Question

An airplane propeller rotates with an angular speed of \(260 \mathrm{rad} / \mathrm{s}\). Through what angle does the propeller rotate in \(5.0 \mathrm{~s}\) ? Give your answer in both radians and degrees.

Short answer

The propeller rotates through 1300 radians or approximately 74509.65 degrees in 5 seconds.

Step-by-step solution

Understand Given Information

We are given that the angular speed of the propeller is \(260 \text{ radians/second}\), and we need to find the angle in both radians and degrees through which the propeller rotates in \(5.0 \text{ seconds}\).

Calculate Angular Displacement in Radians

To find the angle in radians, we multiply the angular speed by the time duration. Use the formula \(\theta = \omega \times t\), where \(\theta\) is the angular displacement, \(\omega = 260 \text{ rad/s}\), and \(t = 5.0 \text{ s}\). Thus, \(\theta = 260 \times 5.0 = 1300 \text{ radians}\).

Convert Radians to Degrees

To convert from radians to degrees, use the conversion factor \(\frac{180}{\pi}\). Thus, the angle in degrees is \(1300 \times \frac{180}{\pi} \approx 74509.65 \) degrees.

Key concepts

Understanding Angular Speed

Angular speed is how fast something rotates or spins. It's similar to the speed of a car, but instead of moving forward, the object is turning. This speed is usually given in radians per second (rad/s).
For example, the airplane propeller spins at 260 rad/s, which is its angular speed. This means that every second, it turns through an angle of 260 radians.
To understand this better:

• Think of it like the hands of a clock moving around its face; the angular speed tells us how quickly this rotation happens.
• Knowing angular speed helps in predicting how fast an object completes a full rotation or part of a rotation.

When dealing with angular speed, always make sure you're using the right units and remember that it tells us the rotation per unit of time.

Converting Radians to Degrees

In mathematics, different units are used to measure angles, with radians and degrees being the most common. Sometimes, it's necessary to convert between these units, especially when interpreting the results in a more familiar way like degrees.
To convert radians to degrees, you use the conversion factor \(\frac{180}{\pi}\). This factor is derived from the fact that a complete circle covers an angle of \(2\pi \) radians, which is equivalent to 360 degrees.

• For instance, if you have an angle of 1300 radians, you multiply by \(\frac{180}{\pi}\) to find the equivalent in degrees.
• Calculation: \(1300 \times \frac{180}{\pi} \approx 74509.65\) degrees.

This conversion is crucial in fields like physics and engineering, where angles need to be expressed in different units depending on the context or preferences.

Calculating Time Duration

Time duration in rotational movements helps us understand how long it takes for something to rotate a certain amount through an angle. To find out how far a rotating object moves, you need to know both its angular speed and how long it has been rotating.
Using the formula for angular displacement \(\theta = \omega \times t\), where \(\theta\) is the angle in radians, \(\omega\) is the angular speed in rad/s, and \(t\) is the time in seconds, helps to calculate this.
To find the angle the airplane propeller rotates in a specific time:

• Plug in the given angular speed (260 rad/s) and time duration (5 seconds) into the formula.
• The result: \(\theta = 260 \times 5.0 = 1300 \text{ radians}\).

This calculation shows you how much the propeller turns in that time frame, offering insight into its performance and efficiency. By understanding these concepts, you can analyze and predict rotational behaviors effectively.