A bicyclist is traveling at an angular speed of 8pi radians per second. How fast is she traveling in miles per hour if her tires are 28 inches in diameter?
To find the linear speed of the bicyclist in miles per hour, we need to convert her angular speed from radians per second to miles per hour.
First, let's convert the tire diameter from inches to miles. We know that there are 12 inches in a foot and 5280 feet in a mile. So, the tire diameter is:
28 inches * 1 foot/12 inches * 1 mile/5280 feet = 28/12/5280 miles = 0.000378788 miles
The circumference of the tire is equal to the diameter times pi (π), so the circumference is:
Circumference = π * diameter = π * 0.000378788 miles = 0.001189269 miles
To find the linear speed in miles per second, we can multiply the angular speed by the circumference:
Linear speed = Angular speed * Circumference = 8π radians/second * 0.001189269 miles/radian = 0.009514154 miles/second
Finally, let's convert the linear speed from miles per second to miles per hour. There are 60 seconds in a minute and 60 minutes in an hour, so:
Linear speed = 0.009514154 miles/second * 60 seconds/minute * 60 minutes/hour = 34.2509544 miles/hour
Therefore, the bicyclist is traveling at a speed of approximately 34.25 miles per hour.
To find the speed of the bicyclist in miles per hour, we can use the relationship between angular speed (ω) and linear speed (v) for a rotating object. The formula is:
v = ω * r,
where ω is the angular speed in radians per second and r is the radius of the circle or the distance from the center to the point of interest.
In this case, the radius is equal to half the diameter of the tires, which is 28 inches / 2 = 14 inches.
Now, we need to convert the units from inches to miles. Since there are 12 inches in a foot and 5280 feet in a mile, we have:
v = (8π radians/second) * (14 inches) * (1 foot/12 inches) * (1 mile/5280 feet).
Let's simplify the expression:
v = (8π * 14) / (12 * 5280) miles/second.
Since there are 3600 seconds in an hour, we can convert from miles/second to miles/hour by multiplying by 3600:
v = [(8π * 14) / (12 * 5280)] * 3600 miles/hour.
Now, we can calculate the value of v:
v ≈ 11.50 miles/hour.
Therefore, the bicyclist is traveling at a speed of approximately 11.50 miles per hour.
horizontal
speed=angularspeed*radius
that will give you speed in inches/second
convert that to mph