Problem 38 - College Physics

Question

An airplane propeller is \(2.08 \mathrm{~m}\) in length (from tip to tip) with mass \(117 \mathrm{~kg}\) and is rotating at \(2400 \mathrm{rpm}\) (rev/min) 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) The rotational kinetic energy is approximately 1.33 million joules. (b) The new angular speed should be about 2914 rpm.

Step 1: Understanding the Problem

In this exercise, we have a rotating airplane propeller modeled as a slender rod. We need to calculate its rotational kinetic energy at given conditions and determine the new angular speed if the mass is reduced while maintaining the same kinetic energy.

Step 2: Calculating Moment of Inertia

The moment of inertia for a slender rod rotating about its center is given by \( I = \frac{1}{12} m L^2 \), where \( m \) is the mass and \( L \) is the length of the rod. Substituting the values, we have \( I = \frac{1}{12} \times 117 \times (2.08)^2 \approx 42.056 \; \text{kg} \cdot \text{m}^2 \).

Step 3: Calculating Rotational Kinetic Energy

The rotational kinetic energy \( KE \) is given by \( KE = \frac{1}{2} I \omega^2 \). First, convert rpm to rad/s using \( \omega = \frac{2400 \times 2\pi}{60} \approx 251.3 \; \text{rad/s} \). Now substitute \( I = 42.056 \) and \( \omega = 251.3 \) to calculate \( KE = \frac{1}{2} \times 42.056 \times (251.3)^2 \approx 1.33 \times 10^6 \; \text{J} \).

Step 4: Finding Required Angular Speed After Mass Reduction

If the mass is reduced to 75% of its original mass, the new mass \( m' = 0.75 \times 117 = 87.75 \; \text{kg} \). The new moment of inertia \( I' = \frac{1}{12} \times 87.75 \times (2.08)^2 \approx 31.54 \; \text{kg} \cdot \text{m}^2 \). To keep the same kinetic energy, set \( KE' = KE = 1.33 \times 10^6 \; \text{J} \) and solve for \( \omega' \) using \( KE = \frac{1}{2} I' \omega'^2 \). This gives \( \omega' = \sqrt{\frac{2 \times 1.33 \times 10^6}{31.54}} \approx 304.9 \; \text{rad/s} \). Convert \( \omega' \) to rpm: \( \frac{304.9 \times 60}{2\pi} \approx 2914 \; \text{rpm} \).

Key Concepts

Moment of Inertia

In rotational dynamics, the moment of inertia is an important concept, often compared to mass in linear motion. It signifies an object's resistance to rotational acceleration, depending on mass distribution relative to the axis of rotation. For a slender rod rotating around its center, the formula is:

• \( I = \frac{1}{12} m L^2 \)

where \( m \) is the mass, and \( L \) is the length. This formula helps quantify how mass located farther from or nearer to the rotation axis affects rotation. In our propeller example, with a mass of \( 117 \mathrm{~kg} \) and length \( 2.08 \mathrm{~m} \), its moment of inertia is calculated as:

• \( I = \frac{1}{12} \times 117 \times (2.08)^2 \approx 42.056 \, \text{kg} \cdot \text{m}^2 \)

The high value indicates considerable resistance to change in spinning motion, which is crucial for balance and performance in objects like propellers. If you reduce mass, as in the exercise, it significantly affects moment of inertia.

Rotational Kinetic Energy

Just like objects in linear motion have kinetic energy, rotating objects have rotational kinetic energy. It defines the energy due to rotation and is calculated using:

• \( KE = \frac{1}{2} I \omega^2 \)

where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. To find this energy for any rotating system, like the propeller:

• First convert angular speed from rpm to rad/s: \( \omega = \frac{2400 \times 2\pi}{60} \approx 251.3 \, \text{rad/s} \)
• Substitute into the kinetic energy formula: \( KE = \frac{1}{2} \times 42.056 \times (251.3)^2 \approx 1.33 \times 10^6 \, \text{J} \)

This rotational form of kinetic energy lets us understand how fast an object is rotating and its mass distribution's effect on energy consumption. Despite reducing mass in our exercise, keeping kinetic energy constant requires adjusting angular velocity.

Angular Speed Calculations

Angular speed, often expressed in revolutions per minute (rpm), tells us how fast an object is rotating. To compute changes like in our exercise, where kinetic energy remains constant but mass decreases, transforms require angular speed recalculations:

• Given propeller changes, first find new moment of inertia: \( I' = \frac{1}{12} \times 87.75 \times (2.08)^2 \approx 31.54 \, \text{kg} \cdot \text{m}^2 \)
• With \( KE = \frac{1}{2} I' \omega'^2 \), isolate for new angular speed: \( \omega' = \sqrt{\frac{2 \times 1.33 \times 10^6}{31.54}} \approx 304.9 \, \text{rad/s} \)
• Convert \( \omega' \) to rpm: \( \frac{304.9 \times 60}{2\pi} \approx 2914 \, \text{rpm} \)

As a result, despite reduced mass, increasing angular speed maintains kinetic energy balance. This calculation underscores the interplay between an object's mass and velocity in rotational dynamics.