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 : ${\vec {F}}_{inertial}=-m{\vec {A}}$