I can do the Ferris Wheel type. This one I can not solve.
A light is attached to a 13 inch radius bicycle tire, at a point 8 inches from the center. If Harvey rides the bike at 15 mph, how fast is the light moving up and down at its fastest?
To solve this problem, we need to apply the concepts of circular motion and trigonometry.
First, let's find the circumference of the bicycle tire. The circumference of a circle is given by the formula C = 2πr, where r is the radius. In this case, the radius is 13 inches, so the circumference is C = 2π(13) = 26π inches.
Since the light is attached 8 inches from the center of the tire, it traces out the path of a circle with a radius of 8 inches. The light's motion can be broken down into two components: the horizontal motion as the bike moves forward and the vertical motion as the light moves up and down.
The horizontal motion of the light is just the same as the bike's velocity, which is 15 mph. So, the horizontal speed of the light is 15 mph.
To find the vertical speed of the light, we need to consider the relationship between the linear speed and angular speed when an object is in circular motion. The linear speed of a point on a rotating object is equal to the product of its angular speed (in radians per unit time) and its distance from the center of rotation. In this case, the light is 8 inches from the center and the bike is moving at 15 mph, which is equivalent to (15 * 5280) / 3600 = 22 feet per second.
To convert this linear speed to angular speed, divide it by the radius of the light's path (8 inches or 2/3 feet). The angular speed is then 22 / (2/3) = 33 revolutions per second.
Now, we can calculate the vertical speed of the light. The vertical speed is given by the product of the angular speed and the radius of the light's path (8 inches). Therefore, the vertical speed is 33 revolutions per second * 8 inches, or 264 inches per second.
Thus, the light is moving up and down at its fastest at a speed of 264 inches per second.