Problem 44 - Calculus

Question

A compact disc spins at a rate of 200 to 500 revolutions per minute. What are the equivalent rates measured in radians per second?

Step-by-Step Solution

Answer

The rates are \( \frac{20\pi}{3} \) to \( \frac{50\pi}{3} \) radians per second.

Step 1: Understand the relationship between revolutions and radians

One complete revolution is equivalent to rotating through an angle of \(2\pi\) radians. Thus, if a disc spins at a certain number of revolutions per minute, we can convert each revolution to radians by multiplying by \(2\pi\).

Step 2: Convert revolutions per minute to revolutions per second

To convert the spin from revolutions per minute to revolutions per second, divide the number of revolutions per minute by 60 (since there are 60 seconds in a minute). For example: \( \frac{200}{60} = \frac{10}{3} \) and \( \frac{500}{60} = \frac{25}{3} \).

Step 3: Convert revolutions per second to radians per second

Multiply the revolutions per second obtained in Step 2 by \(2\pi\) to get the angular velocity in radians per second. For \( \frac{10}{3} \) revolutions/second, the conversion is \( \frac{10}{3} \times 2\pi = \frac{20\pi}{3} \) radians/second. For \( \frac{25}{3} \) revolutions/second, the conversion is \( \frac{25}{3} \times 2\pi = \frac{50\pi}{3} \) radians/second.

Key Concepts

Revolutions Per Minute

Revolutions per minute (RPM) is a measure of the rotational speed of an object. It indicates how many complete turns an object makes in one minute.
RPM is commonly used for engines and other machinery to describe how fast they are operating. To fully understand RPM:

• A higher RPM means a faster spinning object.
• It is a time-based measure, specifically using one minute as the base unit of time.

When dealing with problems involving RPM, it's important to think about the relationship between time and rotation. This understanding can help when converting between different measures of angular speed.

Radians Per Second

Radians per second is another way to describe angular velocity. It measures how fast an object rotates or spins through radians over one second.
The concept of radians is central in this unit:

• One complete circle is equal to an angle of \(2\pi\) radians.
• The radian is a natural unit of angle, allowing calculations in physics and engineering to be incredibly convenient.

Angular velocity in radians per second provides a time-independent way to express rotational speed. This is because radians are a distance measure along the arc of a circle, allowing straightforward integrations into formulas for angular motion.

Unit Conversion

Converting between units is essential in physics and engineering to ensure consistency in calculations. The steps to perform these conversions correctly involve understanding the relationship between the units.
For converting RPM to radians per second, two main steps are vital:

• First, convert the count of revolutions per minute to revolutions per second by dividing by 60, considering there are 60 seconds in a minute.
• Next, convert revolutions per second to radians per second by multiplying the number of revolutions by \(2\pi\). This accounts for the fact that each revolution corresponds to \(2\pi\) radians.

Understanding how to change units leverages mathematical principles, ensuring more precise measurements for various applications like mechanics and kinematics.