| Both sides previous revision Previous revision Next revision | Previous revision |
| 183_notes:youngs_modulus [2015/06/30 17:24] – obsniukm | 183_notes:youngs_modulus [2021/02/18 20:41] (current) – [Young's Modulus] stumptyl |
|---|
| | Section 4.5 and 4.6 in Matter and Interactions (4th edition) |
| | |
| ===== Young's Modulus ===== | ===== Young's Modulus ===== |
| |
| Earlier, you read how to [[183_notes:model_of_a_wire|add springs in parallel and in series]]. In these notes, you will read about how the microscopic measurements of bond length and interatomic spring stiffness relate to macroscopic measures like [[http://en.wikipedia.org/wiki/Young's_modulus|Young's modulus]]. We will continue using Platinum wire as our example. | Earlier, you read how to [[183_notes:model_of_a_wire|add springs in parallel and in series]]. **In these notes, you will read about how the microscopic measurements of bond length and interatomic spring stiffness relate to macroscopic measures like [[http://en.wikipedia.org/wiki/Young's_modulus|Young's modulus]].** We will continue using Platinum wire as our example. |
| |
| ==== Hanging a mass from a platinum wire ==== | ===== Hanging a mass from a platinum wire ===== |
| |
| Consider a 2m long platinum wire ($L = 2m$) with a square cross section. That is, the wire is not "round" when viewed on end, but square. This wire is 1mm thick ($S = 1mm$); each side of the wire is 1mm. If you were to hang a 10kg weight, this 2m wire stretches by 1.166 mm ($s = 1.166mm$). | Consider a 2m long platinum wire ($L = 2m$) with a square cross section. That is, the wire is not "round" when viewed on end, but square. This wire is 1mm thick ($S = 1mm$); each side of the wire is 1mm. If you were to hang a 10kg weight, this 2m wire stretches by 1.166 mm ($s = 1.166mm$). |
| |
| === Determining the interatomic "spring stiffness" === | ==== Determining the interatomic "spring stiffness" ==== |
| |
| If you model the whole wire as a single spring, you can find the spring stiffness of the whole wire. From the momentum principle (momentum not changing), you can determine the this stiffness because the net force is zero. | If you model the whole wire as a single spring, you can find the spring stiffness of the whole wire. From the [[183_notes:momentum_principle|momentum principle]] (momentum not changing), you can determine the this stiffness because the net force is zero. |
| |
| $$\vec{F}_{net} = \vec{F}_{grav} + \vec{F}_{wire} = 0$$ | $$\vec{F}_{net} = \vec{F}_{grav} + \vec{F}_{wire} = 0$$ |
| $$k_{s,wire} = \dfrac{mg}{s} = \dfrac{(10kg)(9.81 m/s^2)}{0.001166m} = 8.41\times10^4 N/m$$ | $$k_{s,wire} = \dfrac{mg}{s} = \dfrac{(10kg)(9.81 m/s^2)}{0.001166m} = 8.41\times10^4 N/m$$ |
| |
| This is very large spring constant because the wire (taken as a whole) is very stiff. | This is very large spring constant because the wire (taken as a whole) is very stiff. //Note: the units of N/m for k.// |
| |
| == Finding the number bonds in the wire == | ==== Finding the number bonds in the wire ==== |
| |
| [{{ 183_notes:mi3e_04-017.png?200|Using the model of a cubic lattice, you can determine the number of bonds in the wire and the number of bonds in a single chain.}}] | [{{ 183_notes:mi3e_04-017.png?150|Using the model of a cubic lattice, you can determine the number of bonds in the wire and the number of bonds in a single chain.}}] |
| |
| To find the interatomic spring stiffness, you will need to know how many chains of atoms are in the wire (how many side-by-side springs) and how many atoms are in the chain (how many end-to-end springs). Those values can be found by using the bond "length", [[183_notes:model_of_a_wire#modeling_the_interatomic_bond_as_spring|which you read about earlier]]. To remind you, the estimated bond length for Platinum is $d=2.47\times10^{-10}m$. | To find the interatomic spring stiffness, you will need to know how many chains of atoms are in the wire (how many side-by-side springs) and how many atoms are in the chain (how many end-to-end springs). Those values can be found by using the bond "length", [[183_notes:model_of_a_wire#modeling_the_interatomic_bond_as_spring|which you read about earlier]]. To remind you, the estimated bond length for Platinum is $d=2.47\times10^{-10}m$. |
