A rod of length L and mass M has a nonuniform mass distribution. The linear mass density (mass per length) is λ = cx² , where x is measured from the center of the rod and c is a constant. Find an expression in terms of L and M for the moment of inertia of the rod for rotation about an axis through the center.

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Understand the problem: The moment of inertia (I) is a measure of an object's resistance to rotational motion about a given axis. For a nonuniform rod, we need to integrate over the mass distribution to calculate I. The linear mass density λ = cx² varies with position x, where x is measured from the center of the rod.

Express the mass element (dm): The mass of a small segment of the rod, dm, can be written as dm = λ dx. Substituting λ = cx², we get dm = c x² dx.

Set up the moment of inertia integral: The moment of inertia for a small mass element is dI = x² dm. Substituting dm = c x² dx, we get dI = c x⁴ dx. To find the total moment of inertia, integrate this expression over the length of the rod, from -L/2 to L/2: I = ∫(from -L/2 to L/2) c x⁴ dx.

Relate the constant c to the total mass M: The total mass of the rod is M = ∫(from -L/2 to L/2) λ dx = ∫(from -L/2 to L/2) c x² dx. Solve this integral to find c in terms of M and L. This will allow us to express the moment of inertia in terms of M and L.

Solve the integral for I: Substitute the value of c obtained in the previous step into the integral for I. Evaluate the integral ∫(from -L/2 to L/2) c x⁴ dx to find the final expression for the moment of inertia in terms of M and L.

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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 rotational motion about a specific axis. It depends on the mass distribution relative to that axis. For a continuous mass distribution, it is calculated by integrating the product of mass elements and the square of their distance from the axis of rotation.

Linear Mass Density

Linear mass density (λ) is defined as the mass per unit length of an object. In this case, it varies with position along the rod, given by λ = cx², where c is a constant and x is the distance from the center. This nonuniform distribution affects how mass is distributed relative to the axis of rotation, influencing the moment of inertia.

Integration in Physics

Integration is a mathematical technique used to calculate quantities that accumulate over a continuous range. In the context of finding the moment of inertia for a nonuniform mass distribution, integration allows us to sum the contributions of infinitesimal mass elements across the length of the rod, taking into account their varying distances from the axis of rotation.

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Textbook Question

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