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 42c

Chapter 10, Problem 42c

Calculate the moment of inertia of the array of point objects shown in Fig. 10–58 about the y axis, and the x axis. Assume m = 22kg, M = 3.2kg, and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the x axis. About which axis would it be harder to accelerate this array?

Verified step by step guidance

Step 1: Understand the problem. The moment of inertia (I) is a measure of an object's resistance to rotational acceleration about a given axis. For point masses, the moment of inertia is calculated using the formula:

I = \sum_{i}^{n} m_{i r^{2}}

, where

m_{i}

is the mass of the i-th object and

r_{i}

is its perpendicular distance from the axis of rotation.

Step 2: Identify the distances of each mass from the y-axis. For the y-axis, the distances are determined by the x-coordinates of the masses. Use the given geometry of the rectangular array to find these distances. For example, if a mass is located at (x, y), its distance from the y-axis is |x|.

Step 3: Calculate the moment of inertia about the y-axis. Substitute the masses and their respective distances from the y-axis into the formula for moment of inertia. Sum the contributions of all the masses to get the total moment of inertia about the y-axis.

Step 4: Repeat the process for the x-axis. For the x-axis, the distances are determined by the y-coordinates of the masses. Use the same formula, but this time substitute the distances from the x-axis (|y|) and calculate the total moment of inertia about the x-axis.

Step 5: Compare the moments of inertia about the two axes. The axis with the larger moment of inertia will be harder to accelerate because it represents greater resistance to rotational motion. Use the calculated values to determine which axis this is.

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Key Concepts

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

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion about a specific axis. It depends on the mass of the object and the distribution of that mass relative to the axis of rotation. For point masses, it is calculated as the sum of the products of each mass and the square of its distance from the axis. Understanding this concept is crucial for solving problems related to rotational dynamics.

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 momentum, torque, and the relationship between linear and angular quantities. In the context of the question, analyzing how the moment of inertia affects the acceleration of the array when subjected to torque is essential for understanding the dynamics of the system.

Axis of Rotation

The axis of rotation is an imaginary line around which an object rotates. The choice of axis significantly influences the moment of inertia and, consequently, the object's rotational behavior. In this problem, comparing the moment of inertia about the x-axis and y-axis will help determine which axis makes it harder to accelerate the array, as the distribution of mass relative to each axis affects the overall resistance to rotational acceleration.

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