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| 183_notes:youngs_modulus [2015/09/19 11:33] – [Hanging a mass from a platinum wire] caballero | 183_notes:youngs_modulus [2021/02/18 20:41] (current) – [Young's Modulus] stumptyl |
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| | Section 4.5 and 4.6 in Matter and Interactions (4th edition) |
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| ===== Young's Modulus ===== | ===== Young's Modulus ===== |
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| 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. |
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| ==== Hanging a mass from a platinum wire ==== | ===== Hanging a mass from a platinum wire ===== |
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| 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 ($s = 1.166mm$). |
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| === Determining the interatomic "spring stiffness" === | ==== Determining the interatomic "spring stiffness" ==== |
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| 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. | 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. |
| $$k_{s,wire} = \dfrac{mg}{s} = \dfrac{(10kg)(9.81 m/s^2)}{0.001166m} = 8.41\times10^4 N/m$$ | $$k_{s,wire} = \dfrac{mg}{s} = \dfrac{(10kg)(9.81 m/s^2)}{0.001166m} = 8.41\times10^4 N/m$$ |
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| 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.// |
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| == Finding the number bonds in the wire == | ==== Finding the number bonds in the wire ==== |
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| [{{ 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.}}] | [{{ 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.}}] |
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| 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. | 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 ==== | ===== Young's Modulus ===== |
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| 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]]. |
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| === Stress and strain === | ==== Stress and strain ==== |
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| [{{ 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) $\Delta L$.}}] | [{{ 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) $\Delta L$.}}] |
| ==== Connecting the microscopic and the macroscopic ==== | ==== Connecting the microscopic and the macroscopic ==== |
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| 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: |
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| $$Y=\dfrac{F_T/A}{\Delta L/L} = \dfrac{(k_{s,interatomic}s)/d^2}{s/d} = \dfrac{k_{s,interatomic}}{d}$$ | $$Y=\dfrac{F_T/A}{\Delta L/L} = \dfrac{(k_{s,interatomic}s)/d^2}{s/d} = \dfrac{k_{s,interatomic}}{d}$$ |
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| 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 ($\dfrac{k_{s,interatomic}}{d}$). |
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| | ==== Examples ==== |
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| | * [[:183_notes:examples:videoswk4|Video Example: Chains and Bonds of a Copper Wire]] |