Classical Mechanics/Non-Inertial Reference Frames

It is very important to acknowledge how to construct equations inside of an inertial frame of reference. (As even the Earth is a non-inertial frame)

Consider an inertial reference frame S and a second reference frame S0 which is moving with respect to S with a velocity $$\vec{V}$$ and accelerating with respect to S at a rate $$\vec{A}$$.

From the inertial reference frame (S) Newton's second law will hold and any object of mass m will be observed to have a force acting on it of $$ \vec{F} = m \ddot{\vec{r}} $$ where $$ \vec{r} $$ is measured from the origin of the frame S.

From the non-inertial frame (S0) we must relate the quantities using the Galilean transformation for a moving reference frame, so that the velocity of the mass in the new reference frame is $$ \dot{\vec{r_0}} = \dot{\vec{r}} - \vec{V} $$. Using this fact we can differentiate ( $$ \ddot{\vec{r_0}} = \ddot{\vec{r}} - \vec{A} $$ ) and then substitute the force in the inertial frame ( $$ \vec{F} = m \ddot{\vec{r}} $$ ) to get an expression for the force measured by an observer in the non-inertial frame : $$ m \ddot{\vec{r_0}} = \vec{F} - m \vec{A} $$.

The conclusion that we can reach is that we may continue to use Newton's laws in the non-inertial frame, so long as we add the additional "force" due to the motion of the frame, which is often called the inertial force : $$ \vec{F}_{inertial} = - m \vec {A} $$