D2. Interaction forces between particles

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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:

• The homogeneity of space in Galilean reference frames does not allow the dependence of [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] on positions [math]\displaystyle{ \overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}} }[/math] and [math]\displaystyle{ \overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}} }[/math] separately, but allows the dependence on its difference [math]\displaystyle{ \overline{\mathbf{P}\Qs}(=\overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}-\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}) }[/math].
  

• The isotropy of space in Galilean reference frames does not allow the dependence of [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] on the velocities of de P and Q separately (as [math]\displaystyle{ \overline{\mathbf{v}}_{\text {Gal1 }}(\mathbf{P}, \mathbf{Q}) \neq \overline{\mathbf{v}}_{\text {Gal2 }}(\mathbf{P}, \mathbf{Q}) }[/math]), but allows the dependence on its difference [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]. Vector [math]\displaystyle{ \Delta \overline{\mathbf{v}}_{\forall \mathrm{Gal}} }[/math] is the same in all Galilean reference frames (this is the meaning of the '[math]\displaystyle{ \forall }[/math]Gal' subscript, that means “for all Gal reference frames”). In general, [math]\displaystyle{ \Delta \overline{\mathbf{v}}_{\forall\text {Gal }} }[/math] has a component parallel to [math]\displaystyle{ \overline{\mathbf{P Q}} }[/math], and another one perpendicular to [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]. But the space isotropy does not allow either the dependence of [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] on a direction different from [math]\displaystyle{ \overline{\mathbf{P Q}} }[/math] (as seen in the presentation of Newton's third law). Therefore, we can accept the dependence on vector [math]\displaystyle{ \Delta \overline{\mathbf{v}}_\rho }[/math] but only on the value [math]\displaystyle{ \left|\Delta \overline{\mathbf{v}}_{\mathrm{n}}\right| }[/math](though this dependence rarely appears).

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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 Action-Reaction Principle: they have the same value and are parallel to [math]\displaystyle{ \overline{\mathbf{PQ}} }[/math], though they have opposite directions (Figure D2.2).

Figure D2.2 Force transmitted through an intermediate element between two particles [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math]

Intermediate elements are treated as a black box: the forces introduced between their endpoints come from deformations and phenomena that are outside the scope of the rigid body model considered in this course (for example, phenomena linked to fluid dynamics, thermodynamics or electromagnetics), and therefore what happens inside them is not studied: only a phenomenological description of their consequences on the particles connected through them is sought.

In this course, we consider four intermediate elements between particles:

• springs: the forces introduced between two particles come from their deformation; these forces can be attractive or repulsive (in this case, they can be conditioned by the guiding of the element so that they do not bend laterally), and they allow relative movements between [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math] of any sign.
  

• dampers: they introduce forces, often associated with the viscosity of fluids, opposed to the relative motion between [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math]; in the absence of initial relative motion ([math]\displaystyle{ \dot{\rho}_\mathrm{inic}=0 }[/math]), these elements do not introduce any force.
  

• linear actuators: their operation is based on various phenomena, depending on whether they are hydraulic, pneumatic, electric or magnetic; they exert forces that can be predetermined (that is, known along time: [math]\displaystyle{ \mathrm{F}_\mathrm{acc.lin}(\ts) }[/math] they are data in the problem) or forces that are suitable to control a predetermined relative motion [math]\displaystyle{ \dot{\rho}(\ts) }[/math], either an approaching or a separating motion.
  

• inextensible threads: they prevent the particles from separating (impedeixen [math]\displaystyle{ \dot{\rho} \gt 0 }[/math] )but not from approaching (they allow [math]\displaystyle{ \dot{\rho} \lt 0 }[/math]). Since they are intermediate elements that forbid movement, the force they introduce is called restriction or constraint force.
  

Figure D2.3 classifies the interactions between particles considered in this course, according to whether they are direct or indirect, formulable or non-formulable.

Figure D2.3 Classification of interactions between two particles.

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D2.3 Gravitational attraction

The gravitational interaction force (law of universal gravitation) was formulated by Newton. It is an attraction force, and is inversely proportional to the square of the distance between particles (Figure D2.4). It is an empirical formulation: it is based on astronomical observations accumulated over many years.

Figure D2.4 Formulation of the gravitational attraction force.

[math]\displaystyle{ \mathrm{G}_0 }[/math] is the constant of universal gravitation, and its value is [math]\displaystyle{ \mathrm{G}_0=6,67\cdot 10^{-11} \mathrm{m}^3/(\mathrm{Kg}\cdot \mathrm{s}^2) }[/math]

The distance between particles ([math]\displaystyle{ \rho }[/math]) must be expressed as a function of the coordinates that have been chosen to describe the system configuration.

