Problem 5 - College Physics

Question

Express the following angles in degrees, radians, - Express the following angles in degrees, radians, and/or revolutions (rev) as appropriate: (a) \(105^{\circ}\) (b) \(1.8 \mathrm{rad},\) and \((\mathrm{c}) 5 / 7\) rev.

Step-by-Step Solution

Answer

(a) \(105^{\circ} = \frac{7\pi}{12} \text{ rad} = \frac{7}{24} \text{ rev}\); (b) \(1.8 \text{ rad} \approx 103.13^{\circ} = 0.2865 \text{ rev}\); (c) \(\frac{5}{7} \text{ rev} \approx 257.14^{\circ} \approx 4.49 \text{ rad}\).

Step 1: Convert Degrees to Radians and Revolutions

To convert from degrees to radians, use the formula: \[\text{radians} = \frac{\pi}{180} \times (\text{degrees})\]To convert from degrees to revolutions, use the formula: \[\text{revolutions} = \frac{\text{degrees}}{360}\]For \(105^{\circ}\): Convert to radians: \[\frac{\pi}{180} \times 105 = \frac{105\pi}{180} = \frac{7\pi}{12} \text{ radians}\]Convert to revolutions: \[\frac{105}{360} = \frac{7}{24} \text{ revolutions}\]

Step 2: Convert Radians to Degrees and Revolutions

To convert from radians to degrees, use the formula: \[\text{degrees} = \frac{180}{\pi} \times (\text{radians})\]Convert 1.8 radians to degrees: \[\frac{180}{\pi} \times 1.8 \approx 103.13^{\circ}\]Convert to revolutions using the conversion from radians:\[\text{revolutions} = \frac{1.8}{2\pi} \approx 0.2865 \text{ revolutions}\]

Step 3: Convert Revolutions to Degrees and Radians

To convert revolutions to degrees, use the formula: \[\text{degrees} = (\text{revolutions}) \times 360\]Convert \(\frac{5}{7}\) revolutions to degrees: \[\frac{5}{7} \times 360 \approx 257.14^{\circ}\]For the radian conversion:\[\text{radians} = (\text{revolutions}) \times 2\pi\]\[\frac{5}{7} \times 2\pi \approx 4.49 \text{ radians}\]

Key Concepts

Degrees

Degrees are the unit of measure most people are familiar with when it comes to angles. Every circle is divided into 360 of these slices, making it easy to understand and visualize angles in practical life. Each degree can be further divided into 60 minutes (not time minutes!), and each minute is divided into 60 seconds, which adds precision if needed in detailed calculations.
For example, if you have an angle of 105 degrees, you can break it down into smaller parts like this:

• 1 degree = 60 minutes
• 1 minute = 60 seconds

When converting degrees to other units of angle measurement, like radians or revolutions, the degree's role is usually the starting point because of its familiarity and ease of use. In many fields, especially in navigation and geography, degrees are like the default language when talking about direction and orientation of angles.

Radians

Radians are another way to measure angles and they're based on the radius of a circle. If you imagine unwrapping the circle’s radius along the circumference, when you trace out an entire circle, you've got what's called "2π radians." Each radian measures the angle created when you take the radius of a circle and stretch it into an arc on the circumference equal in length to that radius. This concept aligns angles directly with arc length, making it very useful in mathematical analysis and calculus.
The conversion between degrees and radians may initially seem tricky, but it's useful to remember:

• 180 degrees equals π radians
• Thus, 1 degree equals π/180 radians

This understanding is fundamental when you need to switch between systems. For instance, an angle of 105 degrees would convert to radians using the formula: \[\frac{105\pi}{180} = \frac{7\pi}{12}\text{ radians}\]. This makes radians particularly elegant for mathematical functions and equations due to their natural relation to π.

Revolutions

Revolutions are probably the most intuitive way to think about angles if you picture rotations or spinning objects. One revolution is simply a full 360-degree turn around a circle, meaning a circle's full completion—from one point around to the same point again. When dealing with machinery, especially things like wheels, pulleys, or gears, revolutions often form the easiest way to express rotational motion.
A full understanding of revolutions involves noting that:

• 1 revolution = 360 degrees
• 1 revolution = 2π radians

This makes conversion straightforward. For instance, converting 105 degrees to revolutions involves dividing by 360, yielding approximately \[0.2916\] revolutions. Alternatively, in a fractional form, it's precisely \[\frac{7}{24}\] revolutions. This system is flexible, moving easily between the ideas of distance traveled around the edges of circles, to changes occurring within cycles or periodic systems.