Section 8.4 Density and Center of Mass

Density Looking to measure density –For example population density given in people per some kind of unit –For example density of substance is the mass per unit volume of the substance We can divide the region into small pieces so density is approximately constant on each piece –Then we add the contributions of the pieces

Example Imagine a city in the shape of a square that is 5 miles on each side There is a beach running along one side of the city and the population density d miles from the highway is 20-4d thousand people per square mile What is the approximate population of this city

Example Imagine a city in the shape of a circle that has a radius of 5 miles The further you get from the center of town, the less populated it is It has a population density (in thousands of people per square mile) of 15r – r 2 What is the approximate population of this city

Examples Find the mass of a rod that has a length of 5 meters and whose density is given by at a distance of x meters away from the left end

Center of Mass Important for study because of behavior of mechanical system when in motion –For example SUVs have high centers of mass Point masses –Masses on a line –First look at point masses then use integrals to extend this definition

Consider 2 items on a seesaw –To find the balance point we use the displacement (signed distance) of each from the pivot to calculate the moment –A moment represents the tendency to turn the system about the pivot point Moment of mass about a pivot is mass x displacement The system balances if the total moment is zero –The center of mass is the point about which the total moment is zero

In general the center of mass of a system of n point masses m 1, m 2, …, m n located at positions x 1, x 2, …x n along the x-axis is given by

Continuous Mass Density Instead of discrete masses arranged along the x-axis, suppose there is an object lying on the x-axis between x = a and x = b –Divide it into n pieces of length Δx –On each piece the density is nearly constant so the mass of each piece is given by density times the length –Mass of i th piece is

Combined with formula for center of mass for the n pieces gives us where is the density (mass per unit length) of the object

Examples Find the center of mass of a 10-meter rod lying on the x-axis with its left end at the origin if –The density is constant and the total mass is 10 kg –The density is

2 and 3-D Regions For a system that lies in the plane the center of mass is and for three dimensions it is Where Withequal to the area of a slice perpendicular to the x-axis and the other two defined similarly Let’s look at #24 in the book