Find parametric equations and a parameter interval for the...

Find parametric equations and a parameter interval for the motion of a particle that starts at $(a, 0)$ and traces the circle $x^{2}+y^{2}=a^{2}$ a. once clockwise. b. once counterclockwise. c. twice clockwise. d. twice counterclockwise. (There are many ways to do these, so your answers may not be the same as the ones in the back of the book.)

Find parametric equations and a parameter interval for the motion of a particle that starts at $(a, 0)$ and traces the circle $x^{2}+y^{2}=a^{2}$
a. once clockwise.
b. once counterclockwise.
c. twice clockwise.
d. twice counterclockwise.
(There are many ways to do these, so your answers may not be the same as the ones in the back of the book.)

Key Concepts

Frequency of Rotation

The frequency or angular speed at which the circle is traced is influenced by multiplying the parameter by a constant. For instance, using t multiplied by an integer in the trigonometric arguments results in multiple rotations within the same parameter interval, providing control over the number of circuits the particle makes.

Parameter Interval

The parameter interval determines the amount of the circle that is traced out as the parameter varies. For a complete rotation of the circle, the interval is often taken as an interval of length 2?, though modifying this range can lead to multiple rotations or a partial circle, depending on the problem requirements.

Direction of Motion

The sign and order of the sine and cosine functions dictate the direction of motion along the circle. A standard parameterization like (a cos(t), a sin(t)) leads to a counterclockwise motion, while changing the sign of one component, such as (a cos(t), -a sin(t)), reverses the direction to clockwise.

Parametric Equations for a Circle

Parametric equations express the coordinates of points on a circle in terms of a parameter, typically using sine and cosine functions. This representation, where x = a cos(t) and y = a sin(t) (or variations with sign changes), encodes the geometric properties of the circle and allows one to describe motion along its circumference.

Trigonometric Functions

Sine and cosine are fundamental functions used to parameterize circular motion. Their periodic nature aligns with the circular symmetry, and by adjusting their arguments, one can control the start point, direction, and speed of traversal around the circle.

Transcript

00:01 So this question is about some particle that starts at a position a comma 0 and traces the circle x squared plus y squared equals a squared, or in other words, the circle of radius a around zero zero, a certain number of times.

00:32 So for part a, we want to do it once, so once the full circle, and we want to go clockwise.

00:44 So in other words, what we want to do is if we draw a graph, this is a, this is a.

00:51 What we want to do is we want to start here and go like this in that direction.

01:00 Okay, so how we're going to do this? we're going to build this up, like step by step.

01:06 So initially a circle is something like this.

01:17 So this is the whole reason goes and sign are so good.

01:21 They give you a circle in parameters.

01:25 Okay, so we want something t here and something t here.

01:29 So traditionally, if you just do like x is cosine t, y is sine t, you get the counterclockwise circle of radius one.

01:39 But we want to go clockwise.

01:41 So how do we fix this? well, we fix this by putting in minus sides.

01:49 So now when we start out, so cosine is symmetric, right? so we could even remove that minus sign.

02:00 So if you remember the cosine looks like this, and then it's symmetric like that.

02:08 Whereas the sign, it's good to occasionally just like draw these guys because they're really, useful to know the picture.

02:18 So what we change by putting in this minus sign for the t's is that when we start out, instead of going to the right, we go to the left.

02:27 So we see that the sign is going to decrease first and then increase.

02:33 So what that does is our y coordinate is going to decrease first, and then it's going to increase.

02:42 And then it's going to decrease again at the end.

02:46 So this fix is this.

02:47 The clockwise thing.

02:49 And then to get radius a, well, we just multiply by a.

02:54 So finally, we want to do it once.

02:57 So we need to say what our domain is.

02:59 So we're going to just start at 0.

03:02 And we're going to go till 2 pi.

03:05 Because 2 pi is a full circle.

03:08 It's a full period.

03:11 So this is 2 pi...

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