Problem 85 - Precalculus Mathematics for Calculus

Question

To measure the speed of a current, scientists place a paddle wheel in the stream and observe the rate at which it rotates. If the paddle wheel has radius \(0.20 \mathrm{m}\) and rotates at \(100 \mathrm{rpm},\) find the speed of the current in \(\mathrm{m} / \mathrm{s}\). (IMAGE CAN NOT COPY)

Step-by-Step Solution

Answer

The speed of the current is approximately 2.0944 m/s.

Step 1: Understand the Problem

We need to find the speed of the current by determining the linear speed of the paddle wheel which has a radius and rotates at a given speed (in rpm).

Step 2: Convert Rotational Speed to Radians Per Second

Given that the wheel rotates at 100 revolutions per minute (rpm), we convert this to radians per second. Since 1 revolution equals \(2\pi\) radians, we calculate: \[100 \times \frac{2\pi \, ext{radians}}{1 \, ext{revolution}} \times \frac{1 \, ext{minute}}{60 \, ext{seconds}} = \frac{100 \times 2\pi}{60} \text{ radians/second} = \frac{10\pi}{3} \text{ radians/second}.\]

Step 3: Use Radius to Find Linear Speed

The linear speed \(v\) (in meters per second) is given by the formula \(v = r \omega\), where \(\omega\) is the angular velocity in radians per second and \(r\) is the radius of the wheel. With \(r = 0.2 \) meters: \[v = 0.2 \times \frac{10\pi}{3} = \frac{2\pi}{3} \, ext{m/s}.\]

Step 4: Calculate Numerical Value

Calculate the numerical value of the speed: \[v \approx \frac{2 \times 3.1416}{3} \approx 2.0944 \, ext{m/s}.\]

Key Concepts

Understanding Linear Speed

Linear speed, often referred to as tangential speed, is a measure of how fast an object is moving along a path. In the context of rotational motion, it specifically describes the rate at which a point on the radius of a wheel is moving in a straight line.
To find linear speed, you need two pieces of information:

• The angular speed (how fast the object is rotating).
• The radius of the circle (distance from the center to the edge).

In our exercise, the paddle wheel has a radius of **0.20 meters**. The linear speed can be found using the formula:\[v = r \omega\]where **\(v\)** is the linear speed, **\(r\)** is the radius, and **\(\omega\)** is the angular speed. This formula neatly ties together the spinning of the wheel with how fast any point on its edge is moving, giving us insight into the current's speed.
The linear speed is what helps us relate rotational movement to straightforward distance and time measurements, such as meters per second, which are intuitive and easy to understand.

Exploring Angular Velocity

Angular velocity is a fundamental concept when dealing with rotational motion. It describes how quickly an object is rotating and is usually expressed in radians per second. This is different from linear speed, as it doesn’t measure distance but how much of the circle’s angle has been covered in a certain time.
In the problem, the paddle wheel rotates at **100 revolutions per minute (rpm)**. To work out the angular velocity, we convert rpm to radians per second. Knowing that one revolution equals **\(2\pi\) radians**, the angular velocity is given by:\[\omega = 100 \times \frac{2\pi}{60} = \frac{10\pi}{3} \text{ radians/second}\]This expression shows how extensive the angle is that the paddle wheel covers per second. Understanding this rotation is crucial for calculating other essential motion characteristics, such as linear speed.
Angular velocity connects the rotational motion of the wheel to the language of physics, providing us with a way to easily manipulate and use these quantities in various calculations.

The Importance of Converting Units

Converting units is essential in physics to ensure that quantities are understood in a consistent and useful manner. Missteps in conversion can lead to errors in calculations and misunderstandings of the results.
In our problem, we start with **100 revolutions per minute (rpm)**, a common way of expressing rotational speed. However, to link this to linear speed in meters per second, we need to convert the units into a form compatible with the other measurements. By converting revolutions to radians and minutes to seconds, we ensure unit consistency:

• 1 revolution = \(2\pi\) radians
• 1 minute = 60 seconds

This transformation is achieved by:\[100 \times \frac{2\pi \text{ radians}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} = \frac{10\pi}{3} \text{ radians/second}\]Now, with consistent units, we can multiply by the radius to find the linear speed accurately.
Understanding and mastering unit conversions helps streamline calculations and enhances the accuracy of the results, making analyses robust and reliable.