Problem 2 - University Physics with Modern Physics

Question

An airplane propeller is rotating at 1900 TPm (rev/min). a) Compute the propeller's angular velocity in rad/s. (b) How many seconds does it take for the propeller to turn through \(35^{\circ}\) ?

Step-by-Step Solution

Answer

Angular velocity is \(\frac{190\pi}{3} \text{ rad/s}\). Time for \(35^\circ\) rotation is approximately \(0.00307 \text{ s}\).

Step 1: Understanding the Problem

We are given that the propeller rotates at 1900 revolutions per minute (RPM) and need to calculate the angular velocity in radians per second. Additionally, we need to find how long it takes for the propeller to rotate through \(35^\circ\).

Step 2: Convert RPM to Radians per Second

First, convert the angular velocity from rev/min to rad/s. One revolution is equal to \(2\pi\) radians. Therefore, the conversion is \( \frac{1900\text{ rev/min} \times 2\pi \text{ rad/rev}}{60 \text{ s/min}} \). Calculate this to find the angular velocity.

Step 3: Calculation of Angular Velocity

Perform the calculation:\[\omega = \frac{1900 \times 2\pi}{60} = \frac{3800\pi}{60} = \frac{380\pi}{6} = \frac{190\pi}{3} \text{ rad/s}\]

Step 4: Convert Degrees to Radians

Since \(35^\circ\) is the angle through which the propeller turns, convert this to radians. The conversion formula is \( \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}\).Thus, \(35^\circ\) to radians is:\[\theta = 35 \times \frac{\pi}{180} = \frac{35\pi}{180}\]

Step 5: Determine Time for Given Rotation

Use the formula \( \theta = \omega \times t \) to find the time \(t\). Rearrange to get \( t = \frac{\theta}{\omega} \).Substitute \(\theta = \frac{35\pi}{180}\) and \(\omega = \frac{190\pi}{3}\) into the formula:\[ t = \frac{\frac{35\pi}{180}}{\frac{190\pi}{3}} \]

Step 6: Simplify to Find Time

Simplify the expression to find \(t\):\[ t = \frac{35\pi}{180} \times \frac{3}{190\pi} = \frac{35 \times 3}{180 \times 190} = \frac{105}{34200} \approx 0.00307 \text{ s} \]

Step 7: Finalizing the Answer

You now have both the angular velocity and the time it takes for the propeller to rotate through \(35^\circ\). The calculations yield: Angular velocity = \(\frac{190\pi}{3} \text{ rad/s} \) Time for \(35^\circ\) rotation \(\approx 0.00307 \text{ s} \).

Key Concepts

Unit Conversion

Unit conversion is a critical skill in physics and engineering, allowing us to express quantities in different units based on what is most convenient or standard for a given problem. For example, when dealing with rotational motion, units like revolutions per minute (RPM) are common, but sometimes, we need to express these in terms of radians per second (rad/s) to perform further calculations.

The conversion between these units involves understanding the relationship between revolutions and radians. One revolution is equivalent to the angular distance covered when moving through a full circle, which is exactly \(2\pi\) radians. This relationship helps us make the conversion:

• Given: \(x \text{ RPM} = x \text{ rev/min}\)
• To convert to rad/s, use: \(x \times \frac{2\pi \text{ rad}}{\text{rev}} \times \frac{1}{60 \text{ s}}\)

By using these factors, we transform the units step-by-step from revolutions to radians and from minutes to seconds, obtaining a unit that is more commonly used in angular velocity analysis.

Radians and Degrees

Radians and degrees are two ways to measure angles, each useful in different contexts. Degrees are perhaps more familiar to most people in everyday settings, given that a circle is split into 360 degrees. Meanwhile, radians provide a more natural unit for many calculations in mathematics and physics, particularly because of their relation to a circle's geometry.

To convert from degrees to radians, we use the fundamental equivalence that \(180^{\circ} = \pi\) radians. This yields the conversion factor:

• To convert degrees to radians: \(\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}\)
• Conversely, to convert radians to degrees: \(\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}\)

This straightforward equation allows us to shift between these two units as needed, facilitating a smooth transition in problem-solving contexts.

Revolutions Per Minute (RPM)

Revolutions per minute (RPM) is a unit of rotational velocity or speed that tells us how many full rotations, or "revolutions," an object completes in one minute. It is widely used in contexts such as engines, turbines, and other machinery where rotating components are involved.

While RPM is excellent for quickly conveying the speed of an engine or wheel, translating RPM into other units such as radians per second can be advantageous for certain calculations, like determining angular motion or integrating with other elements of a rotational system in physics problems.

• RPM helps suit specific applications where the cycle completion or frequency per minute is more intuitive.
• However, for a full dynamical analysis, converting RPM to rad/s might be necessary.

Understanding RPM and how it relates to other measures of angular speed can enhance both theoretical analysis and practical applications where rotation is involved.