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5.3 Elasticity: stress and strain  (Page 6/15)

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Calculating force required to deform: that nail does not bend much under a load

Find the mass of the picture hanging from a steel nail as shown in [link] , given that the nail bends only 1.80 µm size 12{ left (1 "." "80" times "10" rSup { size 8{ - 6} } m right )} {} . (Assume the shear modulus is known to two significant figures.)

Diagram showing the side view a nail in a wall, deformed by the weight of a picture hanging from it. The weight w of the picture is downward. There is an equal force w upward on the nail from the wall. The nail is 1 point five zero millimeters thick. The length of the nail that is outside the wall is five point zero zero millimeters. The deformation delta x of the nail as a result of the picture is 1 point eight zero micrometers.
Side view of a nail with a picture hung from it. The nail flexes very slightly (shown much larger than actual) because of the shearing effect of the supported weight. Also shown is the upward force of the wall on the nail, illustrating that there are equal and opposite forces applied across opposite cross sections of the nail. See [link] for a calculation of the mass of the picture.

Strategy

The force F size 12{F} {} on the nail (neglecting the nail's own weight) is the weight of the picture w size 12{w} {} . If we can find w size 12{w} {} , then the mass of the picture is just w g size 12{ { {w} over {g} } } {} . The equation Δ x = 1 S F A L 0 size 12{Δx= { {1} over {S} } { {F} over {A} } L rSub { size 8{0} } } {} can be solved for F size 12{F} {} .

Solution

Solving the equation Δ x = 1 S F A L 0 size 12{Δx= { {1} over {S} } { {F} over {A} } L rSub { size 8{0} } } {} for F , we see that all other quantities can be found:

F = SA L 0 Δ x . size 12{F= { { ital "SA"} over {L rSub { size 8{0} } } } Δx} {}

S is found in [link] and is S = 80 × 10 9 N/m 2 size 12{S="80" times "10" rSup { size 8{9} } " N/m" rSup { size 8{2} } } {} . The radius r size 12{r} {} is 0.750 mm (as seen in the figure), so the cross-sectional area is

A = πr 2 = 1 . 77 × 10 − 6 m 2 . size 12{A=πr rSup { size 8{2} } =1 "." "77" times "10" rSup { size 8{ - 6} } m rSup { size 8{2} } } {}

The value for L 0 size 12{L rSub { size 8{0} } } {} is also shown in the figure. Thus,

F = ( 80 × 10 9 N/m 2 ) ( 1 . 77 × 10 − 6 m 2 ) ( 5 . 00 × 10 − 3 m ) ( 1 . 80 × 10 − 6 m ) = 51 N. size 12{F= { { \( "80" times "10" rSup { size 8{9} } " N/m" rSup { size 8{2} } \) \( 1 "." "77" times "10" rSup { size 8{ - 6} } " m" rSup { size 8{2} } \) } over { \( 5 "." "00" times "10" rSup { size 8{ - 3} } " m" \) } } times \( 1 "." "80" times "10" rSup { size 8{ - 6} } " m" \) ="51"N} {}

This 51 N force is the weight w of the picture, so the picture's mass is

m = w g = F g = 5 .2 kg . size 12{m= { {w} over {g} } = { {F} over {g} } =5 "." 2" kg"} {}

Discussion

This is a fairly massive picture, and it is impressive that the nail flexes only 1.80 µm —an amount undetectable to the unaided eye.

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Changes in volume: bulk modulus

An object will be compressed in all directions if inward forces are applied evenly on all its surfaces as in [link] . It is relatively easy to compress gases and extremely difficult to compress liquids and solids. For example, air in a wine bottle is compressed when it is corked. But if you try corking a brim-full bottle, you cannot compress the wine—some must be removed if the cork is to be inserted. The reason for these different compressibilities is that atoms and molecules are separated by large empty spaces in gases but packed close together in liquids and solids. To compress a gas, you must force its atoms and molecules closer together. To compress liquids and solids, you must actually compress their atoms and molecules, and very strong electromagnetic forces in them oppose this compression.

A cube with area of cross section A and volume V zero is compressed by an inward force F acting on all surfaces. The compression causes a change in volume delta V, which is proportional to the force per unit area and its original volume. This change in volume is related to the compressibility of the substance.
An inward force on all surfaces compresses this cube. Its change in volume is proportional to the force per unit area and its original volume, and is related to the compressibility of the substance.

We can describe the compression or volume deformation of an object with an equation. First, we note that a force “applied evenly” is defined to have the same stress, or ratio of force to area F A size 12{ left ( { {F} over {A} } right )} {} on all surfaces. The deformation produced is a change in volume Δ V size 12{ΔV} {} , which is found to behave very similarly to the shear, tension, and compression previously discussed. (This is not surprising, since a compression of the entire object is equivalent to compressing each of its three dimensions.) The relationship of the change in volume to other physical quantities is given by

Δ V = 1 B F A V 0 , size 12{ΔV= { {1} over {B} } { {F} over {A} } V rSub { size 8{0} } } {}

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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