\theta_p$$ and $$\theta_s$$ has the same sign ? For $$\theta_p$$ , it is to be rotated clockwise to horizontal axis , right ? For $$\theta_s$$ , it is to be rotated anticlockwise to vertical axis , right ? They are in different...
  6. F

    Maximum shear stress in Mohr's circle

    Homework Statement I am not sure how to get the maximum shear force in mohr's circle [/B] Homework EquationsThe Attempt at a Solution To get the ##\theta_s max ## , we have to 'turn' the B to ##\tau_{max}## , right ? So , it should be like this ( in the figure , orange arrow) ? Am i right ...
  7. Z

    Perimeter/Area of Shaded Region within a Circle

    Homework Statement Find the area and perimeter of shared region in the following diagram: Homework Equations Area of shared Region =x/360 * PI * radius * radius Perimeter of shared Region = x/360 * 2* PI* radius[/B]The Attempt at a Solution I am finding the area of circle & then subtracting...
  8. caters

    I Calculating the Area of an Apollonian Gasket: A Formula for n Layers

    Here is my formula for the area of n layers of appolonian gasket(assuming no circles past the nth layer): $$πR^2 - (πR^2 - (\sum_{0}^{n} x_n*πr_{n}^2))$$ Here R is the radius of the outer circle, r is the radius of an inner circle, x is a function that represents the number of circles in a...
  9. Z

    Comparing angles opposite radius of a circle

    Homework Statement In circle O , BC >CD Compare x & y (which is greater?) Homework Equations There are no eq. Rule: angle opposite larger side is larger The Attempt at a Solution In my view both triangles are isosceles triangle. So x & y should be equal because they are both opposite the...
  10. dfklajsdfald

    Finding the Value of axb on the Unit Circle | Round to the Nearest Thousandths

    Homework Statement the point (log a, log b) exists on the unit circle. find the value of axb. round to the nearest thousandths. Homework Equations x2 + y2 = 1 The Attempt at a Solution x2+y2 = 1 loga2+logb2 =1 2loga+2logb = 1 2(loga+logb) = 1 loga + log b = 0.5 logb = 0.5−loga now i try...
  11. F

    An interior Dirichlet problem for a circle

    Homework Statement $$ \bigtriangledown^2=0 for : 0<r<1 \\ BC : u(1,\Theta)= sin(\Theta), 0<\Theta<\pi \\ u(1,\Theta)= 0, pi<\Theta<2\pi \\ $$ Basically its an interior dirichlet problem for a circle. [/B]Homework EquationsThe Attempt at a Solution The answer is supposed to be $$U(r,\Theta) =...
  12. F

    MHB Find the diameter of a circle given linear velocity?

    [SOLVED] Find the diameter of a circle given linear velocity? Hello all! I need help with a certain type of problem. I do not know how I can find the diameter of a circular object given it's linear velocity. Here is an example problem, and I would love any explanation you could give me! Thanks...
  13. lila12345

    I Surface of revolution of a donuts

    HELP I can't find the surface of revolution! By donuts I mean a circle that doesn't touch the axes (tore in french) y^2+(x-4)^2=2^2 is my function ( y^2+x^2=r^2) and the axe of rotation is y so y= sqrt(r^2-x^2) the formula I know : 2* pi (Integral from a to b (F(x)*sqrt( 1+ (f``(x))^2))...
  14. P

    MHB Proving Nine-Point Circle Theorem w/ Parallelogram & Symmetry

    Hi, I'm stuck on this problem and would like some help. The purpose of this exercise is to prove the Nine-Point Circle Theorem. Let triangleABC be a Euclidean triangle and let points D, E, F, L, M, N, and H be as in Figure 8.46. Let γ be the circumscribed circle for triangleDEF. a) Prove that...
  15. W

