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.

Verified step by step guidance

1. Identify the variable density function of the rod. Since the density increases linearly from one end to the other, you can express the linear mass density, \( \lambda(x) \), as a function of position along the rod, \( x \), where \( x \) ranges from \( -\ell/2 \) to \( \ell/2 \). The function can be written as \( \lambda(x) = \lambda_0 + kx \), where \( k \) is a constant that needs to be determined using the given boundary conditions at the ends of the rod.
2. Determine the constant \( k \) by using the boundary conditions. At \( x = -\ell/2 \), \( \lambda(-\ell/2) = \lambda_0 \) and at \( x = \ell/2 \), \( \lambda(\ell/2) = 3\lambda_0 \). Solve these equations to find the value of \( k \).
3. Calculate the total mass of the rod, \( M \), by integrating the linear density over the length of the rod. The total mass is given by \( M = \int_{-\ell/2}^{\ell/2} \lambda(x) \, dx \). Substitute the expression for \( \lambda(x) \) and integrate.
4. Express the moment of inertia, \( I \), about the center of the rod. The moment of inertia for a thin rod with variable density about an axis perpendicular to the rod through its center is given by \( I = \int_{-\ell/2}^{\ell/2} \lambda(x) x^2 \, dx \). Substitute the expression for \( \lambda(x) \) and perform the integration.
5. Simplify the integral to find the expression for the moment of inertia, \( I \), in terms of \( \lambda_0 \) and \( \ell \). This will involve integrating a polynomial function and possibly using integration techniques such as integration by parts or substitution if necessary.

Key Concepts

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.