Problem 59 - Precalculus

Question

A bicycle with 24 -inch diameter wheels is traveling at 15 mih. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?

Step-by-Step Solution

Answer

The wheels make about 17.48 RPM and have an angular speed of about 109.65 rad/min.

Step 1: Understanding the Relationship

To find the angular speed, we need to understand the relationship between linear speed (mph) and angular speed (rad/min). The bicycle has a linear speed, and each complete revolution of a wheel relates to a certain distance traveled.

Step 2: Calculate the Linear Speed in Inches per Minute

First, convert the linear speed from miles per hour to inches per minute. Since 1 mile = 5280 feet and 1 foot = 12 inches, we have:\[ 15 \text{ mph} = 15 \times 5280 \text{ feet/hour} \]\[ = 15 \times 5280 \times 12 \text{ inches/hour} \]\[ = 15 \times 5280 \times 12 \div 60 \text{ inches/minute} \]Calculate numerical value: \[ 15 \times 88 = 1320 \text{ inches/minute} \]

Step 3: Find the Wheel Circumference

The circumference of the wheel gives the distance traveled in one complete revolution. The formula for circumference is:\[ C = \pi \times \text{diameter} = \pi \times 24 \text{ inches} \]

Step 4: Calculate Revolutions per Minute (RPM)

Revolutions per minute is the number of times the wheel revolves in a minute:\[ \text{RPM} = \frac{\text{Linear speed in inches per minute}}{\text{Circumference}} \]\[ \text{RPM} = \frac{1320}{\pi \times 24} \]Calculate numerical value.

Step 5: Calculate Angular Speed in Radians per Minute

Angular speed (\(\omega\)) relates to RPM by the formula:\[ \omega = \text{RPM} \times 2\pi \text{ radians} \]

Step 6: Final Calculation

Substitute the calculated RPM into the angular speed formula:\[ \omega = \frac{1320}{\pi \times 24} \times 2\pi = \frac{1320 \times 2}{24} \]Calculate numerical value to find the angular speed in radians per minute.

Key Concepts

Linear Speed Conversion

When dealing with problems involving speeds, it's often necessary to convert from one unit of measurement to another. Linear speed, which is typically given in miles per hour (mph), can be converted into inches per minute. This conversion is vital for understanding various rotational dynamics. Here's the process:

• Start by recognizing that 1 mile is equal to 5280 feet.
• Then, convert feet to inches, knowing there are 12 inches in a foot.
• Finally, to go from hours to minutes, divide by 60, since there are 60 minutes in an hour.

For example, to convert 15 mph to inches per minute, follow this calculation: \[15 \text{ miles/hour} \times 5280 \text{ feet/mile} \times 12 \text{ inches/foot} \div 60 \text{ minutes/hour} = 1320 \text{ inches/minute}\] Understanding this conversion not only helps in solving the exercise but also sets the foundation for grasping how linear speed translates into rotational motion.

Revolutions per Minute

Revolutions per minute (RPM) is a unit of rotational speed or the frequency of rotation. It indicates how many rotations a wheel makes in one minute. This metric is crucial in connecting linear motion with angular motion. To find RPM given the linear speed, you need:

• The linear speed in an appropriate unit, such as inches per minute.
• The circumference of the wheel since it represents one full revolution.

The formula to calculate RPM is straightforward: \[ \text{RPM} = \frac{\text{Linear speed in inches per minute}}{\text{Circumference in inches}} \]For the exercise, with a 24-inch wheel, you first determine the circumference \[ C = \pi \times \text{diameter} = \pi \times 24 \text{ inches} \] Then calculate the RPM by inserting the linear speed calculation: \[ \text{RPM} = \frac{1320}{\pi \times 24} \] This gives you the number of revolutions made by the wheel per minute, essential for calculating angular speed.

Circumference Calculation

Circumference is a crucial measurement in understanding rotational dynamics. It represents the distance that a point on the edge of a rotating object, like a wheel, travels in one full revolution.To calculate the circumference of a wheel:- Use the formula: \[ C = \pi \times \text{diameter} \]It's important to ensure that the diameter is in the appropriate units for the problem, typically inches in our case.For example, with a 24-inch diameter wheel, the circumference would be \[ C = \pi \times 24 \text{ inches} \]This means each revolution of the wheel covers this distance, directly linking to RPM calculations. By understanding the concept of circumference, you can easily transition from linear motion calculations to those involving rotational dynamics.

Radian Measure

Radial measure in geometry and trigonometry is vital for connecting linear motion to angular motion. Unlike degrees, which divide a circle into 360 parts, radians are based on the circle's radius.The angle in radians, especially in circular motion, is given by the arc length divided by the radius. For angular speed:- The formula to convert RPM to angular speed in radians per minute is: \[ \omega = \text{RPM} \times 2\pi \text{ radians} \]This formula uses the fact that one full revolution equates to an angle of \(2\pi\) radians. Given an RPM value calculated earlier, you can easily find the angular speed in radians per minute.By understanding radians, you see how rotational movement is measured in terms of the intrinsic geometry of circles, bridging linear speeds and rotational speeds.