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183_notes:model_of_a_wire [2014/09/18 14:35] – pwirving | 183_notes:model_of_a_wire [2015/09/19 11:27] – [Modeling the solid wire] caballero | ||
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To understand how solids exert different forces, you must learn how the microscopic, | To understand how solids exert different forces, you must learn how the microscopic, | ||
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+ | ==== Lecture Video ==== | ||
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+ | {{youtube> | ||
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==== Modeling the interatomic bond as spring ==== | ==== Modeling the interatomic bond as spring ==== | ||
- | [{{ 183_notes: | + | [{{ 183_notes: |
To model the interatomic bond as a spring, we will need to first determine how " | To model the interatomic bond as a spring, we will need to first determine how " | ||
- | $$\rho = 21.45 \dfrac{g}{cm^3} \left(\dfrac{1kg}{10^3g}\right)\left(\dfrac{10^2 cm}{1m}\right)^3 = 21.45 \times 10^3 kg/m^3$$ | + | $$\rho = 21.45 \dfrac{g}{cm^3} \left(\dfrac{1kg}{10^3g}\right)\left(\dfrac{100 cm}{1m}\right)^3 = 21.45 \times 10^3 kg/m^3$$ |
Consider a cubic meter of Pt, which is a cube that is 1m long on each side((This amount of Pt would costs nearly $1, | Consider a cubic meter of Pt, which is a cube that is 1m long on each side((This amount of Pt would costs nearly $1, | ||
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This is roughly the distance between atoms in a solid piece of Pt. We think of this as the " | This is roughly the distance between atoms in a solid piece of Pt. We think of this as the " | ||
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==== Modeling the solid wire ==== | ==== Modeling the solid wire ==== | ||
- | [{{ 183_notes: | + | [{{ 183_notes: |
The simplest model we can use for a wire (beyond a single atomic chain), is to model it as many long parallel chains connected by springs. For you to understand this model, you will need to understand how to model two springs connected end-to-end (in series) and two springs connected side-by-side (in parallel). | The simplest model we can use for a wire (beyond a single atomic chain), is to model it as many long parallel chains connected by springs. For you to understand this model, you will need to understand how to model two springs connected end-to-end (in series) and two springs connected side-by-side (in parallel). | ||
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$$s = \dfrac{mg}{k} = \dfrac{100N}{100N/ | $$s = \dfrac{mg}{k} = \dfrac{100N}{100N/ | ||
- | When we attach a second 100N/m spring to the end of the first and then attach the mass, the springs | + | Hence, a single 100 N/m spring will stretch precisely 1m when a 100N ball is hung from it. |
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+ | When we attach a second 100N/m spring to the end of the first and then attach the mass, both springs | ||
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+ | //In series, each spring stretches as if the mass were attached to just that spring//, and the sum of all those stretches gives the overall stretch. | ||
== Modeling two end-to-end springs as one spring (effective spring constant) == | == Modeling two end-to-end springs as one spring (effective spring constant) == | ||
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$$s = \dfrac{mg}{2k} = \dfrac{100N}{200N/ | $$s = \dfrac{mg}{2k} = \dfrac{100N}{200N/ | ||
- | When we attach a second 100N/m spring to the ball, the springs both stretch 0.5m. That is, the overall stretch of the spring-mass system is half of what it is with one spring. //In parallel, each spring stretches the same amount//. | + | When we attach a second 100N/m spring to the ball, the springs both stretch 0.5m. That is, the overall stretch of the spring-mass system is half of what it is with one spring. |
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+ | //In parallel, each spring stretches the same amount//. | ||
== Modeling two side-by-side springs as one spring (effective spring constant) == | == Modeling two side-by-side springs as one spring (effective spring constant) == | ||
- | You will often find it useful to consider | + | You will often find it useful to consider a bunch of side-by-side springs as one spring. Similar to what was done with springs in series, you can model side-by-side springs as one hypothetical spring that stretches the same amount under the same load as the side-by-side springs. For the example above, you can find the single spring that stretches 0.5m when a 100N ball is attached to it. |
$$\Delta \vec{p} = 0\: | $$\Delta \vec{p} = 0\: | ||
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$$k_{s,eff} = \dfrac{mg}{s} = \dfrac{100N}{0.5m} = 200N/m$$ | $$k_{s,eff} = \dfrac{mg}{s} = \dfrac{100N}{0.5m} = 200N/m$$ | ||
- | Evidently, two springs with the same spring constant in parallel can be modeled by a single (firmer) spring with twice the spring constant of the each of the two springs. We can add springs in parallel to find the effective spring constant using this formula. | + | Evidently, two springs with the same spring constant in parallel can be modeled by a single (stiffer) spring with twice the spring constant of the each of the two springs. We can add springs in parallel to find the effective spring constant using this formula. |
$${k_{s, | $${k_{s, | ||
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$${k_{s, | $${k_{s, | ||
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+ | This way of modeling end-to-end and side-by-side springs will be very useful for modeling [[183_notes: |