Suppose a reflector is attached to a 26 inch diameter bike wheel, 8 inches from the center. If the bike wheel rolls without slipping, find parametric equations for the position of the reflector.
I thing i use parametrics????
To find the parametric equations for the position of the reflector as the bike wheel rolls without slipping, we can use the concept of circular motion.
Let's assume that at time t = 0, the reflector is at the highest point (12 o'clock position) on the wheel. We will use the angle θ, measured in radians, to describe the position of the reflector as it moves around the circumference of the wheel.
First, let's find the equation that relates the distance traveled along the circumference of the wheel, s, to the angle θ. The circumference of the wheel is given by C = 2πr, where r is the radius of the wheel. The radius can be calculated as half the diameter, so in this case, r = 26/2 = 13 inches.
The distance traveled can be found using the formula s = rθ. Now, we need to find the Cartesian coordinates of the reflector in terms of θ. For simplicity, let's assume the positive x-axis is located horizontally and to the right of the center of the wheel, and the positive y-axis is located vertically and above the center of the wheel.
The x-coordinate of the reflector can be found by considering the horizontal displacement of the center of the wheel as it moves along the circumference. Since the reflector is attached 8 inches from the center, the x-coordinate is given by x = rθ - 8sin(θ) (subtracting 8sin(θ) accounts for the offset of the reflector from the center).
The y-coordinate of the reflector can be found by considering the vertical displacement of the center of the wheel as it moves along the circumference. Since the reflector is attached 8 inches from the center, the y-coordinate is given by y = r - 8cos(θ) (subtracting 8cos(θ) accounts for the distance of the reflector below the top of the wheel).
So, the parametric equations of the position of the reflector on the bike wheel as it rolls without slipping are:
x = 13θ - 8sin(θ)
y = 13 - 8cos(θ)
These parametric equations describe the position of the reflector as the wheel rotates through each angle θ. Substitute different values of θ into the equations to get corresponding values for x and y, which will give you the position of the reflector at different points in time.