suppose you are biking down a hill at 24 mi/h. What is the angular speed, in radians per second, of your 27-inich-diameter bicycle wheel?
To find the angular speed of a bicycle wheel, we first need to understand the relationship between linear speed (mi/h) and angular speed (radians/s).
The linear speed of an object moving in a circular path is calculated by multiplying the radius of the circle by its angular speed. In this case, the bicycle wheel acts like a circle, and the linear speed can be calculated by multiplying the radius of the wheel by its angular speed.
Given that the diameter of the bicycle wheel is 27 inches, we can calculate the radius:
Radius = Diameter / 2 = 27 inches / 2 = 13.5 inches
Next, we need to convert the linear speed, which is given in mi/h, into inches per hour. Since the radius is in inches, it is important to have the units consistent. To do this conversion, we need to use the relationship that 1 mile equals 5280 feet, and 1 foot equals 12 inches. Thus, 1 mile equals 5280 * 12 = 63,360 inches.
Linear speed (in inches per hour) = 24 mi/h * 63,360 inches/mile = 1,520,640 inches/hour
Now we have the linear speed and the radius. To find the angular speed, we divide the linear speed by the radius:
Angular speed = Linear speed / Radius = 1,520,640 inches/hour / 13.5 inches = 112,654.81 inches/hour
To convert the angular speed from inches/hour to radians/second, we need to use conversion factors. There are 2π radians in a circle, 60 minutes in an hour, and 60 seconds in a minute.
Angular speed (in radians/second) = (112,654.81 inches/hour * 2π radians) / (60 minutes/hour * 60 seconds/minute)
Simplifying the equation yields:
Angular speed ≈ 3.54 radians/second
Therefore, the angular speed of the bicycle wheel is approximately 3.54 radians per second.