    Parametric equation of a circle intersecting 3 points

    Homework Statement Given the points P0 = (0,a), P1 = (b,0), P2 = (0,0), write the parametric equation of a circle that intersects the 3 points. Assume that b > a and both are positive. Homework Equations X = h + rcos(t) Y = k + rsin (t) r = √((x-h)2 + (y-k)2 Cos (t) = (x-h)/r Sin (t) =...
  16. B

    Gears: when Base circle less than the root circle/dedendum

    Hey guys, Trying to design a spur gear but I am very confused as the root circle/dedendum ends up being greater than the base circle. What do I do in this case? The gear I'm trying to design has a 68.33mm pitch diameter, 60 teeth, the pressure angle a standard 20 degrees. What am I doing...
  17. Adeel Ahmad

    Osculating Circle Homework: Solving for Curvature Limit

    Homework Statement I know I would use the curvature equation |f''| / [1-(f')^2]^3/2 and then take the limit of that to -1. I just don't understand why I have to take the limit of the curvature and when I take the limit of the curvature I get |-1| / (13)^3/2 when the answer should be 2.
  18. parshyaa

    B Exploring the Applications of Trigonometric Functions Beyond 90°

    Why trigonometric functions are defined for unit circle, here "why" refers to what made them to define it this way, they may have defined it for right triangle only , can you give me a application where sin(120°) or sin, cos , tan of more than 90° is used to find some values like in physics or...
  19. S

    B  Given circle, find the line of bearing of tan lines thru 0,0

    This question occurred to me a few days ago and it's been bugging me ever since. Consider a circle in the coordinate plane with center P(x,y) and radius R, where R < D, D being the distance from the origin to the circle's center. There are two lines tangent to the circle (T1 and T2) that pass...
  20. Adamolesiak

    Friction and turning angle relation

    Hi there, Suppose we had a car going in a circle. We know that the turning angle(angle between the movement direction and the wheel axis) and the friction are connected, because friction determines the centripetal force and it determines the radius of the circle that we make with our car. I need...
  21. M

    MHB Proving angles are equal in a triangle in a circle.

    hi can someone help me work this out. i think it has something to do with the exterior angle of a triangle is equal to the sum of the interior angles but i can't work it.
  22. M

    MHB State the angles in terms of x , Circle theorems

    Problem O is the center of the circle and AB is the diameter of this circle , C & D are points on this circle , If $\angle CDB=x^\circ$ ,State the following angles in terms of $x$ $\angle CAB$ $\angle CBA$ Workings & what is known $OC=OB=OA$ radii of the same circle $\therefore \angle...
  23. M

    MHB Prove that center of DG = center of the red circle.

    Let BCED and ACFG square. Prove that center of DG = center of the red circle. I don't know how to start
  24. S

    Find the center of the circle of curvature

    Homework Statement For the curve with equation y={ x }^{ 2 } at the point (1, 1) find the curvature, the radius of curvature, the equation of the normal line, the center of the circle of curvature, and the circle of curvature. Homework EquationsThe Attempt at a Solution \kappa \left( 1 \right)...
  25. Shafia Zahin

    B Understanding the Circumference of a Circle: A Comparison of 2π and 2πr

    I just have a little question that i have read in the book of Thomas/Finney Calculus 9th edition that the circumference of a circle is 2pi,i can be wrong obviously but wasn't it supposed to be 2*pi*radius of the circle? Please help.
  26. nysnacc

    Understanding Relative Motion in Circular Motion

    Homework Statement Homework Equations a= omega^2*r The Attempt at a Solution a_A/B = a_A - a_B a_A = 10^2*r (-î) a_B = 10^2*r (+ĵ) a_A/B = 10^2*r (-î) - 10^2*r (+ĵ) => -200 î -200 ĵ why is the answer 200 î -200 ĵ (accelerate in +î direction) ?
  27. H

    Elasticity of a Wire spinning in a circle

    Homework Statement A 12.0-kg mass, fastened to the end of an aluminum wire with an unstretched length of 0.70 m, is whirled in a vertical circle with a constant angular speed of 120 rev>min. The cross-sectional area of the wire is 0.014 cm2. Calculate the elongation of the wire when the mass is...
  28. artriant

    Weights on circular rotating circle

    circular cable i mean ?:)* Hey all second time to use this forum, hope to get same awesome help. Lets say that we have a cable in a shape of a closed perfect circle rotating in perfect conditions no air zero gravity etc around a center. The cable is 100 m circumference, rotating at 1rpm My...
  29. JustynSC

    What is the area, and approximate uncertainty in a circle....

