Study guide and
3 practice problems
on:
Parameterization of a circle
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The circle of radius $r$, traversed counterclockwise, can be parameterized by $$\bfx(\theta) = \langle r \cos \theta, r \sin \theta \rangle \text{ for } 0 \leq \theta \lt 2 \pi$$
Related topics
Parameterized curves
(5 problems)
Multivariable calculus
(147 problems)
Lines, Planes, and Curves
(13 problems)
Practice problems
Consider a cylinder of radius $r$ rolling up a hill of incline $\theta$ at constant speed $v$. Initially the point of contact is $(0,0)$. Find the trajectory of the point initially contacting the hill.
Solution
A frisbee of radius $r$ translates rightward at speed $v$ meter/second. It rotates clockwise at $\omega$ radian/second. Initially the frisbee is centered at the origin.
Find the trajectory swept by the point initially at $(0,r)$.
Compute the speed as a function of time.
Describe when the speed is largest and smallest.
Solution
An ant is on a merry-go-ground that is rotating clockwise at $\omega$ radians per second. Initially, the ant is at $(R,0)$. From the ant's perspective, it walks toward the center with speed $v$. Several snapshots in time are as follows:
Find the parameterization of the path taken by the ant (relative to the ground)
Compute the speed of the ant as a function of $t$. When is it largest?
Set up, but do not evaluate, an integral for the arc length of the path taken by the ant between $t=0$ and when the ant reaches the origin
Solution