These are Newtons Laws as they apply to point masses. We will use these facts to build up the theory for multiparticle objects that have size and shape. Point particles have no rotation or internal vibrations or deformations.

  1. The Law of Inertia: In the abscence of forces, a particle moves with constant velocity $\bold v$, i.e. its accelaration is zero $\bold a=0$

  2. For any particle of mass $m$, the net force $\bold F$ on the particle is equal to $\bold F=m\bold a$, more specifically $\bold{F=\dot p}=m\bold a+\dot m\bold v$ if mass changes with time

    Validity → Newtons Two Laws hold only in inertial nonaccelerating and nonrotating reference frames

  3. If object 1 exerts a force on object 2, $\bold F_{21}$, then object 2 always exerts a force $\bold F_{12}$ on object one that is equal in magnitude and equal in magnitude. $\bold F_{21}=-\bold F_{12}$

    Validity → In relativity Newton’s third law does not necessarily hold as it would require simultaneity

    Validity→ The Magnetic Force for instance does not necessarily follow Newton’s Third Law Interestingly the magnetic force is very interwined with special relativity.

    Central Forces

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    In this case above the action, reaction forces are shown acting along the line joining 1 and 2

    This is a special case. Forces with this property are called central forces

    Gravity and Electrostatic Force between two charges have this property

    However, in general the action-reaction pair do not need to act along a line joining them (acting central), they are only required to be opposite in direction

    Conservation of Momentum from Newton’s Third Law

    Note: The Conservation of Momentum Principle is still prevalent even in relativity. Thus, this principle at its core is not fundamentally caused by Newton’s Third Law, but we can demonstrate that when Newton’s Third Law applies, it implies a conservation in momentum

    Newtons Law in Other Coordinate Frames

    Polar Coordinates

    Cylindrical

    Aside on Reference Frames vs Coordinate Frames

    When changing coordinate frames the observer still remains in the same reference frame, but reformulates Newton’s Laws using direction vectors that are more apt to the problem. Changing coordinate frames could include:

    Its this last type of coordinate frame change of most interest. Changing into a non-inertial reference frame that is rotating seems to mirror the change from Cartesian to Polar.

    For the coordinate frame case Newton’s 2nd Law is modified with accelerations that account for the derivatives of the unit vector terms. In this point of view, the acclerations are modified to better work with the coordinate frame.

    $\vec a_r=\ddot r-r\dot\phi^2$ where the second added term is the centripetal acceleation

    $\vec a_\phi=r\ddot\phi+2\dot r\dot\phi$ where the second added term is the Coriolis acceleration

    For the reference frame case, these modified terms added to the typical $\ddot{q}$ (for a general position basis vector $\vec q$) are instead on the other side of Newton’s 2nd Law, $\sum \vec F$ as forces. These forces are ficticious and required for Newton’s 2nd Law to remain valid in the non-inertial reference frame

    In the non-inertial reference frame for a 2D rotating frame, observers see two ficticious forces:

    $F_{centrifugal}=mr\omega^2~~~~F_{Coriolis}=-2m\vec v\times\vec\omega$

    In general for reference frames in non-inertial reference frames it seems like we can incorporate a ficiticous force that opposes the accleration on the system that an outside observer sees which allows for Newton’s Second Law to remain valid

    Elevator accelerating down → Ficticious Force acting Up making you weigh less

    Elevator accelerating up → Ficticious Force acting Down making you weigh more

    These two cases here don’t seem apt for Coordinate Frame Changes

    These cases don’t involve changes in direction, the Cartesian Frame works fine but the elevator is translating through the coordinate frame. A coordinate change is not required. An accelerating reference frame might help view the system as stationary

    If an object is constrained or not constrained to accelerate with the reference frame the ficticious force can be moved to the other side of Newtons Second Law, $m\vec a$, this new total acceleration $\vec a+\vec a_{fic}$ is the acceleration that an outside observer sees the object to move where $\vec a$ is the acceleration of the object in the accelerating reference frame.