# 12.3 Stress, strain, and elastic modulus  (Page 5/26)

 Page 5 / 26
$\text{bulk strain}=\frac{\text{Δ}V}{{V}_{0}}.$

The bulk strain results from the bulk stress, which is a force ${F}_{\perp }$ normal to a surface that presses on the unit surface area A of a submerged object. This kind of physical quantity, or pressure p , is defined as

$\text{pressure}=p\equiv \frac{{F}_{\perp }}{A}.$

We will study pressure in fluids in greater detail in Fluid Mechanics . An important characteristic of pressure is that it is a scalar quantity and does not have any particular direction; that is, pressure acts equally in all possible directions. When you submerge your hand in water, you sense the same amount of pressure acting on the top surface of your hand as on the bottom surface, or on the side surface, or on the surface of the skin between your fingers. What you are perceiving in this case is an increase in pressure $\text{Δ}p$ over what you are used to feeling when your hand is not submerged in water. What you feel when your hand is not submerged in the water is the normal pressure     ${p}_{0}$ of one atmosphere, which serves as a reference point. The bulk stress is this increase in pressure, or $\text{Δ}p,$ over the normal level, ${p}_{0}.$

When the bulk stress increases, the bulk strain increases in response, in accordance with [link] . The proportionality constant in this relation is called the bulk modulus, B , or

$B=\frac{\text{bulk stress}}{\text{bulk strain}}=-\frac{\text{Δ}p}{\text{Δ}V\phantom{\rule{0.1em}{0ex}}\text{/}\phantom{\rule{0.1em}{0ex}}{V}_{0}}=\text{−}\text{Δ}p\phantom{\rule{0.2em}{0ex}}\frac{{V}_{0}}{\text{Δ}V}.$

The minus sign that appears in [link] is for consistency, to ensure that B is a positive quantity. Note that the minus sign $\left(–\right)$ is necessary because an increase $\text{Δ}p$ in pressure (a positive quantity) always causes a decrease $\text{Δ}V$ in volume, and decrease in volume is a negative quantity. The reciprocal of the bulk modulus is called compressibility     $k,$ or

$k=\frac{1}{B}=-\frac{\text{Δ}V\phantom{\rule{0.1em}{0ex}}\text{/}\phantom{\rule{0.1em}{0ex}}{V}_{0}}{\text{Δ}p}.$

The term ‘compressibility’ is used in relation to fluids (gases and liquids). Compressibility describes the change in the volume of a fluid per unit increase in pressure. Fluids characterized by a large compressibility are relatively easy to compress. For example, the compressibility of water is $4.64\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\text{/atm}$ and the compressibility of acetone is $1.45\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\text{/atm}.$ This means that under a 1.0-atm increase in pressure, the relative decrease in volume is approximately three times as large for acetone as it is for water.

## Hydraulic press

In a hydraulic press [link] , a 250-liter volume of oil is subjected to a 2300-psi pressure increase. If the compressibility of oil is $2.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\phantom{\rule{0.1em}{0ex}}\text{/}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\text{atm},$ find the bulk strain and the absolute decrease in the volume of oil when the press is operating.

## Strategy

We must invert [link] to find the bulk strain. First, we convert the pressure increase from psi to atm, $\text{Δ}p=2300\phantom{\rule{0.2em}{0ex}}\text{psi}=2300\phantom{\rule{0.1em}{0ex}}\text{/}\phantom{\rule{0.1em}{0ex}}14.7\phantom{\rule{0.2em}{0ex}}\text{atm}\approx \phantom{\rule{0.2em}{0ex}}160\phantom{\rule{0.2em}{0ex}}\text{atm},$ and identify ${V}_{0}=\phantom{\rule{0.2em}{0ex}}250\phantom{\rule{0.2em}{0ex}}\text{L}.$

## Solution

Substituting values into the equation, we have

$\begin{array}{}\\ \\ \text{bulk strain}=\frac{\text{Δ}V}{{V}_{0}}=\frac{\text{Δ}p}{B}=k\text{Δ}p=\left(2.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\text{/atm}\right)\left(160\phantom{\rule{0.2em}{0ex}}\text{atm}\right)=0.0032\hfill \\ \text{answer:}\phantom{\rule{0.5em}{0ex}}\text{Δ}V=0.0032\phantom{\rule{0.2em}{0ex}}{V}_{0}=0.0032\left(250\phantom{\rule{0.2em}{0ex}}\text{L}\right)=0.78\phantom{\rule{0.2em}{0ex}}\text{L.}\hfill \end{array}$

## Significance

Notice that since the compressibility of water is 2.32 times larger than that of oil, if the working substance in the hydraulic press of this problem were changed to water, the bulk strain as well as the volume change would be 2.32 times larger.

