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Ch. 10 - Rotational Motion
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 10 - Rotational MotionProblema 58
Capitolo 10, Problema 58

(III) Integrate to derive the formula for the moment of inertia of a uniform thin rod of length ℓ about an axis through its center, perpendicular to the rod (see Fig. 10–21f).

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Start by recalling the formula for the moment of inertia: \( I = \int r^2 \, dm \), where \( r \) is the perpendicular distance from the axis of rotation to the mass element \( dm \).
Define the mass element \( dm \) in terms of the linear mass density \( \lambda \), where \( \lambda = \frac{M}{\ell} \) (\( M \) is the total mass of the rod and \( \ell \) is its length). Thus, \( dm = \lambda \, dx \), where \( dx \) is a small segment of the rod.
Set up the coordinate system: Place the center of the rod at the origin of the \( x \)-axis, so the rod extends from \( -\ell/2 \) to \( \ell/2 \). The distance \( r \) from the axis of rotation to a small segment of the rod is simply \( |x| \).
Substitute \( r \) and \( dm \) into the moment of inertia formula: \( I = \int_{-\ell/2}^{\ell/2} x^2 \lambda \, dx \). Replace \( \lambda \) with \( \frac{M}{\ell} \), giving \( I = \frac{M}{\ell} \int_{-\ell/2}^{\ell/2} x^2 \, dx \).
Evaluate the integral: \( \int_{-\ell/2}^{\ell/2} x^2 \, dx \). Use the symmetry of the function \( x^2 \) about the origin to simplify the integral to \( 2 \int_{0}^{\ell/2} x^2 \, dx \). Solve this integral and multiply by \( \frac{M}{\ell} \) to find the final expression for \( I \).

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Moment of Inertia

The moment of inertia is a measure of an object's resistance to rotational motion about a specific axis. It depends on the mass distribution relative to that axis, with greater distances from the axis resulting in a higher moment of inertia. For a uniform thin rod, this concept is crucial as it quantifies how the rod will respond to applied torques.
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Intro to Moment of Inertia

Integration in Physics

Integration is a mathematical process used to calculate quantities that accumulate over a continuous range, such as mass distribution along a rod. In deriving the moment of inertia, integration allows us to sum the contributions of infinitesimally small mass elements across the length of the rod, providing a comprehensive measure of its rotational inertia.
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Finding Moment Of Inertia By Integrating

Axis of Rotation

The axis of rotation is the line about which an object rotates. In this case, the axis is through the center of the rod and perpendicular to its length. The choice of axis significantly affects the moment of inertia, as it determines how the mass is distributed relative to that axis, influencing the rod's rotational dynamics.
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Parallel Axis Theorem
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