Question

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

It takes approximately 0.00689 seconds.

Step-by-step solution

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, \[15^{\circ} = 15 \times \frac{\pi}{180} \, \text{radians} \approx 0.2618 \, \text{radians}\]

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.

Solve for Time

Rearrange the equation \(\theta = \omega t\) to solve for the time \(t\):\[t = \frac{\theta}{\omega}\]Substitute the values:\[t = \frac{0.2618 \, \text{radians}}{38 \, \text{rad/s}} \approx 0.00689 \, \text{s}\]

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.

Key concepts

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.

• 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.

Understanding angular speed helps predict and analyze motions in various fields, including mechanics, engineering, and even in sports science.

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:

• \(15^{\circ}\) becomes \(15 \times \frac{\pi}{180} \approx 0.2618 \text{ radians}\)

This conversion is fundamental when working with formulas that involve radians, ensuring that all parts of the equation agree in units. This careful conversion method preserves the accuracy of any rotational calculations you might need.

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.