Problem 10 - Applied Physics

Question

Find the angular speed in Problems \(5-10 .\) Number of revolutions \(=6.25\) \(t=5.05 \mathrm{~s}\) \(\omega=\) \(\mathrm{rad} / \mathrm{s} \mathrm{s}\)

Step-by-Step Solution

Answer

The angular speed is approximately 7.774 rad/s.

Step 1: Identify the Known Variables

In the problem, we are given the number of revolutions, which is 6.25, and the time, which is 5.05 seconds.

Step 2: Convert Revolutions to Radians

Since 1 full revolution is equal to \(2\pi\) radians, we multiply the number of revolutions by \(2\pi\) to get the angular displacement in radians. Consequently, \(6.25 \text{ revolutions} \times 2\pi \text{ radians/revolution} = 12.5\pi\) radians.

Step 3: Apply the Angular Speed Formula

Angular speed \(\omega\) is calculated using the formula \(\omega = \frac{\text{angular displacement}}{\text{time}}\). Here, \(\omega = \frac{12.5\pi}{5.05}\) rad/s.

Step 4: Calculate the Angular Speed

Divide the angular displacement by the time to find \(\omega\). This gives \(\omega \approx \frac{12.5 \times 3.1416}{5.05} \approx 7.774\) rad/s.

Key Concepts

Revolutions to Radians

Converting revolutions to radians is a crucial step in solving problems involving angular motion. When an object completes one full circle, it completes one revolution, equivalent to a displacement of \(2\pi\) radians. The conversion between revolutions and radians is straightforward. You simply multiply the number of revolutions by \(2\pi\).
For example, if an object completes 6.25 revolutions, the angular displacement in radians is calculated as follows:

• Calculate the radians: \(6.25 \times 2\pi = 12.5\pi \text{ radians}\).

This conversion helps in further calculations, such as finding angular speed, which typically uses radians as the standard unit of measurement.

Angular Displacement

Angular displacement refers to the change in the angle as an object moves along a circular path. It tells us how far the object has rotated in terms of angle, starting from its initial position. It's a vector quantity, possessing both magnitude and direction. Often measured in radians, angular displacement is vital for understanding rotational motion.
When we convert revolutions to radians, as detailed earlier, we actually calculate the angular displacement. In the example problem, 6.25 revolutions converted to radians yields an angular displacement of \(12.5\pi\) radians. This value is essential when calculating angular speed and helps track how far an object has turned.

Angular Speed Formula

Angular speed provides a measure of how fast an object rotates, expressed as the rate of change of angular displacement over time. The formula for angular speed \(\omega\) is given by the equation:

• \(\omega = \frac{\text{angular displacement}}{\text{time}}\)

Here, angular displacement is the change in the rotation angle measured in radians, and time is the duration over which this change occurs. In our example, with an angular displacement of \(12.5\pi\) radians over 5.05 seconds, the calculation is:

• \(\omega = \frac{12.5\pi}{5.05}\) rad/s

Understanding this formula is key to calculating rotational speed in various physics and engineering applications.

Radians per Second

Radians per second (rad/s) is the standard unit for measuring angular speed. It tells us how many radians an object rotates through in one second. This unit is more precise in physics and engineering contexts, particularly for rotational dynamics.
In our given problem, after applying the angular speed formula, we found that the angular speed \(\omega\) is approximately 7.774 rad/s. This value means that the object rotates through about 7.774 radians every second. Understanding radians per second as a unit helps in grasping how quickly or slowly an object is spinning, providing insights into its rotational behavior.