    Homework Statement What is the area, and approximate uncertainty in a circle with radius 3.1*10^4 cm (or written: 3.1e4 cm)? Homework Equations Area=Pi*r^2 The Attempt at a Solution My attempt to the solution took some trial and error, and it went as follows: Substitute the circle's radius...
  30. C

    Sign convention in Mohr's circle

    Homework Statement I doubt that is the following notes (in the first picture), correct or not? Because i couldn't find any proof in the other book... Homework EquationsThe Attempt at a Solution Can I say that consider σ positive to right and τ positive downward, orientation (θ) clockwise as...
  31. C

    Misinterpretation of Mohr's Circle Rotation Formula?

    Homework Statement for the τx'y' , i use the formula of -0.5(σx - σy)sin2θ + τxycos2θ , i found the ans is -41.3 rather than positive 41.3 , which part of my working is wrong? or I misunderstood something? Homework EquationsThe Attempt at a Solution
  32. C

    Understanding Mohr's Circle: Shear Stress and Normal Stress Interactions

    Homework Statement http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/plane_stress_principal.cfm in this notes , i couldn't understand that why there exists an angle (θp) where the shear stress (τx'y' ) becomes zero , (only normal stress acting ) is there any proof on this ? for...
  33. T

    How Do Forces Affect Particle Motion in Different Vertical Circle Scenarios?

    Motion in a vertical circle* 1. Homework Statement Case 1: When a particle P of mass m moves in a vertical circle centered on O, its motion is governed by two forces, its weight mg and a force R directed radially inwards constraining it to move in a circle radius r. Resolving parallel to an...
  34. K

    I Complex Analysis, holomorphic in circle.

    Hi, I have a question regarding corollary 2.3. in the uploaded image. it looks very trivial to me becauese Cauchy's theorem states "if f(z) is holomorphic, its closed loop integral will be always 0". Is this what the author is trying to say? what's the necesseity of the larger disk D' at here...
  35. W

    What does a circle with a slash through it mean?

    I am reading European engineering publications, and the circle with a slash shows up about half of the time in front of length measurements. For example, Ø15 x 1.12 mm. What does this mean?
  36. orion

    I Understanding the Transition Functions for S^1 Using Atlas Charts

    I am confused about the procedure for finding the transition functions given an atlas. I understand the theory; it's applying it to real life examples where I have my problem. So for example, take S1 (the circle). I want to use 2 charts given by: U1 = {α: 0 < α < 2π} φ1 = (cos α, sin α) U2...
  37. Z

    Portion of Altitude of a Triangle Inscribed in a Circle

    Homework Statement In the figure Q image2.jpeg (attached), equilateral triangle ABC is inscribed in circle O, whose radius is 4. Altitude BD is extended until it intersects the circle at E. What is the length of DE? Solution figure is attached. They formed a right angled triangle & calling it...
  38. C

    Why is Mohr's circle labeled clockwise and not anticlockwise?

