5.3 Elasticity: stress and strain  (Page 7/15)

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where $B$ is the bulk modulus (see [link] ), ${V}_{0}$ is the original volume, and $\frac{F}{A}$ is the force per unit area applied uniformly inward on all surfaces. Note that no bulk moduli are given for gases.

What are some examples of bulk compression of solids and liquids? One practical example is the manufacture of industrial-grade diamonds by compressing carbon with an extremely large force per unit area. The carbon atoms rearrange their crystalline structure into the more tightly packed pattern of diamonds. In nature, a similar process occurs deep underground, where extremely large forces result from the weight of overlying material. Another natural source of large compressive forces is the pressure created by the weight of water, especially in deep parts of the oceans. Water exerts an inward force on all surfaces of a submerged object, and even on the water itself. At great depths, water is measurably compressed, as the following example illustrates.

Calculating change in volume with deformation: how much is water compressed at great ocean depths?

Calculate the fractional decrease in volume ( $\frac{\Delta V}{{V}_{0}}$ ) for seawater at 5.00 km depth, where the force per unit area is $5\text{.}\text{00}×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}N/{m}^{2}$ .

Strategy

Equation $\Delta V=\frac{1}{B}\frac{F}{A}{V}_{0}$ is the correct physical relationship. All quantities in the equation except $\frac{\Delta V}{{V}_{0}}$ are known.

Solution

Solving for the unknown $\frac{\Delta V}{{V}_{0}}$ gives

$\frac{\Delta V}{{V}_{0}}=\frac{1}{B}\frac{F}{A}.$

Substituting known values with the value for the bulk modulus $B$ from [link] ,

$\begin{array}{lll}\frac{\Delta V}{{V}_{0}}& =& \frac{5.00×{\text{10}}^{7}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}}{2\text{.}2×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}}\\ & =& 0.023=2.3%.\end{array}$

Discussion

Although measurable, this is not a significant decrease in volume considering that the force per unit area is about 500 atmospheres (1 million pounds per square foot). Liquids and solids are extraordinarily difficult to compress.

Conversely, very large forces are created by liquids and solids when they try to expand but are constrained from doing so—which is equivalent to compressing them to less than their normal volume. This often occurs when a contained material warms up, since most materials expand when their temperature increases. If the materials are tightly constrained, they deform or break their container. Another very common example occurs when water freezes. Water, unlike most materials, expands when it freezes, and it can easily fracture a boulder, rupture a biological cell, or crack an engine block that gets in its way.

Other types of deformations, such as torsion or twisting, behave analogously to the tension, shear, and bulk deformations considered here.

Section summary

• Hooke’s law is given by
$F=k\text{Δ}L,$

where $\Delta L$ is the amount of deformation (the change in length), $F$ is the applied force, and $k$ is a proportionality constant that depends on the shape and composition of the object and the direction of the force. The relationship between the deformation and the applied force can also be written as

$\Delta L=\frac{1}{Y}\frac{F}{A}{L}_{0},$

where $Y\phantom{\rule{0.25em}{0ex}}$ is Young’s modulus , which depends on the substance, $A$ is the cross-sectional area, and ${L}_{0}$ is the original length.

• The ratio of force to area, $\frac{F}{A}$ , is defined as stress , measured in N/m 2 .
• The ratio of the change in length to length, $\frac{\Delta L}{{L}_{0}}$ , is defined as strain (a unitless quantity). In other words,
$\text{stress}=Y×\text{strain}.$
• The expression for shear deformation is
$\Delta x=\frac{1}{S}\frac{F}{A}{L}_{0},$

where $S$ is the shear modulus and $F$ is the force applied perpendicular to ${L}_{\text{0}}$ and parallel to the cross-sectional area $A$ .

• The relationship of the change in volume to other physical quantities is given by
$\Delta V=\frac{1}{B}\frac{F}{A}{V}_{0},$

where $B$ is the bulk modulus, ${V}_{\text{0}}$ is the original volume, and $\frac{F}{A}$ is the force per unit area applied uniformly inward on all surfaces.

