Mechanical point

• The center of mass of an isolated system is at rest or in a uniform motion (according to Galileo).

  This principle holds true in an inertial frame of reference.

• According to Newton's third law, when one point exerts a force on a second point, the second point simultaneously exerts a force equal in magnitude and opposite in direction on the first point.

• An inertial frame of reference is one in which the principle of inertia holds true.

• Kepler's frame of reference : (heliocentric frame of reference)

  Its origin is the center of mass of the Sun.

  Its three axises are orientated towards three « fixed » stars.

• Copernicus's frame of reference :

  Centered on the center of mass of the solar system.

• Geocentric frame of reference :

  Its origin is the center of mass of the Earth.

  Its three axises are orientated towards three « fixed » stars.

  The geocentric frame is in circular translation relatively to Kepler's frame.

According to the principle of inertia : at rest or in rectilinear uniform translation.

No : for instance, weight has always the same value, it does not depend on the chosen frame.

It is the angular momentum :

\(\vec L_O=\vec {OM} \wedge m\vec v\)

which gives the moment of the momentum \(\vec p\) with respect to O.

It is the same equation between a force and its torque with respect to O :

\(\vec M_O=\vec {OM} \wedge \vec f\)

The package keeps the same uniform rectilinear translation motion. It stays vertically to the plane.

• \(\vec F = \alpha \;\vec r\;\) : the existence of a force would then depend on the chosen frame of reference !

• \(\vec F=\;\beta \;\frac{{d\vec r}}{{dt}}\;\) : a material point can have constant velocity without having a force exerted upon (first law of motion).

• \(\vec F=\;\gamma \frac{{{d^3}\vec r}}{{d{t^3}}}\; \) : if \(\vec F=\vec 0\), acceleration is constant and the motion is accelerated : by experience, this is false.

Let \(z\) be the horizontal axis of a punctual mass \(m\). If (Oz) is orientated upwards, \(E_p=+mgz\), if it is orientated downwards, \(E_p=-mgz\).

• This potential energy is \(E_p=-\frac{GmM_T}{r}\), \(r\) is the distance between the particle and the center of the Earth.

• The electrostatic potentiel energy of two punctual charged particles is : \(E_p=\frac{1}{4\pi \varepsilon_0}\frac{q_1q_2}{r}\)

This energy is \(E_p=\frac {1}{2}k(\ell-\ell_0)^2\), where \(\ell\) is the length of the spring and \(\ell_0\) is the length when the spring is at rest.

A conservative force \(\vec f\) derived from a potential energy \(E_p\) :

\(\vec f=-\overrightarrow {grad}\; E_p\)

The elementary work is :

\(\delta W_{\vec f}=\vec f . d\vec r=-\;\overrightarrow {grad}\; E_p . d\vec r=-dE_p\)

It follows that the work of a conservative force is opposite to the variation of potential energy.

From the work energy theorem : (in an intertial frame of reference)

\(dE_c=\delta W_{\vec f}=-dE_p\)

If \(E_m=E_c+E_p\) is the mechanical energy of the particle :

\(dE_m=0\;\;\;\;\;so\;\;\;\;E_m=cste\)

It follow that mechanical energy is constant.

This energy is :

\(E_{p,cent}=-\frac{1}{2}m\omega^2r^2\)

If \(r\) is the distance between (Oz) and the point.

• Inertial force : \(\vec f_{i,e} = m\omega^2 \vec {HM}\), if H is the orthogonal projection of M on (Oz).

• Coriolis force : \(\vec f_{ic}=- 2 m \vec \omega \wedge \vec v\;'\), if \(\vec v\;'\) is the relative velocity of M.

Lorentz force is : \(\vec f=q(\vec E + \vec v \wedge \vec B)\)

• Moment of a force : \(\vec M_{\vec F /_0}=\vec {OA} \wedge \vec F\)

• Angular momentum : \(\vec L_O=\vec {OM} \wedge m\vec v\)

• Static friction :

  \(\vec v_g=\vec 0\) and \(T<f_sN\)

• Kinetic friction :

  \(\vec v_g .\vec T<0\) and \(T=f_dN\)

• The gravitational pull is :

  \(\vec f=-GmM_T\frac{\vec r}{r^3}\)

• Weight is the resulting force of the gravitational pull and of the centrifugal force (the Earth spins with angular velocity \(\vec \Omega\)around the North-South axis) :

  \(\vec P=m\vec g=-GmM_T\frac{\vec r}{r^3}+m\Omega^2\vec {HM}\)

  If H is the projection of M on the North-South axis.

Kepler's third law :

\(\frac {T^2}{R^3}=\frac {4\pi^2}{GM_T}\)

• In an inertial reference frame, the work energy theorem for power in a closed system is :

  \(\frac{dE_c}{dt}=P_{\vec f_{ext}}+P_{\vec f_{int}}\)

• For a closed solid system, the power of exterior forces is equal to zero :

  \(\frac{dE_c}{dt}=P_{\vec f_{ext}}\)

The kinetic energy of the vehicle is converted in the work of friction force :

\(\frac{1}{2}mv^2=kmg\ell\)

So :

\(\ell = \frac{v^2}{2kg}=50\;m\)

It is inertial if the velocity vector of the center of mass \(\vec v(G)\) is constant, it means if the system is isolated.