Wednesday, November 4, 1998

Chapter 8: Angular MomentumChapter 9: Density, Stress, Strain,

Young’s Modulus,shear modulus

We can construct a completely analogousargument to define angular momentum

I I

tLt

Where L is the angular momentum L I

A solid cylinder with moment ofinertia I=MR2/2 is given an angularvelocity of 1 rad/s. If the cylinderhas radius 0.5 m and mass 2 kg, whatis its angular momentum?

I MR 12

2 20 5 2 05 0 25( . )( )( . ) . kg m kg m2

L I ( . )( ) . /0 25 1 0 25 kg m rad / s kg m s2 2

Solids & Fluids

We’ve spent a lot of time looking at systemsof point masses, objects with no physicalextent. Let’s now look at more realisticrepresentations: objects that flow, stretch,and compress.

Let’s start by characterizing a solidmass of uniform composition.

What quantities can we directly measure?

If our block was made of copper, and wedoubled its size, what would happen to itsmass?

Remember our scale arguments (way back inthe first week of the course)!!!

L 2L

V0 = L3

V = (2L)3

V = 8L3 = 8V0

M

8M

In fact, if we made a plot of the mass of ourcopper block versus its volume, we’d find

m

V

This line has a slopethat characterizes thetype of material fromwhich the block is made.We define the slope ofthis line as the density() of the material.

slope

=

How would the slope of the line for a block ofstyrofoam compare to that for a block of lead?

MV

The unit of mass, the gram, was chosen tobe the mass of 1 cm3 of liquid water. Sowater has a density of 1 g/cm3 or 1 kg/L or1000 kg/m3.

The term specific gravity refers to theratio of the density of a given substanceto that of water. Objects with a specificgravity less than 1 will float in water;those greater than 1 will sink.

MV

[ ] [ ][ ]

MV

[ ] kgm3

Note: density is oftenwritten in grams percubic centimeter org/cc. There is a factorof 1000 differencebetween the two setsof units.

Stress on an object results in strain.

You guys are looking pretty strained…I meandrained right now!

A stress results when a force is appliedacross a surface of a given object. Stressesresult in deformations of the object knownas strains.

We push on a piece ofcopper with a force F.If we apply the forceover the entire cross-sectional area of theend of the tube, thestress is given by

stress FA

[ ] [ ][ ]

stress FA

Nm2

F

F

compressive stress

F

F

What happens to realobjects when we exertstress such as appliedin this figure?

In this case, the copper rodwill be compressed. Thefractional change in its lengthis known as the strain

strain ll

l lf

UNITLESSQUANTITY!

When stresses are applied, objects undergostrain. The ratio of the stress to the strainturns out to be a characteristic of the materialfrom which the object is made.

We call the property of a material related tothe way it strains under stress

Y stressstrain

F A

//l l

Y stressstrain

F A

//l l

Notice that Young’s moduluswill have the same units asstress: N / m2

We’ve looked at compressive stress. Whathappens when we try to stretch an objectinstead of trying to compress it?

F

F

lf l

tensile stress We name the stress thattries to pull an objectapart the “tensile stress.”

Again, the ratio of thestress to the strain isstill a characteristic ofthe material from whichthe object is made.

In fact, Young’s modulus describes thistype of stress as well as compressive stress.

A 500-kg load is hung from a 3-msteel wire (Y = 2 X 1011 N/m2) witha cross-sectional area of 0.15 cm2.By how much does the wire stretch?

Well, the force exerted by a 500-kg load is

F mg (500 )( . ) . kg m / s N29 8 4 9 103

stress FA

4 9 10015

4 9 10 3 27 103 3

8..

. . N cm

N1.5 10 m

N / m2 -5 22

Y stressstrain

2 10 327 10118

N / m N / m2

2.strain

A 500-kg load is hung from a 3-msteel wire (Y = 2 X 1011 N/m2) witha cross-sectional area of 0.15 cm2.By how much does the wire stretch?

strain

327 10

2 10163 10

8

113. .N / m

N / m

2

2

strain ll 163 10

33. l

m

l 4 9 10 3.

We can conceive of yet another way to tryto deform our copper tube...

F

F

l

shear stress This type of stresstries to move oneend of the tube inone direction andthe other end of thetube in the oppositedirection!

Unfortunately, it is NOT characterized byYoung’s modulus...

Let’s look at a side view of what happens toour copper cylinder under the influence ofshear stress.

Fs

Fsx

l

Shear stressesproduce

shear strains

“shear force”

shear stress FAs

where A is the area of thetop of the cylinder.

Fs

Fsx

l shear strain x l

Note: the units of shearstress and shear strainare the same as ourtensile and compressiveforms of stress and strain.