Problem 1 - Applied Physics

Question

Convert \(6 \frac{1}{2}\) revolutions (a) to radians. (b) to degrees.

Step-by-Step Solution

Answer

(a) 13\pi radians, (b) 2340 degrees.

Step 1: Understand the Problem

We have a mixed number, \(6 \frac{1}{2}\), which represents total revolutions. We need to convert these revolutions to both radians and degrees.

Step 2: Convert Revolutions to a Fraction

Convert the mixed number \(6 \frac{1}{2}\) into an improper fraction. This can be done by multiplying the whole number part by the denominator of the fraction and adding it to the numerator: \(6 \frac{1}{2} = \frac{6 \times 2 + 1}{2} = \frac{13}{2}\).

Step 3: Convert Revolutions to Radians

One complete revolution is equivalent to \(2\pi\) radians. Therefore, \(\frac{13}{2}\) revolutions is equivalent to \(\frac{13}{2} \times 2\pi = 13\pi\) radians.

Step 4: Convert Revolutions to Degrees

One complete revolution is equivalent to 360 degrees. Therefore, \(\frac{13}{2}\) revolutions is equivalent to \(\frac{13}{2} \times 360 = 2340\) degrees.

Key Concepts

Mixed Number Conversion

A mixed number consists of a whole number and a fraction and often makes calculations a bit tricky. For the purpose of conversion, mixed numbers can be changed into improper fractions.
This involves multiplying the whole number by the denominator of the fraction and adding it to the existing numerator.

• In our example, we have the mixed number \(6 \frac{1}{2}\).
• To convert, we multiply the whole number 6 by the denominator 2, yielding 12.
• Then, add the numerator 1 to get 13, making the improper fraction \(\frac{13}{2}\).

This conversion helps to simplify calculations, especially when working with problems involving measurements like revolutions.

Improper Fractions

An improper fraction is a type of fraction where the numerator is greater than or equal to the denominator.
This is different from a proper fraction, where the numerator is less than the denominator.

• Improper fractions are particularly useful when performing multiplication or division, as they work well in mathematical operations.
• In the example given, the improper fraction \(\frac{13}{2}\) efficiently represents the number of revolutions for conversion purposes.

Understanding improper fractions is crucial in many areas of mathematics, providing a consistent format that is often easier to work with than mixed numbers.

Radian Measure

The radian is a pivotal unit of angular measure used in many areas of mathematics.
It is particularly significant in trigonometry and calculus.

• One complete revolution equals \(2\pi\) radians, which naturally ties the concept of angle to the length of the arc on a unit circle.
• In converting revolutions to radians, you multiply the number of revolutions by \(2\pi\).
• For the improper fraction \(\frac{13}{2}\), the conversion gives \(\frac{13}{2} \times 2\pi = 13\pi\) radians.

This system links angle measures directly to the circumference of a circle, offering an elegant and precise method of measurement.

Degree Measure

Degrees are another way to measure angles commonly used in everyday applications and school mathematics.
Degrees provide an easy-to-understand, straightforward method for expressing angles.

• One complete revolution is equated to 360 degrees.
• In our conversion, the calculation becomes \(\frac{13}{2} \times 360 = 2340\) degrees.
• This highlights how rotations can be expressed in a format that might be more intuitive for people familiar with everyday measurements.

Understanding degree measure is essential for surveying, astronomy, navigation, and many other practical applications.