ProphetesAI is thinking...
cycloid
MindMap
Loading...
Sources
cycloid
▪ I. cycloid, n. (a.) (ˈsaɪklɔɪd, ˈsɪk-) [See next.] 1. Math. a. The curve traced in space by a point in the circumference (or on a radius) of a circle as the circle rolls along a straight line. The common cycloid is that traced by a point in the circumference of the circle, and has cusps where this...
Oxford English Dictionary
prophetes.ai
Cycloid
shape as the cycloid it originates from. the original cycloid, it describes a new cycloid (see also cycloidal pendulum and arc length).
wikipedia.org
en.wikipedia.org
Cycloid (disambiguation)
A cycloid is a curve traced by a rolling circle. "Cycloid" can also refer to:
Cyclida (formerly Cycloidea), an order of prehistoric crustaceans
Cycloid scale, a type of scale seen on some fishes
Cycloid-β
wikipedia.org
en.wikipedia.org
How to find the parametric equation of a cycloid? "A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line." - Wikipedia !cycloid animation In many calculus b...
$t$ measures the angle through which the wheel has rotated, starting with your point in the "down" position. Since the wheel is rolling, the distance it has rolled is the distance along the circumference of the wheel from your point to the "down" position, which (since the wheel has radius $r$) is $...
prophetes.ai
The prolate cycloid A cycloid is given by the parametric equations: $ x = 2 - \pi \cos(t)$ and $ y = 2t - \pi \sin(t)$. The problem asks for the slope of the tangents on the cycloid at a point where the cycloid inter...
At $t$ and $t'$ the coordinates repeat: $x(t)=x(t')\land y(t)=y(t')\implies2t-\pi\sin t=2t'-\pi\sin t'\land\cos t=\cos t'$ $\cos t=\cos t'\implies t'=t+k2\pi;\in\mathbb Z-\\{0\\}\lor (t'=-t+k'2\pi,t\neq n\pi);k,n\in\mathbb Z$ a) $2t-\pi\sin t=2(t+k2\pi)-\pi\sin (t+k2\pi)$ $\sin(t+k2\pi)-\sin t=4k$, ...
prophetes.ai
Cycloid gear
A Roots blower is one extreme, a form of cycloid gear where the ratio of the pitch diameter to the generating circle diameter equals twice the number of The resulting cycloid is then called an involute and the gear is called an involute gear.
wikipedia.org
en.wikipedia.org
Parametrization of the curve A cycloid is a flat curve that is traced by point of the rim of a circle while the circle rolls without slippage on the line. Show that if the line is the axis $x$ and the circle has radiu...
By definition of cycloid the arc $\widehat{PQ}$ subtended by the angle $t$ (which is choose as parameter!)
prophetes.ai
A cycloid that goes through the beginning and through a general point Parametric equations of the general cycloid through the beginning $(0,0)$ are $$x(t)=\frac{2t-\sin2t}{2d}$$ $$y(t)=\frac{1-\cos 2t}{2d}$$ How ca...
First eliminate $d$: $$\frac{2t-\sin2t}{1-\cos2t}=\frac ab$$ Solve that for $t$. You will have to do so numerically, since this is a transcendental equation. Once you have a suitable $t$ (there might be several to choose from, or even infinitely many), computing $d$ is easy.
prophetes.ai
Cycloid rolls on another identical Cycloid A common cycloid with parametric equations $$ x = r ( t- \sin t - \pi) \, , y= (3+\cos t) r $$ with center point P $(0,4r)$ of its base at beginning of motion rolls on anot...
At the point $P(t)$ on the second cycloid at parameter $t$, the slope is $\sin(t)/(1-\cos(t))$. The slope at the corresponding point on the first cycloid is $-\sin(t)/(1-\cos(t))$.
prophetes.ai
Area of cycloid We want to calculate the area of the cycloid: $x = a(t-\sin t)$, $y=a(1-\cos t)$, with $t\in [0,2π]$. I know that the solution has to do with the Green's Theorem and calculating the surface integral ...
When calculating the area of a parametric curve you should express the area as follows, $$A=\int y(x)~dx=\int y(t)\dot x(t)~dt$$ In your case we have $\dot x=a(1-\cos t)=y$, therfore $$ \begin{align} A &=a^2\int_0^{2\pi} (1-\cos t)^2 ~dt\\\ &=a^2\left( \frac{3 t}{2} - 2 \sin t + \frac14 \sin 2t\righ...
prophetes.ai
Parameterization Of A Cycloid I need to find the length of $x^{2/3}+y^{2/3}=1$ which is said to be a Cycloid. There is an answer fo for it but how did they get to the parameterization?
This is astroid, not cycloid that you are quoting. You can find all that you need in Wiki and elswhere, it's a well known curve. Cycloid is the curve traced by a point on a circle as it rolls along a straight line.
prophetes.ai
What's the area of one arch of a cycloid? So, the cycloid is given with parametric equations: $$x=r(t-\sin{t})$$ $$y=r(1-\cos{t})$$ The teacher solved it like this: $$P=\int_a^by(x)dx$$ $x=x(t)$; $\alpha<t<\beta$ $$P=...
Here is a picture of cycloid:
<
Do you know that the area under a curve $y(x)$ between an interval $[a,b]$ is
$$\int^b_a y(x) dx$$
By using change
prophetes.ai
Is $\vert\sin{x}\vert$ a cycloid? Forgive this seemingly basic question; I recently found out about cycloids and cannot find any answers on the web. My guess is that it’s not, due to some part of the definition of a c...
One possible explanation is using the parametrization of a cycloid. It is given by $$ x = r(t-\sin t)\\\ y = r(1-\cos t) $$ for some $r>0$. If $y = |\sin x|$ is a cycloid, or more generally, homothetic to cycloid, then it will be possible to find some constant $a\in \mathbb{R}$ such that $$
prophetes.ai
What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid? What is the difference between Tautochrone curve and Brachistochrone curve as both are cycloid? If possible, show some r...
Mathematically, they both are the same curve but they arise from slightly different but related problems. While the Brachistochrone is the path between two points that takes shortest to traverse given only constant gravitational force, the Tautochrone is the curve where, no matter at what height you...
prophetes.ai
x(u,v), y(u,v), z(u,v) parametric equations for a special cycloid I'm trying to find out 3D parametric equations for a cycloid. I know that a cycloid is a 2D curve that is generated by a point on a rolling circle. Bu...
From the two-dimensional parametrization, we simply have $$\begin{align*} x(t) &= R \cos \bigl( \tfrac{r}{R}(t - \sin t) \bigr), \\\ y(t) &= R \sin \bigl( \tfrac{r}{R}(t - \sin t) \bigr), \\\ z(t) &= r - r \cos t. \end{align*}$$ Suitable choices for $r$ and $R$ will yield a closed curve, in which ca...
prophetes.ai