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.

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.

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.

Transcript

00:04 All right.

00:05 So the equation in question is up here at the top of me boxes so that it's highlighted for us.

00:13 Equation is x squared.

00:14 Plus why minus one squared equals four.

00:21 Now, this is an equation of a circle.

00:25 Um, a common form.

00:28 The common equation for a circle is represented right here, which is x minus h.

00:34 Where'd minus use me.

00:36 That's gonna be a plus.

00:37 Less why? minus k squared equals r squared.

00:44 Now this h and k are special because h comma k is the center of our circle on a coordinate plain and r is the radius of our circle.

00:56 So looking at the above equation, we can easily find out what hk and our values are.

01:03 So let's do that h is going to be zero.

01:07 Hey, is one and r is the square root of four, which is just going to be too.

01:14 So now we can write these into our parametric equations thes of the common forms of parametric equations for a circle.

01:25 So our ex equation is going to be x minus x equals h, which is zero plus r, which is two times co sign of tea and why is going to be.

01:38 Why equals a, which is one plus r two.

01:43 I'm sine of t so we write that in right here.

01:49 Those are our equations.

01:51 Here's part a r b, part c.

01:53 For now, they all have the same equations.

01:57 We are likely going after changes up because, as you could see, this are a once it rotating clockwise or heart being see want these after clockwise, so we'll figure that out a bit.

02:12 First things first, let's draw our circle.

02:15 There are a circle.

02:16 We need the center and we need the radius, which we conveniently already have right here.

02:21 Let's mark the center.

02:22 That's gonna be zero comma one and let's mark the top radius of two so the top is going to be zero comma re.

02:33 Let's mark the right right is going to be, too.

02:38 I'm a one with the left being negative, too.

02:44 I'm a one and let's mark the bottom.

02:48 Remember, the center is zero comma one, so the bottom is just zero comma.

02:54 Negative one.

02:55 Remember that radius is you.

02:57 Let's draw that circle nice and easy.