Question

Each of three helicopter rotor blades shown in Fig. 9-42 is \(5.20 \mathrm{~m}\) long and has a mass of \(240 \mathrm{~kg}\). The rotor is rotating at 350 rev/min. What is the rotational inertia of the rotor assembly about the axis of rotation? (Each blade can be considered a thin rod.)

Short answer

The rotational inertia of the rotor assembly about the axis of rotation is \(I = 3 \times \frac{1}{3} \times 240 \mathrm{~kg} \times (5.20 \mathrm{~m})^2 = 8192 \mathrm{~kg} \cdot \mathrm{m}^2\).

Step-by-step solution

Find the Rotational Inertia of One Blade

Start off by calculating the rotational inertia of a single blade (as if it were a thin rod) using the formula \(I = \frac{1}{3} m L^2\). Substitute \(m = 240 \mathrm{~kg}\) and \(L = 5.20 \mathrm{~m}\) into the formula and solve.

Calculate Total Rotational Inertia

As the rotor assembly is made up of three blades, multiply the calculated rotational inertia of a single blade by three to find the total rotational inertia of the assembly.

Key concepts

rotational motion

Rotational motion can be understood as the motion of an object around a central point or axis. Imagine twirling a string around your finger; this is similar to how rotational motion works. There are a few important aspects to understand about this kind of motion:

• Axis of Rotation: The straight line around which the rotation occurs. For something like a helicopter rotor, this would be the centerline around which its blades spin.
• Circular Path: Each point on a rotating object travels along a path that forms a circle. This path's radius is the distance from the axis of rotation to the point on the object.
• Units of Measurement: Rotational motion is typically measured in revolutions per minute (rev/min) or radians per second (rad/s).

Understanding these basics helps you grasp more complex topics like rotational inertia and dynamics, as they all link back to how an object moves along its rotational path.

moment of inertia

The moment of inertia is a key property of rotating objects, much like mass is to linear motion. It represents how difficult it is to change an object's rotational speed. Think of it as rotational "mass," which depends on where the mass is situated relative to the axis of rotation.

• Calculation: For a thin rod rotating around one end, the moment of inertia is calculated with the formula \( I = \frac{1}{3} m L^2 \), where \( m \) is the mass and \( L \) is the length of the rod.
• Distribution of Mass: The farther the mass is from the axis, the higher the moment of inertia, making rotational changes more difficult.
• Sum of Parts: For objects composed of multiple parts, like a helicopter rotor, the total moment of inertia is the sum of the individual moments of inertia of each part.

By understanding how to calculate and apply the moment of inertia, we can better predict how rotational bodies behave in motion.

rotational dynamics

Rotational dynamics is the study of forces and torques and how they affect rotational motion. It involves understanding how different forces influence an object's movement around its axis.

• Torque: Just as force causes linear acceleration, torque causes angular acceleration. It depends on two factors: the force applied and the distance from the axis of rotation.
• Newton's Second Law for Rotation: An analogous form of Newton's second law applies to rotational dynamics: \( \tau = I \alpha \), where \( \tau \) is the torque, \( I \) is the moment of inertia, and \( \alpha \) is the angular acceleration.
• Equilibrium: When the net torque is zero, the object remains in rotational equilibrium, and its rotational velocity remains constant.

By mastering these concepts, you gain insights into how objects start to rotate, how they can stop, and why they rotate at certain speeds. This is especially useful when analyzing mechanical systems like rotor blades or engines.