D2. Interaction forces between particles

[math]\displaystyle{ \newcommand{\uvec}{\overline{\textbf{u}}} \newcommand{\vvec}{\overline{\textbf{v}}} \newcommand{\evec}{\overline{\textbf{e}}} \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}} \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}} \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}} \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}} \newcommand{\cs}{\textrm{c}} \newcommand{\ds}{\textrm{d}} \newcommand{\ms}{\textrm{m}} \newcommand{\ts}{\textrm{t}} \newcommand{\us}{\textrm{u}} \newcommand{\vs}{\textrm{v}} \newcommand{\Fs}{\textrm{F}} \newcommand{\Rs}{\textrm{R}} \newcommand{\Ts}{\textrm{T}} \newcommand{\Ls}{\textrm{L}} \newcommand{\Ns}{\textrm{N}} \newcommand{\Bs}{\textrm{B}} \newcommand{\es}{\textrm{e}} \newcommand{\fs}{\textrm{f}} \newcommand{\is}{\textrm{i}} \newcommand{\js}{\textrm{j}} \newcommand{\rs}{\textrm{r}} \newcommand{\Os}{\textbf{O}} \newcommand{\Gs}{\textbf{G}} \newcommand{\Cbf}{\textbf{C}} \newcommand{\Or}{\Os_\Rs} \newcommand{\Qs}{\textbf{Q}} \newcommand{\Cs}{\textbf{C}} \newcommand{\Ps}{\textbf{P}} \newcommand{\Ss}{\textbf{S}} \newcommand{\deg}{^\textsf{o}} \newcommand{\xs}{\textsf{x}} \newcommand{\ys}{\textsf{y}} \newcommand{\zs}{\textsf{z}} \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}} \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}} \newcommand{\vec}[1]{\overline{#1}} \newcommand{\vecbf}[1]{\overline{\textbf{#1}}} \newcommand{\vecdot}[1]{\overline{\dot{#1}}} \newcommand{\OQvec}{\vec{\Os\Qs}} \newcommand{\QPvec}{\vec{\Qs\Ps}} \newcommand{\OPvec}{\vec{\Os\textbf{P}}} \newcommand{\OCvec}{\vec{\Os\Cs}} \newcommand{\OGvec}{\vec{\Os\Gs}} \newcommand{\abs}[1]{\left|{#1}\right|} \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}} \newcommand{\vector}[3]{ \begin{Bmatrix} {#1}\\ {#2}\\ {#3} \end{Bmatrix}} \newcommand{\vecdosd}[2]{ \begin{Bmatrix} {#1}\\ {#2} \end{Bmatrix}} \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})} \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})} \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})} \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})} \newcommand{\velo}[1]{\vvec_{\textrm{#1}}} \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}} \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}} \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})} \newcommand{\psio}{\dot{\psi}_0} \newcommand{\Pll}{\textbf{P}_\textrm{lliure}} \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)} \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)} \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)} }[/math]

Newton's second law can be used to predict the acceleration of a particle P when all the interaction forces exerted on P by all the other particles (Q) are known (all [math]\displaystyle{ \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}} }[/math]), or to calculate the forces needed to guarantee a predetermined movement ([math]\displaystyle{ \overline{\mathbf{O}_\Rs\mathbf{P}}(\ts) }[/math]).

This section deals with the interaction forces between particles, whether they can be formulated or not. Formulating an interaction is finding a mathematical expression that allows calculating its value at each time instant given the mechanical state of the particles and the constants associated with the type of interaction phenomenon. When a force cannot be formulated, it is an unknown of the dynamic problem.

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D2.1 Kinematic dependence of interaction forces

According to the Principle of Determinacy, the forces of interaction between two particles P and Q at each time instant can only depend on the position and velocity of the particles at that time instant:

[math]\displaystyle{ \ms_\mathrm{p} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{P}, \mathrm{t})=\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}\left(\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}(\mathrm{t}), \overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}(\mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{P}, \mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{Q}, \mathrm{t})\right)=-\mathrm{m}_{\mathbf{Q}} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{Q}, \mathrm{t}) . }[/math]

Galileo's Principle of Relativity (equivalence of Galilean reference frames for the formulation of dynamics) imposes restrictions on the type of dependence of [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] on positions and velocities:

El Principi de Relativitat de Galileu (equivalència de les referències galileanes per a laformulació de la dinàmica) imposa restriccions al tipus de dependència de [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] en posicions i velocitats:

• La homogeneitat de l’espai en referències galileanes impedeix que [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] depengui en les posicions [math]\displaystyle{ \overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}} }[/math] i [math]\displaystyle{ \overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}} }[/math] per separat, però permet la dependència en la seva diferència [math]\displaystyle{ \overline{\mathbf{P}\Qs}(=\overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}-\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}) }[/math]
  

