Problem 106 - Precalculus

Question

Find the angular speed (radians/second) associated with rotating a central angle \(\theta\) in time \(t\). $$\theta=60^{\circ}, t=0.2 \mathrm{sec}$$

Step-by-Step Solution

Answer

The angular speed is \( \frac{5\pi}{3} \) radians per second.

Step 1: Convert Degrees to Radians

First, we need to convert the angle from degrees to radians. We know that \[ 1^{\circ} = \frac{\pi}{180} \text{ radians} \] so, \[ 60^{\circ} = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians} \] This will be the angle in radians for our calculation.

Step 2: Find the Angular Speed Formula

The formula to find angular speed \( \omega \) is \[ \omega = \frac{\theta}{t} \] where \( \theta \) is the angle in radians and \( t \) is the time in seconds.

Step 3: Substitute Values into the Formula

Substitute \( \theta = \frac{\pi}{3} \) and \( t = 0.2 \) seconds into the formula:\[ \omega = \frac{\frac{\pi}{3}}{0.2} \] Simplify the expression to find the angular speed.

Step 4: Calculate the Angular Speed

Perform the division:\[ \omega = \frac{\pi}{3} \times \frac{1}{0.2} = \frac{\pi}{3} \times 5 = \frac{5\pi}{3} \text{ radians/second} \]Thus, the angular speed is \( \frac{5\pi}{3} \) radians per second.

Key Concepts

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which are more familiar from everyday usage, radians provide a natural way to describe angles in mathematical terms. This is because one radian is the angle created when the length of the arc of a circle is equal to the circle's radius. In simple terms:

• One full circle is always equal to \(2\pi\) radians.
• Half a circle would thus be \(\pi\) radians.

The conversion from degrees to radians is essential for many mathematical problems, especially those involving circular motion or periodic functions.
In physics and engineering, radians are often preferred because they simplify the mathematics involved. For the conversion from degrees to radians, use the factor: \(\frac{\pi}{180}\). This means \(1^{\circ} = \frac{\pi}{180}\) radians.

Conversion of Units

Unit conversion is a fundamental skill in physics and mathematics, allowing you to translate measurements from one set of units to another. When dealing with angles, converting degrees to radians is frequently required because many formulas assume angles are in radians.
Here's how you can easily convert degrees to radians:

• Identify the degree measure to be converted.
• Use the known conversion factor \(1^{\circ} = \frac{\pi}{180}\) radians.
• Multiply the degree measure by \(\frac{\pi}{180}\).

For instance, to convert \(60^{\circ}\) to radians, multiply by the conversion factor: \(60 \times \frac{\pi}{180} = \frac{\pi}{3}\) radians. This step ensures that the angle is correctly expressed in radians, which is crucial for calculations involving rotational motion.

Formula for Angular Speed

Angular speed tells us how quickly an object rotates or revolves about a central point. It's measured in radians per second (rad/s). Angular speed is essential in understanding rotational dynamics and is calculated using the formula:

• \(\omega = \frac{\theta}{t}\)
• Where \(\omega\) is the angular speed, \(\theta\) is the angle in radians, and \(t\) is the time taken in seconds.

To find angular speed:

• First, make sure the angle \(\theta\) is in radians.
• Use the time \(t\) in seconds for the duration of rotation.
• Divide the angle by the time to get angular speed.

In the exercise example, once you've converted the angle \(60^{\circ}\) to \(\frac{\pi}{3}\) radians, the formula becomes \(\omega = \frac{\frac{\pi}{3}}{0.2} = \frac{5\pi}{3}\). This calculation shows that the angular speed is \(\frac{5\pi}{3}\) radians per second, indicating the rate of rotation.