    Homework Statement a.)why the angle θ p is in clockwise direction? why shouldn't it be in anticlockwise direction? b.) I have labelled A,B ,C and D...Why the maximum normal stress is labelled at location A? while the min stress is labelled at location B? Can I label the min and max stress at...
  39. F

    Mohr's Circle Formula: Understanding Why σ_xl Disappears

    Homework Statement https://web.iit.edu/sites/web/files/departments/academic-affairs/Academic%20Resource%20Center/pdfs/Mohr_Circle.pdf[/B] why will σ_xl disappear in the equation ? [ [σ_xl - (σx -σy)/ 2 ] ^2 ] = ( [ (σx -σy)/ 2] ^2 ) is it wrong ? Homework EquationsThe Attempt at a...
  40. H

    I Show the envelope of circles around a circle is a cardioid

    Is there a shorter way to get the answer, the polar equation of a cardioid, directly? My solution involves some tedious work and doesn't give the polar equation directly: The equation of the member curves (circles) is ##f(x, y...
  41. M

    MHB Find the ratio of lines in a circle

    I have no idea how to solve this problem. ABCD is just an irregular Quadrilateral so nothing too special with that figure. We are looking for the ratio BC:CD and we only have that two angles. I know that the answer is 1:√2, but I have no idea how to find it.
  42. I

    I Great Circle Distance Derivation

    I derived the shortest distance between two points on a spherical surface (Great Circle Distance) , using the definition of the spherical coordinates and the dot product of the position vectors r1 and r2 where r1 = ( R cosθ1 cosφ1 , R cosθ1 sinφ1 , R sinθ1 ) r2 = ( R cosθ2 cosφ2 , R cosθ2 sinφ2...
  43. LotusTK

    Bucket of water spinning in vertical circle

    Homework Statement The water stays in the bucket, even at the top of the circular path, as long as the speed exceeds a certain value. Explain why. I think i have a good answer, but not 100% sure. My answer: There is a centripetal force acting on the bucket and the water since they are...
  44. P

    Find Centre of Mass of Semi-Circle: Solving Integrals

    Homework Statement Find the centre of mass of a semi-circle Homework Equations ##y_{cm}=\frac{1}{M} \int y dm ## The Attempt at a Solution So ## y= R cos \theta ## where theta is measured from the vertical, and the base of the semi-circle is along the horizontal Now apparently from here...
  45. D

    Proofing that All Arcs of a Circle = 360 Degrees

    Im doing a proof. For instance all sums of a triangle add upp to 180 degrees. But how to i motivate that all arcs on a circle add up to 360 degrees. A part of my proof is that Arc A + B + C = 360 degrees. But i don't know what to write in the column that motivates every step. ! Like a lap is...
  46. K

    Instrument Design to Deliver Energy to 50mm Circle

    I didn't know where to put this, I figured I'd start here since this area sees traffic that other areas might not see. Someone can move it if there's a better section for it. I'm thinking of an instrument design that has a light path as follows: UV-Vis light source, monochromator or filter that...
  47. S

    Line Integral over circle region

    Homework Statement Evaluate ∫c (x + y) ds, where C is the circle centred at (1/2, 0) with radius 1/2. Homework EquationsThe Attempt at a Solution parametrise x=1/2cos(t) y=1/2sin(t) 0≤t≤2π ds=√dx2+dy2 =√(1/2)2-sin2(t)+(1/2)2cos2(t) =√-(1)2(1/2)2sin2(t)+(1/2)2cos2(t)...
  48. prashant singh

    I Incentre and circle formed by it

    Why circle formed by incentre can not lie outside the triangle. I think because perpendicular drawn from incentre to each side will always be tangent to the circle. What's your opinion.
  49. mr.tea

    Complex numbers on the unit circle

    Homework Statement Let ##z_1,z_2,z_3## be three complex numbers that lie on the unit circle in the complex plane, and ##z_1+z_2+z_3=0##. Show that the numbers are located at the vertices of an equilateral triangle. Homework EquationsThe Attempt at a Solution As far as I understand, I need to...
  50. C

    Differentiating an Implicit Function: a Circle

    Homework Statement Find the expression for the slope on the lower half of the circle y^2 + x^2 = 25. 2. Attempt at a solution. The text says you get 2x + 2y(dy/dx) = 0. I got this and then solved for dy/dx to get dy/dx = -2y - 2x. Then, I substituted for y the x value-expression for the...
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