PC1672 Advanced dynamics

2.1 Accelerating frames

In some situations it is convenient to look at motion from the viewpoint of a noninertial frame of reference. Relativity principles (Galileo's or Einstein's) do not apply to these frames and so the laws of physics look different in them.

For example if we measure an object's position with respect to an origin that moves with a varying velocity, then the acceleration we observe ( ${\bf a}'$ ) will differ from that seem by an inertial observer ( ${\bf a}$ )

$\begin{displaymath}{\bf a'}={\bf a}-{\bf A}\end{displaymath}$

where ${\bf A}$ is the acceleration of our frame with respect to the inertial one. Newton's second law

$\begin{displaymath}m{\bf a}={\bf F}\end{displaymath}$

becomes in our noninertial frame

$\begin{displaymath}m{\bf a}'={\bf F}-m{\bf A}\end{displaymath}$

From our (accelerating) point of view, it is a though there is an extra force $-m{\bf A}$ acting on the object. This sort of term in an equation of motion is called an inertial force (or sometimes a ``fictitious force''). For the inertial observer, there is no such force, it is just a consequence of the object's inertia and our use of an accelerating frame of reference. However it certainly feels like a force to us in our accelerating frame!

Inertial forces are always proportional to an object's mass (its ``inertia''). They can be treated like extra contributions to the gravitational force on the object.

Equivalence: Einstein turned this similarity into something much deeper. The principle of equivalence says that, at least locally, there is no way to distinguish between a uniform gravitational force and a frame with a constant acceleration. In General Relativity, a gravitational force can be thought of as a consequence of using a noninertial frame in a warped space-time.

Textbook references

K&K 8.3 (8.4)
F&C 5.1
(B&O 7.1)
(M 11.1)
(M&T 10.1)

Mike Birse
17th May 2000