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 $\overrightarrow{V}$ and accelerating with respect to S at a rate $\overrightarrow{A}$.

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 $\overrightarrow{F}=m\ddot{\overrightarrow{r}}$ where $\overrightarrow{r}$ 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 $\dot{\overrightarrow{r_0}}=\dot{\overrightarrow{r}}-\overrightarrow{V}$. Using this fact we can differentiate ( $\ddot{\overrightarrow{r_0}}=\ddot{\overrightarrow{r}}-\overrightarrow{A}$ ) and then substitute the force in the inertial frame ( $\overrightarrow{F}=m\ddot{\overrightarrow{r}}$ ) to get an expression for the force measured by an observer in the non-inertial frame : $m\ddot{\overrightarrow{r_0}}=\overrightarrow{F}-m\overrightarrow{A}$.

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 : ${\overrightarrow{F}}_{inertial}=-m\overrightarrow{A}$ Template:BookCat