Problem 2 - Applied Physics

Question

Convert \(2880^{\circ}\) (a) to revolutions. (b) to radians.

Step-by-Step Solution

Answer

(a) 8 revolutions, (b) 16π radians.

Step 1: Understand the Problem Requirements

We are given an angle of \(2880^{\circ}\) and need to convert it first into revolutions and then into radians.

Step 2: Converting Degrees to Revolutions

To convert degrees to revolutions, use the conversion factor that one revolution is \(360^{\circ}\). Calculate the number of revolutions:\[\text{Number of revolutions} = \frac{2880^{\circ}}{360^{\circ}} \]\[= 8 \] So, \(2880^{\circ}\) is equal to 8 revolutions.

Step 3: Converting Degrees to Radians

To convert degrees to radians, use the conversion factor \(180^{\circ} = \pi\, \text{radians}\).Calculate the number of radians:\[\text{Number of radians} = 2880^{\circ} \times \frac{\pi}{180^{\circ}} \]\[= 2880 \times \frac{\pi}{180} = 16\pi \, \text{radians} \]So, \(2880^{\circ}\) is equal to \(16\pi\) radians.

Key Concepts

Degrees to Revolutions

Converting degrees to revolutions is a simple process once you understand the basic relationship between these two units of measurement. A full revolution is equivalent to a circle, or more precisely, it is equivalent to 360 degrees.
To convert an angle calculated in degrees into revolutions, you utilize this fundamental relationship:

• One revolution = 360 degrees

To calculate the number of revolutions from a given number of degrees, you simply divide the degrees by 360. For instance, if you have 2880° and want to know how many complete revolutions that represents, you compute:\[\text{Number of revolutions} = \frac{2880}{360} = 8\]This shows there are 8 full revolutions in 2880 degrees. Easy peasy!

Degrees to Radians

Radians offer another powerful method for expressing angles. Unlike degrees, radians are based on the radius of a circle. Specifically, one full revolution of a circle is equal to \(2\pi\) radians.
To convert degrees into radians, you use the relationship:

• 180 degrees is equal to \(\pi\) radians

This conversion is handy because a circle's circumference relates directly to its radius via the formula \(C = 2\pi r\). To make the transition from degrees to radians:\[\text{Number of radians} = \text{Degrees} \times \frac{\pi}{180}\]For example, converting 2880° to radians involves:\[2880 \times \frac{\pi}{180} = 16\pi\]Therefore, 2880 degrees is equivalent to \(16\pi\) radians.

Revolutions

A revolution describes a complete circular path, akin to turning a wheel or drawing a full circle. When something revolves, it covers 360 degrees of motion.
Revolutions prove useful in various applications, including:

• Engineering, when analyzing rotating machinery
• Astronomy, to study planetary orbits around the sun
• Computer graphics, for rotating objects smoothly

By converting degrees into revolutions, you gain a clear understanding of how many full circular movements occur. This makes revolutions a preferred choice in scenarios where you want to represent full cycles in the simplest way.

Radians

Radians stand out as a fundamentally important unit in mathematics and physics. They provide a direct link between a circle's radius and its circumference. Unlike degrees, which divide a circle into 360 parts, radians use the circle's own geometry.
Why is this significant? Here are a couple of reasons:

• Mathematical simplicity: Trigonometric functions like sine and cosine work seamlessly with radians, often simplifying calculations.
• Natural occurrence: Many natural phenomena and mathematical formulas inherently use radians.

In the context of the circle, when you travel an arc equal to the radius of the circle, you have moved one radian. This connection creates a natural and intuitive measurement that many mathematicians and scientists prefer.