Find parametric equations for the path of a particle that moves along the circle x^2+(y-1)^2=4 i

Find parametric equations for the path of a particle that...

Key Concepts

Adjusting the Parameter for a Given Starting Point

When the starting point of the motion is specified, the standard parameterization may need a phase shift. The phase shift is determined by finding the angle t for which the cosine and sine values yield the initial coordinates relative to the circle's center. This adjustment ensures that as the parameter t begins its interval, the particle is at the designated start position, thereby aligning the parametric equations with the given initial condition.

Scaling the Angular Parameter for Multiple or Partial Rotations

This concept deals with modifying the parameter t so that the motion covers a specified number of rotations or a fraction of a rotation. Multiplying t by a constant factor causes the particle to travel around the circle multiple times (if the factor is greater than 1) or less than one full rotation (if the factor is a fraction). It is essential to adjust the interval over which t varies to match the intended number of complete or partial revolutions along the circle.

Trigonometric Parameterization of a Circle

This concept involves using sine and cosine functions to represent the points on a circle. For a circle with center (h, k) and radius r, the typical parametric equations x = h + r*cos(t) and y = k + r*sin(t) describe the circle as t varies over an interval. This method leverages the periodic nature of trigonometric functions to represent the continuous and closed path of the circle.

Orientation and Direction of Rotation

Understanding orientation is essential when parameterizing a circle, as you must decide whether the particle moves clockwise or counterclockwise. The standard parameterization using (cos(t), sin(t)) typically results in a counterclockwise motion as the parameter increases. To reverse the direction (i.e. get clockwise motion), one can replace t with -t or adjust the trigonometric functions accordingly. This adjustment affects the sign in one of the sine or cosine components, changing the direction of travel along the curve.

Transcript

00:01 We want to calculate three different parameterizations for this circle, which i've drawn here.

00:07 The first one, we want to move once clockwise around this circle, starting at the point to 1, which is the rightmost point of this circle.

00:18 So we want to use the fact that 2 cosine t squared plus 2 sine t squared, where t is our parameter, is equal to 4.

00:32 So after we plug in whatever our parameterization of x and y are, this is what we want to have left.

00:39 For x, that's simple.

00:41 We just set x equal to 2 cosine of t directly.

00:46 We set x equal to cosine and not sign because we want to start from the right most value of x.

00:53 So when t is equal to zero, we want x to be 2, which two cosine of t satisfies.

01:01 Now, we set y equal to 2 sine of t.

01:05 If we left it at that and plugged it in, we'd have 2 sine of t minus 1.

01:09 So we add one in order to cancel out the negative 1.

01:13 So this is a parameterization of that circle.

01:17 And additionally, we specify the interval of parameterization.

01:22 We start at 0, as mentioned, and we complete a rotation every 2 pi.

01:27 So we want to have the interval end to 2 pi.

01:32 But this parameterization isn't quite done because this moves counterclockwise around the circle.

01:40 It moves in this direction.

01:44 So how do we change that? we want to move down instead of up at the beginning.

01:48 So we want to negate the sign turn here because then as t increases, sign will increase, it will decrease.

01:57 We don't negate the one because otherwise the unit would be translated down one instead of up one.

02:03 The circle would be centered at negative 1.

02:06 So this parameterization moves once clockwise around the circle and starts at the point 2 .1.

02:14 For our second parameterization, we want to move three times counterclockwise around the circle.

02:20 And again, we're starting at the point 2 .1...

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