Problem 7 - Applied Physics

Question

Find the angular speed in Problems \(5-10 .\) Number of revolutions \(=4.00\) \(t=3.00 \mathrm{~s}\) \(\omega=\quad \mathrm{rad} / \mathrm{s}\)

Step-by-Step Solution

Answer

The angular speed \( \omega \) is approximately \( 8.38 \) rad/s.

Step 1: Understand the Problem

We need to find the angular speed \( \omega \) given that the object makes 4.00 revolutions in 3.00 seconds.

Step 2: Convert Revolutions to Radians

One full revolution is equal to \( 2\pi \) radians. Therefore, 4.00 revolutions is equal to \( 4.00 \times 2\pi = 8\pi \) radians.

Step 3: Use the Formula for Angular Speed

Angular speed \( \omega \) is defined as the total angular displacement divided by time, \( \omega = \frac{\theta}{t} \), where \( \theta \) is the angular displacement and \( t \) is the time.

Step 4: Calculate Angular Speed

Substitute the values into the formula: \( \omega = \frac{8\pi}{3.00} \). Simplifying this, we get \( \omega = \frac{8\pi}{3} \approx 8.38 \) rad/s.

Key Concepts

angular displacement

Angular displacement is a measure of the angle through which an object moves on a circular path. It is the change in the angular position of an object and is expressed in terms of radians. For an object that makes multiple revolutions, angular displacement can be found by multiplying the number of revolutions by the angle of one full revolution.

One complete revolution corresponds to an angular displacement of \( 2\pi \) radians. Thus, if an object completes multiple revolutions, the total angular displacement can be calculated by the formula:

• \( \theta = \text{{number of revolutions}} \times 2\pi \)

This angular displacement helps us understand how far the object has rotated in a specific time frame. It is essential to express angular displacements in radians when performing further calculations related to angular speed.

revolutions to radians conversion

Revolutions to radians conversion is an essential step when dealing with problems involving rotations or circular motion. Since many formulas, especially those concerning angular speed, use radians as their unit of angular displacement, conversion becomes necessary.

We know that one revolution is \( 360^{\circ} \). In terms of radians, this is \( 2\pi \) radians. Therefore, to convert revolutions to radians, you simply multiply the number of revolutions by \( 2\pi \).

• Example: 4 revolutions = \( 4 \times 2\pi = 8\pi \) radians

Converting revolutions to radians ensures consistency in units and enables straightforward calculation when substituting values into angular speed or other related equations. It makes mathematical operations easier and standardized, preventing mistakes during calculations.

angular speed formula

The angular speed formula is crucial in understanding how fast an object rotates or spins around a central point. Angular speed, commonly represented by the Greek letter \( \omega \), tells us the rate at which angular displacement occurs.

The formula for calculating angular speed \( \omega \) is:

• \( \omega = \frac{\theta}{t} \)

where:

• \( \omega \) is the angular speed in radians per second (rad/s)
• \( \theta \) is the angular displacement in radians
• \( t \) is the time period over which the displacement occurs, measured in seconds

Using this formula, we can determine how quickly an object is rotating. In our example, we had 8\( \pi \) radians of displacement over 3.00 seconds. Substituting these values in gives:

• \( \omega = \frac{8\pi}{3} \approx 8.38 \) rad/s

This result provides a direct insight into the rotation speed of the object, a fundamental concept in physics and engineering. By understanding and utilizing this formula, you can precisely analyze and interpret rotational motion.