• La isotropia de l'espai en referències galileanes impedeix que [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] depengui de les velocitats de [math]\displaystyle{ \mathbf{P} }[/math] i [math]\displaystyle{ \mathbf{Q} }[/math] per separat (doncs [math]\displaystyle{ \overline{\mathbf{v}}_{\text {Gal1 }}(\mathbf{P}, \mathbf{Q}) \neq \overline{\mathbf{v}}_{\text {Gal2 }}(\mathbf{P}, \mathbf{Q}) }[/math]), però permet que depengui de la seva diferència: [math]\displaystyle{ \overline{\mathbf{v}}_{\text {Gal }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {Gal }}(\mathbf{Q}) \equiv \Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}} }[/math]. El vector [math]\displaystyle{ \Delta \overline{\mathbf{v}}_{\forall \mathrm{Gal}} }[/math] és el mateix independentment de quina sigui la referència Gal (per això s'ha posat el subíndex ' [math]\displaystyle{ \forall }[/math]Gal', que vol dir “per a tota referència Gal”). En general, [math]\displaystyle{ \Delta \overline{\mathbf{v}}_{\forall\text {Gal }} }[/math] té una component parallela a [math]\displaystyle{ \overline{\mathbf{P Q}} }[/math], i una de perpendicular a [math]\displaystyle{ \overline{\mathbf{P Q}} }[/math] : [math]\displaystyle{ \Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}}=\Delta \overline{\mathbf{v}}_{\| \overline{\mathbf{P Q}}}+\Delta \overline{\mathbf{v}}_{\perp \overline{\mathbf{P Q}}} \equiv \Delta \overline{\mathbf{v}}_\rho+\Delta \overline{\mathbf{v}}_{\mathrm{n}} }[/math]. Però la isotropia de l'espai tampoc no permet que [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] depengui d'una direcció que no sigui [math]\displaystyle{ \overline{\mathbf{P Q}} }[/math](com s'ha vist quan s'ha presentat la Tercera llei de Newton). Per tant, s'accepta la dependència en el vector [math]\displaystyle{ \Delta \overline{\mathbf{v}}_\rho }[/math] però només en el valor de [math]\displaystyle{ \left|\Delta \overline{\mathbf{v}}_{\mathrm{n}}\right| }[/math](tot i que usualment aquesta última dependència no apareix).

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D2.2 Classification of interaction forces

If the universe consisted only of dimensionless particles with no connecting elements between them, the list of possible interactions would be very short: in the mechanics, one could only speak of the force of gravitational attraction, which occurs “at a distance” ([math]\displaystyle{ \rho \gt 0 }[/math]). Talking about contact interaction between particles ([math]\displaystyle{ \rho = 0 }[/math]) is not possible: if [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math] “come into contact”, since the location of each of them is defined by a single point, the direction of the interaction is not determined. On the other hand, two particles cannot occupy the same point in space.

Strictly speaking, it is only possible to talk about contact interactions between a particle [math]\displaystyle{ \mathbf{P} }[/math] and a particle [math]\displaystyle{ \mathbf{Q} }[/math] that belongs to a rigid body [math]\displaystyle{ \mathrm{S}_\mathbf{Q} }[/math], or between particles [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math] that belong to rigid bodies [math]\displaystyle{ \mathrm{S}_\mathbf{P} }[/math] and [math]\displaystyle{ \mathrm{S}_\mathbf{Q} }[/math], respectively. These interactions are dealt with in unitat D3.

When [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math] interact at a distance ([math]\displaystyle{ \rho \gt 0 }[/math]), we talk of direct interaction. If they are not in contact ([math]\displaystyle{ \rho \gt 0 }[/math]) but them there is an element acting as an intermediate between them, we talk of indirect interaction through the element. In dynamics, all elements with negligible mass compared to that of [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math] (if they are not part of a rigid body) or to the rigid bodies to which they belong are considered intermediate elements (IE).

Forces transmitted between [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math] through those elements fulfil the

Les forces que es transmeten entre [math]\displaystyle{ \mathbf{P} }[/math] i [math]\displaystyle{ \mathbf{Q} }[/math] per mitjà d’aquests elements verifiquen el Principi d’Acció-Reacció: tenen el mateix valor, de direcció [math]\displaystyle{ \overline{\mathbf{PQ}} }[/math] però de sentits oposats (Figura D2.2).

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D2.8 Friction

© Universitat Politècnica de Catalunya. Tots els drets reservats

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