   Chapter 14, Problem 60RE

Chapter
Section
Textbook Problem

Center of Mass In Exercises 59 and 60, find the mass and the indicated coordinate of the center of mass of the solid region Q of density ρ bounded by the graphs of the equations.Find y ¯ using ρ ( x , y , z ) = k x . Q : z = 5 − y ,     z = 0 ,     y = 0 ,     x = 0 ,     x = 5

To determine

To calculate: The mass and the indicated coordinate y¯ of center of mass of solid region Q with density given as (ρ=kx). The solid region Q is bounded by the graph of equation Q:z=5y,x=0,y=0,z=0,x=5.

Explanation

Given:

The density of solid region Q is ρ(x,y,z)=kx and the solid is bounded by the graph of equation,

Q:z=5y,x=0,y=0,z=0,x=5.

Formula used:

Mass (m) of a solid region Q having density ρ is,

m=Qρ(x,y,z)dV

First moment about yz plane is,

Myz=Qxρ(x,y,z)dV

x Coordinate (x¯) of center of mass of a solid region Q is,

y¯=Mxzm

Calculation:

A 3-D plot is made with the help of maple software,

The limits of x,y and z can be calculated with the help of this 3-D plot.

Take the provided equation of the solid Q:z=5y,x=0,y=0,z=0,x=5 into consideration.

In the above equation, the bounds of x is 0x5 and bounds of z is, 0z(5y)

Take the xy plane into consideration.

In this case, since the value of z=0, the equation of the line will be given as,

0=5yy=5

The lower limit of x,y,z is provided in the question, i.e.,

x=0,y=0,z=0

On applying the formula for the mass (m) of solid body Q, we get,

m=Qρ(x,y,z)dV=050505ykxdzdydx=kx0505[z]05ydydx=kx0505[(5y)0]dydx

On integrating with respect to y, we get,

kx0505(5y)dydx=kx05[5yy22]05dx=kx05[5(5)5220]dx=252k05xdx

Further, on integrating with respect to x, we get,

252k05xdx=252k[x22]05=252k=6254k

Thus, mass (m) of the solid body Q is, 6254k

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find more solutions based on key concepts 