a car move 6m. what is the acceleration?
depends how long
Peter
What is the simplest explanation on the difference of principle, law and a theory
how did the value of gravitational constant came give me the explanation
how did the value of gravitational constant 6.67×10°-11Nm2kg-2
Varun
A steel ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.45 m. (a) Calculate its velocity just before it strikes the floor.
9.8m/s?
Sqrt(2*1.5m*9.81m/s^2)
Richard
0.5m* mate.
0.05 I meant.
Guess your solution is correct considering the ball fall from 1.5m height initially.
Sqrt(2*1.5m*9.81m/s^2)
Deepak
How can we compare different combinations of capacitors?
find the dimension of acceleration if it's unit is ms-2
lt^-2
b=-2 ,a =1
M^0 L^1T^-2
Sneha
what is botany
Masha
it is a branch of science which deal with the study of plants animals and environment
Varun
what is work
a boy moving with an initial velocity of 2m\s and finally canes to rest with a velocity of 3m\s square at times 10se calculate it acceleration
Sunday
.
Abdul
6.6 lol 😁😁
Abdul
show ur work
Sunday
Abdul
Abdul
If the boy is coming to rest then how the hell will his final velocity be 3 it'll be zero
Abdul
re-write the question
Nicolas
men i -10 isn't correct.
Stephen
using v=u + at
Stephen
1/10
Happy
ya..1/10 is very correct..
Stephen
hnn
Happy
how did the value 6.67×10°-11Nm2kg2 came tell me please
Varun
Work is the product of force and distance
Kym
physicist
Michael
what is longitudinal wave
A longitudinal wave is wave which moves parallel or along the direction of propagation.
sahil
longitudinal wave in liquid is square root of bulk of modulus by density of liquid
harishree
Is British mathematical units the same as the United States units?(like inches, cm, ext.)
We use SI units: kg, m etc but the US sometimes refer to inches etc as British units even though we no longer use them.
Richard
Thanks, just what I needed to know.
Nina
What is the advantage of a diffraction grating over a double slit in dispersing light into a spectrum?
yes.
Abdul
Yes
Albert
sure
Ajali
yeap
Sani
yesssss
bilal
hello guys
Ibitayo
when you will ask the question
Ana
bichu
is free energy possible with magnets?
joel
no
Mr.
you could construct an aparatus that might have a slightly higher 'energy profit' than energy used, but you would havw to maintain the machine, and most likely keep it in a vacuum, for no air resistance, and cool it, so chances are quite slim.
Mr.
calculate the force, p, required to just make a 6kg object move along the horizontal surface where the coefficient of friction is 0.25
Gbolahan
Albert
if a man travel 7km 30degree east of North then 10km east find the resultant displacement
11km
Dohn
disagree. Displacement is the hypotenuse length of the final position to the starting position. Find x,y components of each leg of journey to determine final position, then use final components to calculate the displacement.
Daniel
1.The giant star Betelgeuse emits radiant energy at a rate of 10exponent4 times greater than our sun, where as it surface temperature is only half (2900k) that of our sun. Estimate the radius of Betelgeuse assuming e=1, the sun's radius is s=7*10exponent8metres
2. A ceramic teapot (e=0.20) and a shiny one (e=0.10), each hold 0.25 l of at 95degrees. A. Estimate the temperature rate of heat loss from each B. Estimate the temperature drop after 30mins for each. Consider only radiation and assume the surrounding at 20degrees
James
Is our blood not red
If yes than why when a beam of light is passing through our skin our skin is glowing in red colour
because in our blood veins more red colour is scattered due to low wavelength also because of that scattered portion comes on skin and our skin act as a thinscreen.
so you saying blood is not red?
Donny
blood is red that's why it is scattering red colour!
like if u pass light frm red colour solution then it will scatter red colour only.. so same it is with our skin..red colour blood is moving inside the veins bcz of thinkness of our fingers.. it appears to be red.
No I am not saying that blood is not red
then ur qtn is wrong buddy.. 😊
Blood is red. The reason our veins look blue under our skin, is because thats the only wavelength on light that can penetrate our skin.
Mr.
Red light is reflected from our blood but because of its wavelength it is not seen. While in the other hand blue light has a longer wavelength allowing it to pass the our skin and to our eyes.
Nina
Thus, our veins appear blue while they are really red... THE MORE YOU KNOW...(;
Nina
So in conclusion our blood is red but we can only see blue spectrum because of our skin. The more longer a wavelength is the more durable it is to reflection, so blue light cant pass thew skin completely causing a reflection which causes veins to appear blue. While the red light is scatter around.
Nina
the reason why when we shine a light at our skin it appears red is because the red light is increased and more goes to your eyes. So in other words it increases the amount of red light vs it being scatterd around everywhere.
Nina
I think the blood is only a mixture of colors but red is predominant due to high level of haemoglobin.
stanley
As a side note, the heme part of hemoglobin is why blood is red
Sedlex
a car starts from rest acceleration and moves with a uniform acceleration a, in time t. the distance covered during the motion is expressed as?.
distance=a×(t^2)
Emmanuel
1/2at.t
David