At time to = 0 the current to the DC motor is reverse, resulting in angular displacement of the motor shafts given by angle = (198rad/s)t - (24rad/s^2)t^2 - (2rad/s^3)t^3 At what time is the angular velocity of the motor shaft zero
3s
Basit
what is angular velocity
In three experiments, three different horizontal forces are ap- plied to the same block lying on the same countertop. The force magnitudes are F1 " 12 N, F2 " 8 N, and F3 " 4 N. In each experi- ment, the block remains stationary in spite of the applied force. Rank the forces according to (a) the
state Hooke's law of elasticity
Hooke's law states that the extension produced is directly proportional to the applied force provided that the elastic limit is not exceeded. F=ke;
Shaibu
thanks
Aarti
You are welcome
Shaibu
thnx
Junaid
what is drag force
Junaid
A backward acting force that tends to resist thrust
Ian
solve:A person who weighs 720N in air is lowered in to tank of water to about chin level .He sits in a harness of negligible mass suspended from a scale that reads his apparent weight .He then dumps himself under water submerging his body .If his weight while submerged is 34.3N. find his density
Ian
The weight inside the tank is lesser due to the buoyancy force by the water displaced. Weight of water displaced = His weight outside - his weight inside tank = 720 - 34.3 = 685.7N Now, the density of water = 997kg/m³ (this is a known value) Volume of water displaced = Mass/Density (next com)
Sharath
density or relative density
Shaibu
density
Ian
Upthrust =720-34.3=685.7N mass of water displayed = 685.7/g vol of water displayed = 685.7/g/997 hence, density of man = 720/g / (685.7/g/997) =1046.6 kg/m3
1046.8
R.d=weight in air/upthrust in water =720/34.3=20.99 R.d=density of substance/density of water 20.99=x/1 x=20.99g/cm^3
Shaibu
Kg /cubic meters
Shaibu
Upthrust = 720-34.3=685.7N vol of water = 685.7/g/density of water = 685.7/g/997 so density of man = 720/g /(685.7/g/997) =1046.8 kg/m3
is there anyway i can see your calculations
Ian
Upthrust =720-34.3=685.7
Upthrust 720-34.3
=685.7N
Vol of water = 685.7/g/997
Hence density of man = 720/g / (685.7/g/997)
=1046.8 kg/m3
so the density of water is 997
Shaibu
Yes
Okay, thanks
Shaibu
try finding the volume then
Ian
Vol of man = vol of water displayed
I've done that; I got 0.0687m^3
Shaibu
okay i got it thanks
Ian
u welcome
Shaibu
HELLO kindly assist me on this...(MATHS) show that the function f(x)=[0 for xor=0]is continuous from the right of x->0 but not from the left of x->0
I do not get the question can you make it clearer
Ark
Same here, the function looks very ambiguous. please restate the question properly.
Sharath
please help me solve this problem.a hiker begins a trip by first walking 25kmSE from her car.she stops and sets her tent for the night . on the second day, she walks 40km in a direction 60°NorthofEast,at which she discovers a forest ranger's tower.find components of hiker's displacement for each day
Take a paper. put a point (name is A), now draw a line in the South east direction from A. Assume the line is 25 km long. that is the first stop (name the second point B) From B, turn 60 degrees to the north of East and draw another line, name that C. that line is 40 km long. (contd.)
Sharath
Now, you know how to calculate displacements, I hope? the displacement between two points is the shortest distance between the two points. go ahead and do the calculations necessary. Good luck!
Sharath
thank you so much Sharath Kumar
Liteboho
thank you, have also learned alot
Duncan
No issues at all. I love the subject and teaching it is fun. Cheers!
Sharath
cheers!
Liteboho
cheers too
Duncan
hii
Lakshya
hii too
Siciid
haye
Siciid
yes
Siciid
yes
Lakshya
shggggg
Lakshya
you mean
Siciid
solution problems
Siciid
what is the definition of model
please is there any way that i can understand physics very well i know am not support to ask this kind of question....
matthew
yes
Duncan
prove using vector algebra that the diagonals of a rhombus perpendicular to each other.
A projectile is thrown with a speed of v at an angle of theta has a range of R on the surface of the earth. For same v and theta,it's range on the surface of moon will be
0
Keshav
what is soln..
Keshav
o
Duncan
Using some kinematics, time taken for the projectile to reach ground is (2*v*g*Sin (∆)) (here, g is gravity on Earth and ∆ is theta) therefore, on Earth, R = 2*v²*g*Sin(∆)*Cos(∆) on moon, the only difference is the gravity. Gravity on moon = 0.166*g substituting that value in R, we get the new R
Sharath
Some corrections to my old post. Time taken to reach ground = 2*v*Sin (∆)/g R = (2*v²*Sin(∆)*Cos(∆))/g I put the g in the numerator by mistake in my old post. apologies for that. R on moon = (R on Earth)/(0.166)
Sharath
state Newton's first law of motion
Every body will continue in it's state of rest or of uniform motion in a straight line, unless it is compelled to change that state by an external force.
Kumaga
if you want this to become intuitive to you then you should state it
Shii
changing the state of rest or uniform motion of a body
koffi
if a body is in rest or motion it is always rest or motion, upto external force appied on it. it explains inertia
Omsai
what is a vector
smith
a ship move due north at 100kmhr----1 on a River flowing be due east on at 25kmperhr. cal the magnitude of the resultant velocity of the ship.
The result is a simple vector addition. The angle between the vectors is 90 degrees, so we can use Pythagoras theorem to get the result. V magnitude = sqrt(100*100 + 25*25) = 103.077 km/hr. the direction of the resultant vector can be found using trigonometry. Tan (theta) = 25/100.
Kumar
103.077640640442km/h
Peter
state Newton's first law of motion
An object continues to be in its state of rest or motion unless compelled by some external force
Alem
First law (law of inertia)- If a body is at rest, it would remain at rest and if the body is in the motion, it would be moving with the same velocity until or unless no external force is applied on it. If force F^=0 acceleration a^=0 or v^=0 or constant.
Govindsingh
how would you measure displacement in your car?
what is constellation
The product of a. (vector b× vector a)
I want to join the conversation
ok
Kamal
Ok
Bishal
ok
Rohit
Two charges 1uc and 3uc are separated 4m apart. find the point on the line connecting them at which their electric field intensity balances each other
Chukwurah
hmmm
JMPSCL