Ch. 10 - Rotational Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 10 - Rotational Motion

Problem 45a

Chapter 10, Problem 45a

Let us treat a helicopter rotor blade as a long thin rod, as shown in Fig. 10–60. If each of the three rotor helicopter blades is 3.75 m long and has a mass of 135 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.

Verified step by step guidance

The moment of inertia for a single thin rod rotating about one end is given by the formula:

I = \frac{1}{3} m L^{2}

, where

m

is the mass of the rod and

L

is its length.

Substitute the given values for a single blade:

m = 135

kg and

L = 3.75

m into the formula. This gives the moment of inertia for one blade as:

I = \frac{1}{3} (135) ({3.75}^{2})

Since there are three identical blades, the total moment of inertia is the sum of the moments of inertia of all three blades. This can be expressed as:

I total = 3 \times I

, where

I

is the moment of inertia of one blade.

Substitute the calculated value of

I

for one blade into the total moment of inertia formula to find the total moment of inertia of the three blades.

Simplify the expression to obtain the final value for the total moment of inertia. Ensure the units are consistent and the result is expressed in kg·m².

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation. For a thin rod rotating about one end, the moment of inertia can be calculated using the formula I = (1/3)ml², where m is the mass and l is the length of the rod.

Parallel Axis Theorem

The parallel axis theorem allows us to calculate the moment of inertia of an object about any axis parallel to an axis through its center of mass. It states that I = I_cm + md², where I_cm is the moment of inertia about the center of mass, m is the mass, and d is the distance between the two axes. This theorem is useful when dealing with multiple objects or components.

Rotational Dynamics

Rotational dynamics is the study of the effects of forces and torques on the motion of rotating bodies. It encompasses concepts such as angular velocity, angular acceleration, and the relationship between torque and moment of inertia. Understanding these principles is essential for analyzing the motion of objects like helicopter rotor blades.

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Related Practice

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