Problem 105 - Precalculus

Question

Find the angular speed (radians/second) associated with rotating a central angle \(\theta\) in time \(t\). $$\theta=200^{\circ}, t=5 \mathrm{sec}$$

Step-by-Step Solution

Answer

The angular speed is \( \frac{2\pi}{9} \) radians/second.

Step 1: Convert Degrees to Radians

To find the angular speed in radians per second, first convert the angle from degrees to radians. The conversion factor is \( \frac{\pi}{180} \), because \( 180^{\circ} \) equals \( \pi \) radians. Calculate \( \theta \) in radians: \( \theta = 200^{\circ} \times \frac{\pi}{180} = \frac{200\pi}{180} = \frac{10\pi}{9} \) radians.

Step 2: Calculate Angular Speed

The angular speed is defined as the angle rotated per unit of time. Using the converted angle from Step 1, the angular speed \( \omega \) can be calculated using the formula \( \omega = \frac{\theta}{t} \). Substitute \( \theta = \frac{10\pi}{9} \) radians and \( t = 5 \) seconds into the formula: \[ \omega = \frac{\frac{10\pi}{9}}{5} = \frac{10\pi}{45} = \frac{2\pi}{9} \text{ radians/second} \].

Key Concepts

Radian Conversion

When dealing with rotations and angles, it is important to understand that angles can be expressed in different units, namely degrees and radians. Radians are the standard unit of angular measurement in mathematics.
To convert degrees to radians, use the conversion factor \( \frac{\pi}{180} \). This is derived from the fact that a full circle is \( 360^{\circ} \), which is equivalent to \( 2\pi \) radians. Therefore, half a circle—\( 180^{\circ} \)—is \( \pi \) radians.
To convert an angle in degrees to radians, multiply the degree measure by \( \frac{\pi}{180} \). If you have an angle of \( 200^{\circ} \), the calculation will be:

• \( 200^{\circ} \times \frac{\pi}{180} \)

This simplifies to \( \frac{10\pi}{9} \) radians. Understanding this conversion is crucial as it allows you to work seamlessly with angles in various mathematical and physical contexts.

Central Angle

A central angle is an angle whose vertex is the center of a circle. It subtends an arc on the circle's circumference, meaning it "opens up" to form a part of the circle's outer edge.
The central angle is fundamental in the study of circles and is often used when calculating angular displacement or speed. When expressed in radians, the central angle can easily relate to other measurements such as arc length and sector area.
For a circle with radius \( r \) and a central angle \( \theta \) in radians, the arc length \( s \) is given by the formula:

• \( s = r\theta \)

This property makes radians a more natural unit for angular measures in mathematics, as it directly relates linear and angular distances. In the context of our exercise, understanding the central angle helps in calculating how much of the circle is traversed over a given period, leading to calculations of angular speed.

Formula for Angular Speed

Angular speed is a measure of how quickly an object rotates or revolves around a central point. It is crucial in fields ranging from physics to engineering, as it helps describe rotational motion.
The formula for calculating angular speed \( \omega \) is:

• \( \omega = \frac{\theta}{t} \)

where \( \theta \) is the angle the object rotates through in radians, and \( t \) is the time taken.
In our specific problem, we used the converted angle \( \theta = \frac{10\pi}{9} \) radians and a time \( t = 5 \) seconds. Plugging these values into the formula gives you:

• \( \omega = \frac{\frac{10\pi}{9}}{5} = \frac{2\pi}{9} \text{ radians/second} \)

Understanding this formula helps in predicting the rotational speed of objects in real-world scenarios, such as the wheels of a car or the blades of a fan.