| $$k_{s,interatomic} = \dfrac{N_{bonds\:in\:chain}}{N_{chains\:in\:wire}} k_{s,wire} $$ | $$k_{s,interatomic} = \dfrac{N_{bonds\:in\:chain}}{N_{chains\:in\:wire}} k_{s,wire} $$ |
| |
| ==== Young's Modulus ==== | The value that we found for the interatomic spring stiffness of Platinum (41.52 N/m) is typical of most pure metals, which have a range from about 5 to about 50 N/m. |
| | ===== Young's Modulus ===== |
| |
| Like density, the interatomic spring stiffness ($k_{s,interatomic}$) is an [[http://en.wikipedia.org/wiki/Intensive_and_extensive_properties#Intensive_properties|intensive property]] of an object, it doesn't depend on the length or shape of the object. Other properties are [[http://en.wikipedia.org/wiki/Intensive_and_extensive_properties#Extensive_properties|extensive]] such as mass, volume, and the spring stiffness of the whole wire ($k_{s,wire}$). Scientists and engineers often prefer working with intensive properties because they characterize the material and not the object. However, the interatomic spring stiffness is not a property that scientists and engineers often use. When discussing the compression and extension of materials, they often use the bulk modulus or Young's modulus. | Like density, the interatomic spring stiffness ($k_{s,interatomic}$) is an [[http://en.wikipedia.org/wiki/Intensive_and_extensive_properties#Intensive_properties|intensive property]] of an object, it doesn't depend on the length or shape of the object. Other properties are [[http://en.wikipedia.org/wiki/Intensive_and_extensive_properties#Extensive_properties|extensive]] such as mass, volume, and the spring stiffness of the whole wire ($k_{s,wire}$). Scientists and engineers will often work with intensive properties because they characterize the material and not the object. However, the interatomic spring stiffness is not a property that scientists and engineers often use. When discussing the compression and extension of materials, they often use the bulk modulus or [[https://en.wikipedia.org/wiki/Young%27s_modulus|Young's modulus]]. |
| |
| === Stress and strain === | ==== Stress and strain ==== |
| |
| [{{ 183_notes:mi3e_04-018.png?200|Hanging a mass $m$ on the end of a wire with relaxed length $L$ and cross-sectional area $A$ results in an elongation (stretch) $\Delta L$.}}] | [{{ 183_notes:mi3e_04-018.png?100|Hanging a mass $m$ on the end of a wire with relaxed length $L$ and cross-sectional area $A$ results in an elongation (stretch) $\Delta L$.}}] |
| |
| To understand Young's modulus, you must learn about stress and strain. | To understand Young's modulus, you must learn about stress and strain. |
| ==== Connecting the microscopic and the macroscopic ==== | ==== Connecting the microscopic and the macroscopic ==== |
| |
| Because the Young's modulus is an intensive quantity, it applies to our model at any scale. So we can apply the Young's modulus equation to a pair of atoms sharing a single bond of length $d$ (the $s$ is both the relaxed length and the scale of the atoms). If the atomic bond (with interatomic spring stiffness $k_{s,interatomic}$) stretches a small amount $s$, then the Young's modulus for that pair of atoms is given by: | Because the Young's modulus is an intensive quantity, it applies to our model at any scale. So we can apply the Young's modulus equation to a pair of atoms sharing a single bond of length $d$ (the $d$ is both the relaxed length and the scale of the atoms). If the atomic bond (with interatomic spring stiffness $k_{s,interatomic}$) stretches a small amount $s$, then the Young's modulus for that pair of atoms is given by: |
| |
| $$Y=\dfrac{F_T/A}{\Delta L/L} = \dfrac{(k_{s,interatomic}s)/d^2}{s/d} = \dfrac{k_{s,interatomic}}{d}$$ | $$Y=\dfrac{F_T/A}{\Delta L/L} = \dfrac{(k_{s,interatomic}s)/d^2}{s/d} = \dfrac{k_{s,interatomic}}{d}$$ |
| |
| Here, you have connected a macroscopic measurement ($Y$) to a microscopic model ($\dfrac{k_{s,interatomic}}{d}$). | Here, you have connected a macroscopic measurement ($Y$) to a microscopic model ($\dfrac{k_{s,interatomic}}{d}$). |
| | |
| | ==== Examples ==== |
| | |
| | * [[:183_notes:examples:videoswk4|Video Example: Chains and Bonds of a Copper Wire]] |