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.

  1. 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 = 5280×12=63,3605280 \times 12 = 63,360 inches.
    • There are 60 minutes in an hour.

    Therefore, the speed in inches per minute is: Speed in inches per minute=13mi/h×63,360in1mi×160min=13×63,36060=13,728in/min.\text{Speed in inches per minute} = 13 \, \text{mi/h} \times \frac{63,360 \, \text{in}}{1 \, \text{mi}} \times \frac{1}{60 \, \text{min}} = \frac{13 \times 63,360}{60} = 13,728 \, \text{in/min}.

Step 2: Calculate the circumference of the wheel.

The diameter of the wheels is 20 inches, so the radius rr is: r=202=10in.r = \frac{20}{2} = 10 \, \text{in}. The circumference CC of a circle (the distance traveled in one full revolution of the wheel) is given by: C=2πr=2π×10=20πin.C = 2\pi r = 2\pi \times 10 = 20\pi \, \text{in}.

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: Revolutions per minute=Speed in inches per minuteCircumference=13,72820π=13,72862.8319218.53rev/min.\text{Revolutions per minute} = \frac{\text{Speed in inches per minute}}{\text{Circumference}} = \frac{13,728}{20\pi} = \frac{13,728}{62.8319} \approx 218.53 \, \text{rev/min}.

Step 4: Convert revolutions per minute to radians per minute.

Each revolution corresponds to 2π2\pi radians. Therefore, the angular speed in radians per minute is: Angular speed=218.53rev/min×2πrad/rev=218.53×2π1373.1rad/min.\text{Angular speed} = 218.53 \, \text{rev/min} \times 2\pi \, \text{rad/rev} = 218.53 \times 2\pi \approx 1373.1 \, \text{rad/min}.

Thus, the exact angular speed of the wheels is: 13,72810rad/min=1372.8rad/min.\boxed{\frac{13,728}{10} \, \text{rad/min}} = 1372.8 \, \text{rad/min}.

Would you like more details or clarifications on any step?

Here are 5 follow-up questions:

  1. How do you convert from linear speed to angular speed?
  2. What is the relationship between radius, circumference, and revolutions?
  3. How would the angular speed change if the wheel size increased?
  4. Can you convert the angular speed to radians per second?
  5. 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