a circle of radius 1 rolls around the outside of a circle of radius 2 without slipping. the curve traced by a point on the circumfarence of the smaller circle is callled an epicycloid. use the angle theta to find a set of parametric equations for this curve.
To find a set of parametric equations for the epicycloid curve, we can consider the motion of the smaller circle with respect to the larger circle.
Let's denote the center of the larger circle as point O, and the center of the smaller circle as point P. The point on the circumference of the smaller circle can be called point Q.
Now, suppose the smaller circle has completed an angle of θ with respect to the larger circle. This means that the distance between points O and P is equal to 2 units (the radius of the larger circle), and the distance between points P and Q is equal to 1 unit (the radius of the smaller circle).
To find the position of point Q, we can use trigonometry. Let's denote the coordinates of point P as (R*cos(θ), R*sin(θ)), where R is the radius of the larger circle (R = 2 in this case).
Using the cos and sin functions, we can express the coordinates of point Q as follows:
x = (R + r) * cos(θ) - r * cos((R + r) * θ / r)
y = (R + r) * sin(θ) - r * sin((R + r) * θ / r)
In this case, r represents the radius of the smaller circle (r = 1).
So, the parametric equations for the epicycloid curve are:
x = (2 + 1) * cos(θ) - cos((2 + 1) * θ / 1)
y = (2 + 1) * sin(θ) - sin((2 + 1) * θ / 1)
Simplifying further, we get:
x = 3cos(θ) - cos(3θ)
y = 3sin(θ) - sin(3θ)
These are the parametric equations for the epicycloid curve traced by a point on the circumference of the smaller circle as it rolls around the larger circle.
To find a set of parametric equations for the curve traced by a point on the circumference of the smaller circle (epicycloid), we can use the angle theta.
Let's start by considering the motion of the smaller circle. As it rolls around the outside of the larger circle, it completes one full revolution when it has traveled a distance equal to its circumference. Now, the circumference of a circle can be calculated using the formula C = 2πr, where C is the circumference and r is the radius.
In this case, the radius of the smaller circle is 1, so its circumference is 2π(1) = 2π. Therefore, when the smaller circle completes one full revolution, its center has traveled a distance equal to the circumference of the larger circle.
Now, let's introduce the angle theta as a parameter to describe the motion of the smaller circle. As the angle theta increases from 0 to 2π, we can think of it as representing the fraction of the smaller circle's revolution completed.
To determine the position of the point on the circumference of the smaller circle, we need to consider both the rotational motion of the smaller circle and the translational motion along the circumference of the larger circle.
Since the circumference of the larger circle is 2π times that of the smaller circle, the distance traveled along the larger circle's circumference can be expressed as 2π times the fraction of the smaller circle's circumference completed, which is 2πθ.
Now, we can define the parametric equations for the point on the epicycloid:
x = (2 + 1) * cos(theta) = 3 * cos(theta)
y = (2 + 1) * sin(theta) = 3 * sin(theta)
By substituting different values for theta, you can obtain the corresponding (x, y) coordinates on the epicycloid.