Understanding the Lawn Roller Problem

The problem asks us to find the total angle, measured in radians, that a lawn roller turns through in 25 minutes, given that it completes 20 revolutions in one hour. To solve this, we need to understand the relationship between revolutions, time, and radians.

Converting Revolutions to Radians

One full revolution corresponds to a complete circle, which is equivalent to \(2\pi\) radians. This is a fundamental conversion factor we will use.

Calculating Revolutions per Unit Time

We are told the lawn roller makes 20 revolutions in one hour. One hour is equal to 60 minutes. So, the rate of revolution is:

\( \text{Revolutions per minute} = \frac{\text{Total revolutions}}{\text{Total time in minutes}} \)

\( \text{Revolutions per minute} = \frac{20 \text{ revolutions}}{60 \text{ minutes}} \)

\( \text{Revolutions per minute} = \frac{1}{3} \text{ revolutions/minute} \)

Finding Total Revolutions in 25 Minutes

Now that we know the roller makes \( \frac{1}{3} \) of a revolution every minute, we can find out how many revolutions it makes in 25 minutes:

\( \text{Total revolutions in 25 minutes} = \text{Revolutions per minute} \times \text{Time in minutes} \)

\( \text{Total revolutions in 25 minutes} = \frac{1}{3} \text{ revolutions/minute} \times 25 \text{ minutes} \)

\( \text{Total revolutions in 25 minutes} = \frac{25}{3} \text{ revolutions} \)

Converting Total Revolutions to Radians

Finally, we convert the total number of revolutions in 25 minutes into radians. Since 1 revolution is \(2\pi\) radians:

\( \text{Total radians} = \text{Total revolutions} \times \text{Radians per revolution} \)

\( \text{Total radians} = \frac{25}{3} \text{ revolutions} \times 2\pi \text{ radians/revolution} \)

\( \text{Total radians} = \frac{50\pi}{3} \text{ radians} \)

Therefore, the lawn roller runs through \( \frac{50\pi}{3} \) radians in 25 minutes.

Step-by-Step Calculation Summary

1. Identify the given information: 20 revolutions in 1 hour (60 minutes).
2. Determine the target time: 25 minutes.
3. Calculate the revolution rate per minute: \( \frac{20}{60} = \frac{1}{3} \) revolutions/minute.
4. Calculate the total revolutions in 25 minutes: \( \frac{1}{3} \times 25 = \frac{25}{3} \) revolutions.
5. Convert total revolutions to radians using the conversion \(1 \text{ revolution} = 2\pi \text{ radians}\): \( \frac{25}{3} \times 2\pi = \frac{50\pi}{3} \) radians.

Revision Table: Key Concepts for Revolutions and Radians

Additional Information: Angular Displacement

The total angle turned through is also known as angular displacement. It is a measure of how much an object has rotated relative to a reference point. In this problem, the lawn roller undergoes angular displacement as it rolls.

• Angular displacement can be measured in degrees, radians, or revolutions.
• Radians are the standard unit for angular measurement in physics and many mathematical contexts because they simplify formulas involving circular motion.
• The rate of angular displacement is called angular velocity, often measured in radians per second. In this problem, we first found the rate in revolutions per minute and then converted the total displacement to radians.

Understanding the relationship between linear motion (like the distance covered by the roller's circumference) and angular motion (the angle it turns) is crucial in problems involving rotating objects.