Question

Set up the triple integrals for finding the mass and the center of mass of the solid bounded by the graphs of the equations. $$ \begin{array}{l} x=0, x=a, y=0, y=b, z=0, z=c \\ \rho(x, y, z)=k z \end{array} $$

Short answer

The total mass of the solid is given by \(m = \int_0^a \int_0^b \int_0^c k z \, dz\, dy\, dx\). The coordinates of the center of mass are computed using \(\bar{x} = \frac{1}{m} \int_0^a \int_0^b \int_0^c x k z \, dz\, dy\, dx\), \(\bar{y} = \frac{1}{m} \int_0^a \int_0^b \int_0^c y k z \, dz\, dy\, dx\), and \(\bar{z} = \frac{1}{m} \int_0^a \int_0^b \int_0^c z k z \, dz\, dy\, dx\).

Step-by-step solution

Calculate the Total Mass

The total mass \(m\) of the solid is given by the triple integral of the density function over the volume of the solid. For this solid, that equation becomes:\[m = \int \int \int \rho(x, y, z) \, dV\]Since the density function is \(\rho(x, y, z) = k z\), and considering the limits of the integrals, \(x = 0\) to \(a\), \(y = 0\) to \(b\), and \(z = 0\) to \(c\), this equation becomes:\[m = \int_0^a \int_0^b \int_0^c k z \, dz\, dy\, dx\]

Find the Coordinates of the Center of Mass

The coordinates \((\bar{x}, \bar{y}, \bar{z})\) of the center of mass are given by:\[\bar{x} = \frac{1}{m} \int \int \int x \rho(x, y, z) \, dV\]\[\bar{y} = \frac{1}{m} \int \int \int y \rho(x, y, z) \, dV\]\[\bar{z} = \frac{1}{m} \int \int \int z \rho(x, y, z) \, dV\]Using the provided boundaries and the density function, these can be written as:\[\bar{x} = \frac{1}{m} \int_0^a \int_0^b \int_0^c x k z \, dz\, dy\, dx\]\[\bar{y} = \frac{1}{m} \int_0^a \int_0^b \int_0^c y k z \, dz\, dy\, dx\]\[\bar{z} = \frac{1}{m} \int_0^a \int_0^b \int_0^c z k z \, dz\, dy\, dx\]

Key concepts

Mass Calculation Using Integrals

When studying the physical properties of an object, one fundamental characteristic is its mass. To calculate the mass of a three-dimensional solid with variable density, we turn to calculus and the use of triple integrals. Consider a solid with a density function \( \rho(x, y, z) \) which can vary throughout the solid. The mass can be mathematically expressed as a triple integral:

calculates a sum of infinitesimally small 'pieces' of the solid, each with a tiny volume \( dV \) and a corresponding density given by \( \rho(x, y, z) \) . By integrating over the entire volume, we account for the varying density and find the solid's total mass. For a solid with constant density, this process simplifies to the product of the density and the volume.

Center of Mass Coordinates

The center of mass of a solid is a point that represents the average location of all the mass in the object. The coordinates of the center of mass \( (\bar{x}, \bar{y}, \bar{z}) \) are determined by computing the moments about the coordinate planes and dividing by the total mass. These moments are found using triple integrals:

This process weighs the position of each infinitesimal volume element by its density before integrating over the entire volume of the solid. Each coordinate of the center of mass reflects the balance point in terms of the respective axis. In scenarios where a solid has a uniform density, the center of mass simply coincides with the geometric center (centroid).

Density Function Integration

In the context of physics and engineering, density function integration is the act of calculating physical quantities of a non-uniform material by incorporating its density distribution throughout the solid. The density function \( \rho(x, y, z) \) describes how density varies at different points.

When we integrate this function over the solid's volume, what we're actually doing is summing up the densities of all infinitesimal volume elements to compute properties like mass, center of mass, and moments of inertia. Triple integrals allow us to extend this process to three dimensions, where the density can change in any direction, and accurate calculations are crucial for design and analysis in fields such as biomechanics, aerospace, and materials science.

Calculus in Physics

Calculus is an invaluable tool in physics, providing a mathematical framework for understanding and predicting the behavior of physical systems. It enables us to deal with change and motion and to tackle problems involving variable forces and energies. Through differentiation, we capture instantaneous rates of change, like velocity and acceleration.

Integration, on the other hand, allows us to aggregate infinitesimal contributions to calculate quantities over space, such as work done by a force or, as in the exercise discussed, the mass of an object with varying density. Embracing calculus is key to grasping the intricacies of the universe, from celestial mechanics to quantum mechanics, making it the language in which much of physics is written.