Problem 35 - Calculus

Question

A solid cylinder of radius \(r\) is assembled from hollow cylinders of length \(l,\) radius \(x,\) and volume \((2 \pi x)(l)(d x) .\) The solid cylinder has mass \(M=\int_{0}^{r} 2 \pi x l \rho d x\) and \(I=\int_{0}^{r} x^{2} 2 \pi x l \rho d x\) With \(\rho=7\) find \(M\) and \(I\) and \(J=I / M r^{2}\).

Step-by-Step Solution

Answer

Mass \(M = 7 \pi l r^2\), Inertia \(I = \frac{7 \pi l r^4}{2}\), Ratio \(J = \frac{1}{2}\).

Step 1: Understanding the Problem

The problem provides the formulas for mass \(M\) and moment of inertia \(I\) for a composite cylinder built from hollow cylinders. We are given that \(\rho = 7\). Our task is to find the total mass \(M\), moment of inertia \(I\), and the ratio \(J = \frac{I}{M r^2}\).

Step 2: Setting Up the Integral for Mass

The mass \(M\) is given by \(M = \int_{0}^{r} 2 \pi x l \rho \, dx\). Substitute \(\rho = 7\) into the formula: \[M = \int_{0}^{r} 2 \pi x l \cdot 7 \, dx = 14\pi l \int_{0}^{r} x \, dx.\]

Step 3: Calculating the Mass Integral

Now compute the integral: \[14\pi l \int_{0}^{r} x \, dx = 14 \pi l \left[\frac{x^2}{2}\right]_{0}^{r} = 14 \pi l \left(\frac{r^2}{2}\right) = 7 \pi l r^2.\] Thus, the mass \(M = 7 \pi l r^2.\)

Step 4: Setting Up the Integral for Moment of Inertia

The moment of inertia \(I\) is given by \(I = \int_{0}^{r} x^2 2 \pi x l \rho \, dx\). Substitute \(\rho = 7\) into the formula: \[I = \int_{0}^{r} x^2 \cdot 2 \pi x l \cdot 7 \, dx = 14\pi l \int_{0}^{r} x^3 \, dx.\]

Step 5: Calculating the Moment of Inertia Integral

Now compute the integral: \[14\pi l \int_{0}^{r} x^3 \, dx = 14 \pi l \left[\frac{x^4}{4}\right]_{0}^{r} = 14 \pi l \left(\frac{r^4}{4}\right) = \frac{7 \pi l r^4}{2}.\] Thus, the moment of inertia \(I = \frac{7 \pi l r^4}{2}.\)

Step 6: Calculating the Ratio \(J\)

Now calculate the ratio \(J = \frac{I}{M r^2}\):\[J = \frac{\frac{7 \pi l r^4}{2}}{7 \pi l r^2 \cdot r^2} = \frac{\frac{7 \pi l r^4}{2}}{7 \pi l r^4} = \frac{1}{2}.\] Hence, \(J = \frac{1}{2}.\)

Key Concepts

Mass of Solid Cylinder

The mass of a solid cylinder is a fundamental concept in physics and engineering. When calculating the mass of a solid cylinder, you consider the entire volume and the density of the material. In this problem, the cylinder is made up of tiny hollow cylinders stacked together.
To find the total mass of this composite cylinder, you need to integrate over the volume. This is done by setting up the integral for mass using the formula:

• \(M = \int_{0}^{r} 2 \pi x l \rho \, dx\)

Here, \(2 \pi x\) represents the circumference of each hollow cylinder, \(l\) is the length, and \(\rho\) is the density, set to 7 in this case. You substitute these values to calculate the integral, resulting in the expression for total mass, which simplifies to:

• \(M = 7 \pi l r^2\)

This calculation shows how integrating across all the tiny cylinders gives the mass of the whole solid cylinder.

Integral Calculus

Integral calculus is a powerful mathematical tool used to calculate quantities like areas, volumes, and other accumulative quantities. In this exercise, integral calculus is essential for determining both the mass and the moment of inertia of a composite shape.
Integrals help in summing up infinitesimally small parts to get the total for a bulky object, as in the case of our solid cylinder. The key integrals involved are:

• To find mass \(M\): \(M = \int_{0}^{r} 2 \pi x l \rho \, dx\)
• To find moment of inertia \(I\): \(I = \int_{0}^{r} x^2 2 \pi x l \rho \, dx\)

Each calculation uses the definite integral to accumulate the mass and mechanical inertia from all the small parts of the cylinder. The results are exact values representing the physical properties of the entire solid cylinder. Integral calculus thus simplifies complex physical problems by transforming them into solvable mathematical equations.

Composite Shapes

Composite shapes play a significant role when dealing with objects made from simpler components. In this exercise, the solid cylinder is a composite shape formed from many hollow cylinders. Understanding composite shapes allows us to solve complex problems by analyzing each component separately first.
The beauty of studying composite shapes lies in their simplicity due to partitioning. Each hollow cylinder can be handled with basic formulas, and their properties can be summed using integration to understand the whole form. This approach helps break down complex structures:

• Start with the basic geometric and physical properties of each shape
• Use integration to combine these properties
• Arrive at a comprehensive understanding of the entire composite structure

Working with composite shapes not only makes calculations more manageable but also enhances comprehension of how individual parts contribute to the whole. This encapsulation is crucial in fields ranging from mechanical design to architecture.