✏️ Exemple D2.1: gravitational attraction between two satellites

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Two satellites [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math], modelled as particles of mass [math]\displaystyle{ \ms_\Ps }[/math] and [math]\displaystyle{ \ms_\mathrm{Q} }[/math] , describe circular orbits of radii [math]\displaystyle{ \rs_\Ps }[/math] and [math]\displaystyle{ \rs_\mathrm{Q} }[/math], respectively, around a planet in the same plane. The configuration of the system is described by the angles [math]\displaystyle{ \theta_\Ps }[/math] and [math]\displaystyle{ \theta_\mathrm{Q}. }[/math]

The gravitational force they exert on each other is:

[math]\displaystyle{ \rho=|\overline{\mathbf{P Q}}|=|\overline{\mathbf{O Q}}-\overline{\mathbf{O P}}|=\sqrt{\left(\rs_\mathrm{Q} \sin \theta_\mathrm{Q}-\rs_\Ps \sin \theta_\Ps\right)^2+\left(\rs_\mathrm{Q} \cos \theta_\mathrm{Q}-\rs_\Ps \cos \theta_\Ps\right)^2}=\sqrt{\rs_\mathrm{Q}^2+\rs_\Ps^2-2 \rs_\mathrm{Q} \rs_\Ps \sin \left(\theta_\mathrm{Q}+\theta_\Ps\right)} }[/math]

[math]\displaystyle{ \mathrm{F}_{\Ps \leftrightarrow \mathrm{Q}}^{\text {grav }}=\mathrm{G}_0 \frac{\ms^2}{\rs_\mathrm{Q}^2+\rs_\Ps^2-2 \rs_\mathrm{Q} \rs_\Ps \cos \left(\theta_\Ps-\theta_\mathrm{Q}\right)} }[/math]

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D2.4 Interaction through springs

Springs introduce attractive or repulsive forces between their ends depending on their deformation. From their natural length [math]\displaystyle{ \rho_\mathrm{nat} }[/math] (for which no force is produced between the spring ends), an elongation ([math]\displaystyle{ \rho-\rho_\mathrm{nat} \gt 0 }[/math]) causes attractive forces while a shortening ([math]\displaystyle{ \rho-\rho_\mathrm{nat} \lt 0 }[/math]) causes repulsive forces.

The mathematical formulation of these forces is obtained empirically from tests that measure the force as a function of the length change. Usually, we start from a static configuration in which the spring length [math]\displaystyle{ \rho_0 }[/math] does not have to coincide with the natural one. If [math]\displaystyle{ \rho_0 \gt \rho_\mathrm{nat} }[/math] , the force [math]\displaystyle{ \mathrm{F}_0 }[/math] between the spring ends for that configuration is attractive. Otherwise, ([math]\displaystyle{ \rho_0 \lt \rho_\mathrm{nat} }[/math]), [math]\displaystyle{ \mathrm{F}_0 }[/math] is a repulsive force.

The linear springs considered in this course have a linear behaviour: The variation in force [math]\displaystyle{ \Delta\mathrm{F} }[/math] from the reference value ([math]\displaystyle{ \Delta\mathrm{F}=\mathrm{F}-\mathrm{F}_0 }[/math]) )is proportional to the variation in length [math]\displaystyle{ \Delta\rho=\rho-\rho_0 }[/math] through a constant k.

A spring that is part of a mechanical system can introduce attractive and repulsive forces throughout its operation. Despite this, these forces are drawn with a single criterion (attractive or repulsive), and are formulated so that their value can have a positive or negative sign. In this way, both attractive and repulsive forces can be reproduced with a single drawing (Figure D2.5).

Figure D2.5 Formulation of the attractive (a) or repulsive (b) force of a spring with linear behaviour.

✏️ Exemple D2.2: força d’atracció d’una molla de comportament lineal

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La molla de comportament lineal actua entre les partícules [math]\displaystyle{ \mathbf{P} }[/math] i [math]\displaystyle{ \mathbf{Q} }[/math] que es mouen dins de dues guies paral·leles. Per a [math]\displaystyle{ \mathrm{x}_1=\mathrm{x}_2 }[/math] , la molla està estirada i la força que exerceix entre els seus extrems és [math]\displaystyle{ \mathrm{F}_0 }[/math].

Per a [math]\displaystyle{ \mathrm{x}_1=\mathrm{x}_2 }[/math] , la llargària de la molla és L i la força [math]\displaystyle{ \mathrm{F}_0 }[/math] és d’atracció. Per a [math]\displaystyle{ \mathrm{x}_1 \neq \mathrm{x}_2 }[/math], la llargària augmenta i la força d’atracció també.

L’expressió de la força de la molla per a una configuració general com a força atractiva és:

[math]\displaystyle{ \mathrm{F}_\mathrm{at}^\mathrm{molla}=\mathrm{F}_0+k\Delta\rho=\mathrm{F}_0 + k [\rho(\mathrm{x}_1 \neq \mathrm{x}_2)-\rho(\mathrm{x}_1=\mathrm{x}_2)]=\mathrm{F}_0 + k[\sqrt{\mathrm{L}^2+(\mathrm{x}_1-\mathrm{x}_2)^2}-\mathrm{L}] }[/math].

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D2.8 Friction

© Universitat Politècnica de Catalunya. Tots els drets reservats

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