3.20: Untitled Page 46

Chapter 3

3.4 Problems

Section 3.1

3‐1. In Figure 3.1 we have illustrated a body in the shape of a sphere located in the center of the tube. The flow in the tube is laminar and the velocity profile is parabolic as indicated in the figure. Indicate how the shape of the sphere will change with time as the body is transported from left to right. Base your sketch on a cut through the center of the sphere that originally has the form of a circle.

Keep in mind that the body does not affect the velocity profile.

Figure 3.1. Body flowing and deforming in a tube

3‐2a. If the straight wire illustrated in Figure 3.2a has a uniform mass per unit length equal to o , the total mass of the wire is given by mass  o L

If the mass per unit length is given by (

 x) ,the total mass is determined by the

following line integral:

x  L

mass 

( x) dx

 x0

For the following conditions

( x)     x  L

2

2

1

o

3

  0 0065

.

kg/m,   0 0017

.

kg/m , L  1 4

. m

o

determine the total mass of the wire.

Figure 3.2a. Wire having a uniform or non‐uniform mass density

Single component systems

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3.2b. If the flat plate illustrated in Figure 3.2b has a uniform mass per unit area equal to  , the total mass of the plate is given by

o

mass  

 

o A

o L L

1

2

If the mass per unit area is given by ( x, y) ,the total mass is determined by the Figure 3.2b. Flat plate having a uniform or non‐uniform mass density area integral given by

y  L



2

x

L 1

mass 

 dA 

 ( x,y) dx dy

 

A

y  0

x  0

For the following conditions

( x,y)     xy

o

2

4

 

 





o

0 0065

.

kg/m ,

0 00017

.

kg/m ,

1

L

1 4

. m, L

2 7 m

2

.

determine the total mass of the plate.

3‐2c. If the density is a function of position represented by

     x  L   y  L

2





3

2 2

1

1

o

1

develop a general expression for the mass contained in the region indicated by 0  x  L

0  y  L

0  z  L 3

1

2

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