Chapter 8: Problem 6
How much time does it take for a spinning baseball with an angular speed of \(38 \mathrm{rad} / \mathrm{s}\) to rotate through \(15^{\circ}\) ?
Short Answer
Expert verified
It takes approximately 0.00689 seconds.
Step by step solution
01
Convert Degrees to Radians
The first step is to convert the angle from degrees to radians since the initial angular speed is in radians per second. To do this, use the conversion factor: \[1^{\circ} = \frac{\pi}{180} \, \text{radians}\]Therefore, \[15^{\circ} = 15 \times \frac{\pi}{180} \, \text{radians} \approx 0.2618 \, \text{radians}\]
02
Use the Angular Speed Formula
Now, use the formula for angular displacement, which relates angular speed \(\omega\), angular displacement \(\theta\), and time \(t\):\[\theta = \omega t\]Here, \(\theta\) is the angular displacement, and \(\omega\) is the angular speed.
03
Solve for Time
Rearrange the equation \(\theta = \omega t\) to solve for the time \(t\):\[t = \frac{\theta}{\omega}\]Substitute the values:\[t = \frac{0.2618 \, \text{radians}}{38 \, \text{rad/s}} \approx 0.00689 \, \text{s}\]
04
Interpret the Result
The result indicates that it takes approximately 0.00689 seconds for the baseball to rotate through an angle of 15 degrees at an angular speed of 38 radians per second.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angular Speed
Angular speed is like the speedometer for objects that spin or rotate. It tells us how fast an object turns around a specific point. The unit for angular speed is typically radians per second (rad/s), allowing us to measure the rotation speed in a standard mathematical way. Just as linear speed might be measured in meters per second, angular speed focuses on how quickly an object can complete a full circular path, but on a rotational axis.
This quantity is crucial in understanding movements of rotating objects, like wheels, planets, or even our simple spinning baseball.
This quantity is crucial in understanding movements of rotating objects, like wheels, planets, or even our simple spinning baseball.
- If an object has a high angular speed, it means it's rotating very fast around its center or axis.
- A low angular speed indicates slower rotation.
Radian Conversion
Radian conversion is a key step in working with angular measurements. While degrees can be easy to visualize because they're often taught in geometry, radians are the math-pro's favorite for calculations. This is because radians provide a direct link between the circumference of a circle and its radius.
To convert degrees to radians, we use the factor where \(1^{\circ} = \frac{\pi}{180}\ text{radians}\) computes an equivalent measure in radians. With this conversion:
To convert degrees to radians, we use the factor where \(1^{\circ} = \frac{\pi}{180}\ text{radians}\) computes an equivalent measure in radians. With this conversion:
- \(15^{\circ}\) becomes \(15 \times \frac{\pi}{180} \approx 0.2618 \text{ radians}\)
Angular Displacement Formula
The angular displacement formula is essential in understanding rotational movement over time. It connects three important elements: angular displacement \(\theta\), angular speed \(\omega\), and the time \(t\) over which the rotation occurs. The formula is expressed as: \[\theta = \omega t\]
Here:- \(\theta\) represents the angle through which an object has rotated in radians- \(\omega\) is the angular speed in radians per second- \(t\) is the time in seconds over which the rotation happens.
To find the time it takes for a specific amount of rotation, we rearrange the formula to solve for \(t\): \[t = \frac{\theta}{\omega}\]
By inserting the known values (for example, an angular displacement of \(0.2618 \text{ radians}\) and angular speed of \(38 \text{ rad/s}\)), you can calculate the time taken for such rotation as \(0.00689\) seconds - a concise result for quick rotational events. Understanding this formula is pivotal in connecting rotational properties and predicting rotational behaviors.
Here:- \(\theta\) represents the angle through which an object has rotated in radians- \(\omega\) is the angular speed in radians per second- \(t\) is the time in seconds over which the rotation happens.
To find the time it takes for a specific amount of rotation, we rearrange the formula to solve for \(t\): \[t = \frac{\theta}{\omega}\]
By inserting the known values (for example, an angular displacement of \(0.2618 \text{ radians}\) and angular speed of \(38 \text{ rad/s}\)), you can calculate the time taken for such rotation as \(0.00689\) seconds - a concise result for quick rotational events. Understanding this formula is pivotal in connecting rotational properties and predicting rotational behaviors.