Forest UCM MiNF
Mechanics in Noninertial Reference Frames
Linearly accelerating reference frames
Let [math]\displaystyle{ \mathcal S_0 }[/math] represent an inertial reference frame and \mathcal S represent an noninertial reference frame with acceleration [math]\displaystyle{ \vec A }[/math] relative to [math]\displaystyle{ \mathcal S_0 }[/math].
Ball thrown straight up
Consider the motion of a ball thrown straight up as viewed from [math]\displaystyle{ \mathcal S }[/math].
Using a Galilean transformation (not a relativistic Lorentz transformation)
At some instant in time the velocities add like
- [math]\displaystyle{ \dot {\vec {r}_0} = \dot {\vec r}+ \vec V }[/math]
where
- [math]\displaystyle{ \vec V }[/math] = velocity of moving frame [math]\displaystyle{ \mathcal S }[/math] with respect to [math]\displaystyle{ \mathcal S_0 }[/math] at some instant in time
- [math]\displaystyle{ \Rightarrow \dot {\vec r} = \dot {\vec {r}_0} - \vec V }[/math]
taking derivative with respect to time
- [math]\displaystyle{ \Rightarrow \ddot {\vec r} = \ddot {\vec {r}_0} - \dot \vec V= \ddot {\vec {r}_0} - \vec A }[/math]
- [math]\displaystyle{ \Rightarrow m\ddot {\vec r} = m\ddot {\vec {r}_0} - m \vec A= \vec F - m\vec A= \vec F - \vec {F}_{\mbox {inertial}} }[/math]
where
- [math]\displaystyle{ \vec {F}_{\mbox {inertial}} = m \vec A \equiv }[/math] inertial force
- in your noninertial frame, the ball appears to have a force causing it to accelerate in the [math]\displaystyle{ \vec A }[/math] direction.
The inertial force may also be referred to as a fictional force
an example is the "fictional" centrifugal force for rotational acceleration.
The observer in a noninertial reference frame will feel these frictional forces as if they are real but they are really a consequence of your accelerating reference frame
example
- A force pushes you back into your seat when your Jet airplane takes off
- you slam on the brakes and hit your head on the car's dashboard