Math Problem Statement
a bicycle with 20-inch diameter wheels is traveling at 13 mi/h. find the angular speed of the wheels in rad/min. give exact answer
Solution
To find the angular speed of the bicycle's wheels in radians per minute, we can follow these steps:
Step 1: Convert the linear speed from miles per hour to inches per minute.
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The bicycle travels at 13 miles per hour. We need to convert this to inches per minute:
- 1 mile = 5280 feet, and 1 foot = 12 inches, so 1 mile = inches.
- There are 60 minutes in an hour.
Therefore, the speed in inches per minute is:
Step 2: Calculate the circumference of the wheel.
The diameter of the wheels is 20 inches, so the radius is: The circumference of a circle (the distance traveled in one full revolution of the wheel) is given by:
Step 3: Calculate the number of revolutions per minute.
To find the number of wheel revolutions per minute, we divide the linear speed (in inches per minute) by the circumference:
Step 4: Convert revolutions per minute to radians per minute.
Each revolution corresponds to radians. Therefore, the angular speed in radians per minute is:
Thus, the exact angular speed of the wheels is:
Would you like more details or clarifications on any step?
Here are 5 follow-up questions:
- How do you convert from linear speed to angular speed?
- What is the relationship between radius, circumference, and revolutions?
- How would the angular speed change if the wheel size increased?
- Can you convert the angular speed to radians per second?
- How does the formula for angular speed apply to other circular objects, such as gears?
Tip: Remember that angular speed is related to both the speed of travel and the size of the wheel. Larger wheels need fewer revolutions to maintain the same linear speed.
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Math Problem Analysis
Mathematical Concepts
Angular Speed
Unit Conversion
Circular Motion
Formulas
Linear speed (v) = distance/time
Circumference of a circle: C = 2πr
Angular speed (ω) = (Linear speed) / (Radius)
Revolutions to radians: 1 revolution = 2π radians
Theorems
Conversion between linear and angular speed
Relation between circumference and revolutions
Suitable Grade Level
Grades 10-12
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