Problem 40 - College Physics

Question

An airplane propeller is 2.08 m in length (from tip to tip) with mass 117 kg and is rotating at 2400 rpm v about an axis through its center. You can model the propeller as a slender rod. (a) What is its rotational kinetic energy? (b) Suppose that, due to weight constraints, you had to reduce the propeller's mass to 75.0\(\%\) of its original mass, but you still needed to keep the same size and kinetic energy. What would its angular speed have to be, in rpm?

Step-by-Step Solution

Answer

(a) 1,341,982.36 J; (b) 2889.39 rpm.

Step 1: Understanding the Problem

We need to find the rotational kinetic energy of the propeller modeled as a slender rod, and determine the new angular speed if its mass is reduced but kinetic energy remains the same.

Step 2: Calculate the Moment of Inertia

For a slender rod rotating about its center, the moment of inertia is given by \( I = \frac{1}{12} m L^2 \), where \( m \) is the mass and \( L \) is the length of the rod. Here, \( m = 117 \) kg and \( L = 2.08 \) m. Calculate \( I \): \[ I = \frac{1}{12} \times 117 \times (2.08)^2 = 42.4592 \, \text{kg}\cdot\text{m}^2 \].

Step 3: Convert Angular Speed to Radians per Second

The propeller rotates at 2400 revolutions per minute (rpm). Convert this to radians per second (rad/s) using: \[ \omega = \frac{2400 \times 2\pi}{60} = 251.33 \; \text{rad/s} \].

Step 4: Calculate Rotational Kinetic Energy

Rotational kinetic energy is given by \( K = \frac{1}{2} I \omega^2 \). Using \( I = 42.4592 \) kg\cdotm\(^2\) and \( \omega = 251.33 \) rad/s, compute:\[ K = \frac{1}{2} \times 42.4592 \times (251.33)^2 = 1,341,982.36 \; \text{J} \].

Step 5: Calculate New Moment of Inertia with Reduced Mass

If the mass is reduced to 75%, the new mass \( m' \) will be \( 0.75 \times 117 = 87.75 \) kg. The new moment of inertia is: \[ I' = \frac{1}{12} \times 87.75 \times (2.08)^2 = 31.8444 \; \text{kg}\cdot\text{m}^2 \].

Step 6: Find New Angular Speed for Same Kinetic Energy

To keep the kinetic energy the same, \( \frac{1}{2} I' \omega'^2 = 1,341,982.36 \; \text{J} \). Solving for \( \omega' \): \[ \omega' = \sqrt{\frac{2 \times 1,341,982.36}{31.8444}} = 302.32 \; \text{rad/s} \].

Step 7: Convert New Angular Speed Back to RPM

Convert \( \omega' = 302.32 \) rad/s back to rpm: \[ \text{rpm}' = \frac{302.32 \times 60}{2\pi} = 2889.39 \; \text{rpm} \].

Key Concepts

Moment of Inertia

In physics, moment of inertia is a critical concept used when dealing with rotational motion. It serves as a measure of how much torque is needed for a body to rotate about an axis. Think of it as the rotational equivalent of mass in linear motion. The formula for the moment of inertia depends on the object's shape and how the mass is distributed around the axis of rotation. For a slender rod, like an airplane propeller, rotating about its center, the moment of inertia is calculated using: \[ I = \frac{1}{12} m L^2 \] where:

• \( I \) is the moment of inertia,
• \( m \) is the mass of the rod, and
• \( L \) is its length.

To compute the moment of inertia for the propeller in question, we inserted the specific values: mass \( m = 117 \; \text{kg} \) and length \( L = 2.08 \; \text{m} \), leading us to \( I = 42.4592 \; \text{kg} \cdot \text{m}^2\). This value is crucial to further calculate the propeller's rotational kinetic energy.

Angular Speed

Angular speed, denoted as \( \omega \), indicates how fast an object rotates or revolves relative to another point, often the center of rotation. It tells us how much angle is swept per unit of time, typically measured in radians per second (rad/s). Initial data provided the angular speed in revolutions per minute (rpm), hence the need for conversion to a more standardized unit in physics. Thus, angular speed is often computed from rpm using the conversion:\[\omega = \frac{\text{rpm} \times 2\pi}{60}\]When the airplane propeller rotates at 2400 rpm, its angular speed becomes:\[\omega = \frac{2400 \times 2\pi}{60} = 251.33 \; \text{rad/s}\].
Remember, accurate unit conversion is essential in obtaining correct calculations in physics; using rad/s makes it easier to perform additional computations related to forces and energy.

Slender Rod Model

The slender rod model is an idealization often used in physics to simplify the analysis of a body's rotation. In reality, objects can be complex, but modeling them as a slender rod makes calculations more manageable while still providing reasonably accurate results.
For a slender rod rotating about its center, the mass is assumed to be uniformly distributed along its length, allowing us to use the previously discussed formula for the moment of inertia: \[ I = \frac{1}{12} m L^2 \].
This simplification is beneficial for practical problems, like calculating the rotational kinetic energy of an airplane propeller, where the precise shape can be approximated as a rod. By using this model, it becomes easier to apply mathematical equations, facilitating the overall problem-solving process.

Unit Conversion

Unit conversion is a fundamental skill in physics, enabling consistency and accuracy in problem-solving. Different units are often used interchangeably in calculations, necessitating conversion to a common system.
In the given exercise, recognizing the need to convert the angular speed from revolutions per minute (rpm) to radians per second (rad/s) was crucial. This conversion bridges everyday practical units (rpm) with the scientific units required for precise calculations (rad/s).The formula thus is:\[\omega = \frac{\text{rpm} \times 2\pi}{60}\]By performing these conversions accurately, you ensure that all calculations are in sync, especially when integrating angular measurements with linear ones. This practice of unit conversion helps unify all parts of the solution, preventing errors that stem from unit inconsistencies.