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 98

Chapter 10, Problem 98

The density (mass per unit length) of a thin rod of length ℓ increases uniformly from λ₀ at one end to 3λ₀ at the other end. Determine the moment of inertia about an axis perpendicular to the rod through its geometric center.

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Define the linear mass density of the rod as a function of position. Since the density increases uniformly from λ₀ to 3λ₀, the linear mass density can be expressed as λ(x) = λ₀ + (3λ₀ - λ₀)(x/ℓ) = λ₀(1 + 2x/ℓ), where x is the distance from one end of the rod.

Set up the expression for the moment of inertia. The moment of inertia about an axis perpendicular to the rod through its geometric center is given by I = ∫(x² dm), where dm = λ(x) dx is the mass element at a distance x from the axis.

Adjust the limits of integration to account for the axis being at the geometric center. The rod extends from -ℓ/2 to +ℓ/2 relative to the center, so the integral becomes I = ∫[(-ℓ/2) to (ℓ/2)] x² λ(x) dx.

Substitute λ(x) = λ₀(1 + 2x/ℓ) into the integral. This gives I = ∫[(-ℓ/2) to (ℓ/2)] x² λ₀(1 + 2x/ℓ) dx. Factor out λ₀ since it is constant, resulting in I = λ₀ ∫[(-ℓ/2) to (ℓ/2)] x² (1 + 2x/ℓ) dx.

Split the integral into two parts: I = λ₀ [∫[(-ℓ/2) to (ℓ/2)] x² dx + (2/ℓ) ∫[(-ℓ/2) to (ℓ/2)] x³ dx]. Evaluate each integral separately using standard integration techniques, keeping in mind the symmetry of the limits (e.g., odd powers of x will integrate to zero over symmetric limits).

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Key Concepts

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

Density Distribution

In this problem, the density of the rod varies linearly from λ₀ to 3λ₀. This means that the mass per unit length is not constant, and understanding how this distribution affects the overall mass and moment of inertia is crucial. The density function can be expressed as a linear function of position along the rod.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to rotational motion about an axis. It depends on the mass distribution relative to the axis of rotation. For a rod with varying density, the moment of inertia must be calculated by integrating the contributions of each infinitesimal mass element along the length of the rod.

Integration in Physics

Integration is a mathematical tool used to sum up infinitesimal contributions over a continuous distribution. In this context, it allows us to calculate the total moment of inertia by integrating the product of the mass elements and the square of their distances from the axis of rotation. This is essential for accurately determining the moment of inertia for objects with non-uniform density.

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