Problem 3 - Applied Physics

Question

Convert \(25 \pi\) rad (a) to revolutions. (b) to degrees.

Step-by-Step Solution

Answer

(a) 12.5 revolutions. (b) 4500 degrees.

Step 1: Understand the conversion from radians to revolutions

Revolutions are a complete rotation, equivalent to \(2\pi\) radians. The formula to convert radians to revolutions is \( \text{revolutions} = \frac{\text{radians}}{2\pi} \).

Step 2: Calculate revolutions from radians

We start by applying the conversion formula: \[ \text{revolutions} = \frac{25\pi}{2\pi} \] Simplifying this gives us \[ \text{revolutions} = \frac{25}{2} = 12.5 \] So, \(25\pi\) radians is equal to 12.5 revolutions.

Step 3: Understand the conversion from radians to degrees

There are \(180\) degrees in \(\pi\) radians. Therefore, the conversion formula is \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \).

Step 4: Calculate degrees from radians

Use the conversion formula to determine degrees: \[ \text{degrees} = 25\pi \times \frac{180}{\pi} \] The \(\pi\) cancels out, so we have: \[ \text{degrees} = 25 \times 180 = 4500 \] Thus, \(25\pi\) radians is equal to 4500 degrees.

Key Concepts

Revolutions

In mathematics, a revolution is a complete turn around a circle. Imagine a full spin, going all the way around a circle and ending at the starting point. This full circle is known as one revolution.

• Every complete revolution has an angle of 360 degrees.
• In terms of radians, one revolution is exactly equivalent to \(2\pi\) radians.

To convert from radians to revolutions, you need to understand that \(2\pi\) radians make one full revolution. The formula to switch from radians to revolutions is: \[ \text{revolutions} = \frac{\text{radians}}{2\pi} \]This formula says that if you know the angle in radians, you can find out how many complete 360-degree spins you have by dividing by \(2\pi\). This makes it easy to calculate revolutions from radians.

Radians

Radians are a way to measure angles, just like degrees. They are based on the idea of a circle's radius. A radian measures how far around a circle we have gone in terms of the radius of the circle.

• One radian is the angle you make when you wrap the radius of a circle along the outer edge (or circumference) of the circle.
• In a full circle, there are \(2\pi\) radians because the circumference of a circle is \(2\pi\) times its radius.

Radians are often more convenient in math, especially in trigonometry and calculus, due to their relationship with the circle. For example, \(\pi\) radians are equal to 180 degrees, showing a strong link between these two units of measurement.

Degrees

Degrees are perhaps the most familiar way of measuring angles. A full circle is divided into 360 equal parts, known as degrees, making it easy to visualize and estimate angles.

• Each degree is \(\frac{1}{360}\) of a circle.
• A right angle, or the angle of a square corner, is \(90\) degrees.

Degrees are commonly used in everyday life, in navigation, engineering, and even in describing directions. Since a circle is 360 degrees, it's straightforward to understand and relate to; a half-circle covers \(180\) degrees, and a quarter-circle, like a corner of a square, is \(90\) degrees.

Conversion Formulas

Converting between radians and degrees is straightforward if you remember the key fact: \(\pi\) radians is equal to 180 degrees. From this, we derive two main conversion formulas:

• To convert radians to degrees, multiply by \(\frac{180}{\pi}\): \[ \text{degrees} = \text{radians} \times \frac{180}{\pi} \]
• To convert degrees to radians, multiply by \(\frac{\pi}{180}\): \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]

These formulas help translate between the two systems and are foundational in trigonometry. Mastering these conversions is crucial when working on trigonometric problems and understanding circular motion.

Trigonometry

Trigonometry, a branch of mathematics, deals with the relationships between the sides and angles of triangles. The concepts of revolutions, radians, and degrees play a vital role in this field.

• Radians are especially useful in trigonometry because many trigonometric functions are based on the unit circle, where the angle is measured in radians.
• The unit circle is a circle with a radius of one, centered at the origin on a coordinate plane, making it a powerful tool for visual understanding and calculations.

In trigonometry, angles are often measured in radians because they simplify many formulas and expressions. For example, the sine and cosine functions are periodic with a period of \(2\pi\). Understanding these concepts makes solving trigonometric problems much easier.