Problem 122 - Precalculus

Question

Explain the mistake that is made. If a bicycle has tires with radius 10 inches and the tires rotate \(90^{\circ}\) per \(\frac{1}{2}\) second, how fast is the bicycle traveling (linear speed) in miles per hour? Solution: Write the formula for linear speed. \(v=r \omega\) Let \(r=10\) inches and \(\omega=180^{\circ}\) per second. $$v=(10 \text { in. })\left(\frac{180^{\circ}}{\sec }\right)$$ Simplify. \( v=\frac{1800 \mathrm{in} .}{\mathrm{sec}}\) Let 1 mile \(=5280\) feet \(=63,360\) inches and 1 hour \(=3600\) seconds. $$v=\left(\frac{1800 \cdot 3600}{63,360}\right) \mathrm{mph}$$ Simplify. \(v \approx 102.3 \mathrm{mph}\) This is incorrect. The correct answer is approximately 1.8 mph. What mistake was made?

Step-by-Step Solution

Answer

The mistake was not converting angular speed to radians, leading to an incorrect unit calculation.

Step 1: Understanding the Problem

We need to calculate the linear speed of a bicycle given the radius of its tires and the rotational speed of the tires. The linear speed formula is given as \( v = r \omega \), where \( v \) is the linear speed, \( r \) is the radius, and \( \omega \) is the angular speed.

Step 2: Convert Angular Speed to Radians per Second

The angular speed, \( \omega \), given is \( 90^{\circ}\) per \( \frac{1}{2} \) second, which equals \( 180^{\circ} \) per second. However, angular speed should be in radians per second for the formula: \( \omega = 180^{\circ} \times \frac{\pi}{180} = \pi \) radians per second.

Step 3: Apply Linear Speed Formula Correctly

Substitute \( r = 10 \) inches and \( \omega = \pi \) rad/sec into the formula: \( v = r \omega = 10 \cdot \pi = 10\pi \) inches per second.

Step 4: Convert Units to Miles per Hour

Convert the speed from inches per second to miles per hour. There are 63,360 inches in a mile and 3600 seconds in an hour. So, use the conversion: \( v = 10\pi \times \frac{1}{63,360} \times 3600 \) miles/hour.

Step 5: Simplify and Calculate Final Speed

Simplify the expression: \( v = 10\pi \times \frac{3600}{63,360} = \pi \times \frac{1}{5.28} \approx 1.8 \) mph.

Key Concepts

Angular Speed

Angular speed is a measure of how fast an object rotates or revolves around a central point, often the center of a circle. It describes how much an angle is swept out over a specific period of time. In exercises, angular speed ( \(\omega\) ) is typically given in degrees per second or radians per second.
- **Degrees Per Second**: A common unit that describes rotation in terms of angles, measured in degrees. However, it’s not the standard unit for angular calculations. - **Radians Per Second**: The standard unit of angular speed in mathematics and physics. It provides a direct measure of rotation equivalent to linear distance.
For effective use in calculations, angular speed should be converted to radians per second, since radian measures are integrated into the general formulas for motion and forces.

Unit Conversion

Unit conversion is essential when performing calculations involving different measurement systems. In the given problem, converting units correctly was crucial to finding the accurate linear speed.
Here’s a guide to converting units effectively:

• **Degrees to Radians**: Use the conversion factor \(\pi/180\) to change degrees to radians since 180 degrees equal \(\pi\) radians.
• **Inches to Miles**: There are 63,360 inches in a mile, so convert by dividing the number of inches by 63,360.
• **Seconds to Hours**: With 3,600 seconds in an hour, multiply by 3,600 to convert seconds to hours.

Accurate unit conversions ensure that all components of a formula work together without error. A misstep in conversion can lead to dramatically incorrect results, as seen in the original problem's high-speed miscalculation.

Radians

Radians are a unit of angular measure used in mathematics to describe angles. Unlike degrees, which divide a circle into 360 equal parts, radians divide it in terms of the radius itself.
- **Definition**: One radian is the angle created when the arc length is equal to the radius of the circle.- **Conversion**: To convert from degrees to radians, multiply by \(\pi/180\) because a complete circle is \(2\pi\) radians.
Understanding radians is important because they simplify many mathematical expressions, especially in trigonometry and calculus. Many formulas for things like angular speed, area of circles, and periodic functions rely on radians for their calculations.

Linear Speed Formula

The linear speed formula is used to calculate the speed at which an object moves along a circular path. The formula is \(v = r \omega\) , where \(v\) is the linear speed, \(r\) is the radius of the path, and \(\omega\) is the angular speed.
Using the formula efficiently demands that both \(r\) and \(\omega\) be in compatible units. For example, if the radius is measured in inches and angular speed is given in radians per second, the resultant linear speed will be in inches per second.
- **Calculate:** Substitute the values into the formula correctly to find the linear speed.- **Convert:** If necessary, convert the resulting linear speed into a different unit, like miles per hour, for practical applications.
Misapplication of this formula often stems from incorrect unit handling, leading to errors like those seen in the original exercise where a bicycle’s speed was drastically overestimated.