Problem 58 - 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 \(k>0\). \(t\) minutes, \(1 \mathrm{rpm}\)

Step-by-Step Solution

Answer

Answer: The angle through which the wheel turns is \(2\pi t\) radians.

Step 1: Convert rotations per minute to radians per minute

To convert the given rotation speed in rpm to radians per minute, we use the following conversion factor: 1 revolution = \(2\pi\) radians Given the rotation speed is 1 rpm, so we multiply it by the conversion factor: \(1\mathrm{rpm} \times 2\pi\mathrm{radians/revolution} = 2\pi\mathrm{radians/minute}\)

Step 2: Calculate the angle turned in radians

Now that we have the rotation speed in radians per minute, we can find the angle turned in radians. To do this, we simply multiply the rotational speed in radians per minute by the given time in minutes. Angle in radians = Rotation speed in radians per minute × Time in minutes Given \(t\) minutes and rotation speed in radians per minute is \(2\pi\mathrm{radians/minute}\), we have: Angle in radians = \(2\pi\mathrm{radians/minute} \times t\mathrm{minutes} = 2\pi t\mathrm{radians}\) Thus, the angle through which the wheel turns in the given time at 1 rpm is \(2\pi t\) radians.

Key Concepts

Radians

Radians are a measure of angles used commonly in trigonometry and calculus. They offer a natural way to think about angles as they relate to the geometry of a circle.
Unlike degrees, which divide a circle into 360 parts, radians define one complete revolution as \(2\pi\) radians.
This makes the radian measure simpler and more convenient in many mathematical calculations, especially involving circular and periodic patterns.
For example:

• Quarter a circle: \(\frac{\pi}{2}\) radians
• Halfway around: \(\pi\) radians
• Full circle: \(2\pi\) radians

When using radians, the key is to understand how angles relate to the length of the arc they form. This relationship is given by the formula: \(\theta = \frac{s}{r}\), where \(\theta\) is the angle in radians, \(s\) is the arc length, and \(r\) is the radius of the circle.
Thus, the radian provides a straightforward connection between linear and angular motion.

Rotational Motion

Rotational motion refers to the movement of an object around a center or point of rotation, such as a wheel turning around its axle.
This concept is fundamental in physics and engineering, as many systems and machines involve rotating parts. The key aspect of rotational motion is the rotational speed, which is often measured in revolutions per minute (rpm).
In physics, particularly when working with angular quantities, it's often more convenient to express this speed in radians per minute.
This allows for easier integration with other mathematical equations involving angular displacement. Understanding rotational motion also involves:

• Angular displacement, which measures how much an object has rotated and is often given in radians.
• Angular velocity, indicating how fast the object is rotating, commonly in radians per second.
• Understanding the relationship between linear and angular quantities, such as velocity and displacement.

By considering these aspects, one can predict and calculate the behavior of rotating systems accurately.

Conversion Factors

Conversion factors are used to transform measurements from one unit system to another, ensuring that calculations are precise and consistent.
In this context, conversion factors are essential when moving between angular velocity in revolutions per minute (rpm) and radians per minute.Here's how it works:To convert from revolutions to radians, we use the fact that one complete revolution around a circle is \(2\pi\) radians.
When a problem provides a measurement in rpm (like this exercise), multiplying by \(2\pi\) converts it to radians per minute:

• 1 rpm = \(1 \times 2\pi = 2\pi\) radians/minute

Whether you're dealing with time, distance, or any other measurement, conversion factors maintain the dimensional integrity of your calculations.
They are foundational in ensuring that the resulting equations and predictions are based on consistent units, thus enabling accurate and meaningful results.