Problem 59 - Contemporary Precalculus

Question

A wheel is rotating around its axle. Find the angle (in radians) through which the wheel tums in the given time when it rotates at the given mumber of revolutions per minute ( \(r p m\) ). Assume that \(t>0\) and \(k>0\). 1 minute, 2 rpm

Step-by-Step Solution

Answer

Answer: The wheel turns through an angle of \(4\pi\) radians.

Step 1: Convert minutes to seconds

First, we need to convert the given time in minutes to seconds, as we want to find the angle in radians. Time in minutes: \(t = 1\) minute To convert 1 minute to seconds, we will use the following conversion factor: 1 minute = 60 seconds So, the time in seconds: \(t = 1 \cdot 60 = 60\) seconds.

Step 2: Convert rpm to radians per second

Now, let's convert the given revolutions per minute (rpm) to radians per second. Revolutions per minute: \(k = 2\) rpm Here's the conversion process in two steps: 1. Convert rpm to revolutions per second 2. Convert revolutions per second to radians per second First, convert rpm to revolutions per second using the following conversion factor: 1 revolution per minute = \(\frac{1}{60}\) revolutions per second So, \(k = 2 \cdot \frac{1}{60}\) revolutions per second Now, we need to convert revolutions per second to radians per second. Since one revolution corresponds to an angle of \(2\pi\): 1 revolution = \(2\pi\) radians So, the angular speed in radians per second is: \(\omega = 2 \cdot \frac{1}{60} \cdot 2\pi\) radians per second

Step 3: Find the angle in radians

Now that we have the angular speed in radians per second and the time in seconds, we can find the angle in radians by multiplying both values. Angle in radians \(= \omega \cdot t = \left( 2 \cdot \frac{1}{60} \cdot 2\pi \right) \cdot 60\) Simplifying the expression, we get: Angle in radians \(= 2 \cdot 2\pi = 4\pi\) radians. So, the wheel turns through an angle of \(4\pi\) radians in 1 minute at 2 rpm.

Key Concepts

Radians

In the realm of circular motion and angular motion, radians are super important! They help us measure angles in terms of the radius of a circle. Unlike degrees, which divide the circle into 360 parts, radians take a different approach. One radian is simply the angle at the center of a circle formed when the arc length is equal to the radius of the circle.
To visualize this, think about wrapping the radius of the circle along its circumference. This neat fit is what defines one radian. A complete circle, therefore, consists of an angle of 2π radians, which is equivalent to roughly 6.283 radians. To convert degrees to radians, use the formula:

• Radians = Degrees × \( \frac{\pi}{180} \)

The beauty of radians lies in their simplicity for calculations involving periodic and trigonometric functions. Understanding radians is crucial for analyzing anything spinning or rotating, like wheels or planets!

Revolutions per Minute (RPM)

Revolutions per Minute, often abbreviated as RPM, measures how many complete turns an object makes in one minute. This is a common metric in mechanics and engineering, especially when dealing with anything that spins. For example, motors, CD players, and bicycle wheels are often analyzed using RPM to determine their speed.
An important note is that RPM gives a measure of frequency, which is the number of cycles per unit time, just like how we measure hearing frequency in Hertz (cycles per second). To convert RPM to radians per second, a crucial step in many physics problems, use these steps:

• First, convert RPM to revolutions per second: \( \text{Revolutions per second} = \text{RPM} \times \frac{1}{60} \)
• Then, convert it to radians per second by multiplying by \(2\pi\).

This dual conversion captures both the cyclical nature of the rotation and the distance covered by such motion.

Time Conversion

Time conversion involves changing time units into other units more suitable for our calculations. In physics, seconds are often more useful than minutes because they align seamlessly with the International System of Units.
Converting minutes to seconds is straightforward:

• 1 minute = 60 seconds

Even though this seems trivial, converting time is fundamental when calculating velocities, accelerations, or any dynamic quantity because it provides a base in consistent units. Always make sure your time units match those specified in the problem or target units of your calculation for accuracy!

Angular Velocity

Angular velocity refers to the rate at which an object rotates around a point or axis. It tells us how fast something is turning. When looking at a wheel or a planet, you need to know how swiftly it rotates to understand its motion.
The angular velocity \(\omega\) is usually expressed in radians per second. The relationship between RPM and angular velocity involves both revolution count and circle geometry. Here's how you do the conversion:

• First, convert RPM to revolutions per second: \( \text{Revolutions per second} = \text{RPM} \times \frac{1}{60} \)
• Next, multiply by \(2\pi\) to get radians per second: \( \text{Angular velocity} \omega = \text{Revolutions per second} \times 2\pi \)

For motions involving angular displacement, such as in mechanical systems, angular velocity is fundamental to determining the dynamics and kinetics of the system. It simplifies the ability to predict how long it will take for a rotating object to reach a particular angle or to assess its rotational stability.