For t>0, a particle moves along the curve x= a+b sink t,...

For $t>0,$ a particle moves along the curve $x=$ $a+b \sin k t, y=a+b \cos k t,$ where $a, b, k$ are positive constants. (a) Describe the motion in words. (b) What is the effect on the curve of the following changes? (i) Increasing $b$ (ii) Increasing $a$ (iii) Increasing $k$ (iv) Setting $a$ and $b$ equal

For $t>0,$ a particle moves along the curve $x=$ $a+b \sin k t, y=a+b \cos k t,$ where $a, b, k$ are positive
constants.
(a) Describe the motion in words.
(b) What is the effect on the curve of the following changes?
(i) Increasing $b$
(ii) Increasing $a$
(iii) Increasing $k$
(iv) Setting $a$ and $b$ equal

Key Concepts

Parametric Equations

Parametric equations describe a curve by expressing the coordinates of points on the curve as functions of an independent parameter, often time. This approach is particularly useful for modeling the trajectories of moving objects and for representing curves that cannot be easily written as a single function in Cartesian coordinates.

Circular Motion

Circular motion refers to movement along a path that forms a circle, and it is frequently modeled using sine and cosine functions due to their periodic properties that naturally satisfy the circular equation. These equations capture the constant distance (radius) from a fixed center and the continuous rotational behavior.

Graph Transformations

Graph transformations involve modifications such as translations (shifting the curve in the plane) and dilations (scaling the size) of a given curve. In the context of parametric equations, adding constants to the functions translates the entire curve, while multiplying by constants scales its dimensions or alters the speed at which the path is traced.

Angular Frequency

Angular frequency represents the rate of change of the angle in rotational motion and determines how quickly a particle completes one cycle of its path. It is a key parameter in any trigonometric representation of motion, influencing the period and speed of the motion along the circular path.

Transcript

00:01 So for positive t, you have a particle that's moving in the xy plane here with x equals a plus b times sine kt, y equals a plus b cosine kt.

00:17 Let's see, so if we just to get a handle on this, set a to be zero and b and k to both be one, what we have here sine and cosine, this looks like circular motion, right? but it's not your conventional circular motion because when t is say zero, our x is going to be zero and our y is going to be one.

00:40 This is t equals zero, limbo our x and y, and then when t is, let's say pi over two, right? sine of pi over two is one and cosine of pi over two is zero.

00:56 So here we have t equals pi over two, and we see that our circle actually goes around like this, clockwise as opposed to counterclockwise.

01:09 By increasing k, by changing a, b, and k, we're not actually going to change that basic fact that it's going counterclockwise and starting at the top.

01:22 So this is a circle, i said counterclockwise, but i meant it's going clockwise starting at the top, with a couple more things, right? x actually should start, the center of x should be a as opposed to zero, likewise y should be starting in a, so it's centered at the point a, a.

01:49 The radius we can see is b, and the frequency is, or let's say the period is two pi over k.

02:04 These are all just things that we pull out of the general circular motion formula.

02:10 There's the motion in words, and then now that we've described it like that, it's relatively easy to see what the changes would do.

02:18 Increasing b will increase the radius, right? this b we see is affecting the sine and cosine, so like the oscillatory, the circle parts of the motion.

02:33 If we increase a, we move the circle, and then because it's centered at a, a, it's going to be somewhere on this line y equals x, and as we increase a, move it that way.

02:50 So move circle, i'm just going to call that northeast, that is up and to the right.

02:54 If we increase k, we increase the frequency, or decrease the period...

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