Problem 42 - University Physics with Modern 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 \(\mathrm{rpm}(\mathrm{rev} / \mathrm{min})\) 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) Rotational KE is approximately 1.33 × 10^6 J. (b) New angular speed is approximately 3076.44 rpm.

Step 1: Identify the Problem

We need to calculate the rotational kinetic energy of an airplane propeller modeled as a slender rod and find the new angular speed if the mass is reduced to 75% while maintaining the same rotational kinetic energy.

Step 2: Compute Moment of Inertia

For a slender rod rotating about its center, the moment of inertia \( I \) is given by \( I = \frac{1}{12} m L^2 \), where \( m \) is the mass and \( L \) is the length. Here, \( m = 117 \space kg \) and \( L = 2.08 \space m \). Calculate \( I \) as follows:\[ I = \frac{1}{12} \times 117 \space kg \times (2.08 \space m)^2 \approx 42.256 \space kg \cdot m^2 \].

Step 3: Convert RPM to Radians Per Second

The angular speed \( \omega \) in radians per second is related to rpm by the formula \( \omega = \frac{2\pi \times \, rpm}{60} \). For 2400 rpm, compute:\[ \omega = \frac{2\pi \times 2400}{60} \approx 251.33 \space rad/s \].

Step 4: Calculate Rotational Kinetic Energy

Rotational kinetic energy \( K \) is calculated using the formula \( K = \frac{1}{2} I \omega^2 \). Substitute \( I = 42.256 \space kg \cdot m^2 \) and \( \omega = 251.33 \space rad/s \):\[ K = \frac{1}{2} \times 42.256 \times (251.33)^2 \approx 1.33 \times 10^6 \space J \].

Step 5: Find New Mass and Moment of Inertia

If the mass is reduced to 75%, the new mass \( m' = 0.75 \times 117 \space kg = 87.75 \space kg \). The new moment of inertia \( I' \) is:\[ I' = \frac{1}{12} \times 87.75 \times (2.08)^2 \approx 31.692 \space kg \cdot m^2 \].

Step 6: Solve for New Angular Speed

We need to maintain the same kinetic energy \( 1.33 \times 10^6 \space J \). Use the equation \( K = \frac{1}{2} I' \omega'^2 \) to find new \( \omega' \):\[ 1.33 \times 10^6 = \frac{1}{2} \times 31.692 \omega'^2 \]Solve for \( \omega' \):\[ \omega' = \sqrt{\frac{2 \times 1.33 \times 10^6}{31.692}} \approx 322.02 \space rad/s \].

Step 7: Convert New Angular Speed to RPM

Convert \( \omega' \) back to rpm using \( rpm = \frac{\omega' \times 60}{2\pi} \):\[ rpm = \frac{322.02 \times 60}{2\pi} \approx 3076.44 \space rpm \].

Key Concepts

Moment of Inertia

In rotational kinetics, the moment of inertia is a key concept that refers to how an object’s mass is distributed relative to the axis around which it rotates. This is analogous to mass in linear motion and affects how easily an object can change its rotational motion. For a slender rod, like our airplane propeller, the formula to calculate the moment of inertia (I) when it rotates about its center is given by:\[I = \frac{1}{12} m L^2\]Here, m is the mass of the rod and L is its length. For example, with a mass of 117 kg and length 2.08 m, you plug these values into the formula to get approximately 42.256 kg \cdot m^2.
This means that the propeller has a certain resistance to changes in its rotational speed, based on how its mass is spread out across its length.

Angular Speed

Angular speed describes how fast an object rotates or spins around a specific point or axis. It’s often measured in radians per second (rad/s) or revolutions per minute (rpm). Converting between these two units is crucial for solving problems in rotational mechanics.
To change rpm to rad/s, use the relation:\[\omega = \frac{2\pi \times rpm}{60}\]In this problem, the propeller rotates at 2400 rpm. By substituting into the formula, we get an angular speed of approximately 251.33 rad/s.
Understanding angular speed helps us grasp how rapidly the object moves in its circular path, which directly impacts other calculations, like kinetic energy.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. In the context of rotational motion, the formula for rotational kinetic energy is:\[K = \frac{1}{2} I \omega^2\]Where I is the moment of inertia and \(\omega\) is the angular speed. In the exercise, the calculated moment of inertia is 42.256 kg \cdot m^2 and the angular speed is 251.33 rad/s. Substituting these into the formula gives a rotational kinetic energy (\(K\)) of about 1.33 \(\times 10^6\) J.
This energy is vital in understanding how rotating systems behave and is crucial in real-world applications that require energy management and efficiency.

Physics Calculations

Physics calculations often involve solving for unknown variables through mathematical manipulation and understanding physical principles. In this problem, we calculated rotational kinetic energy and then had to find the new angular speed after reducing the mass to 75% of its original value while keeping the same energy.

• Re-calculate the moment of inertia for the reduced mass of 87.75 kg .
• Use the same energy value to solve for the new angular speed, recognizing that reducing mass requires a new configuration to maintain energy.
• Convert the new angular speed from rad/s back to rpm for practical understanding.

These exercises highlight how different physics concepts weave together to solve complex problems, emphasizing both theoretical understanding and practical application.