6.12: Relative motion and inertial reference frames
We will now discuss what happens if two persons (observers), that move relative to each other, both apply Newton’s laws. We first introduce the concept of a reference frame.
The points that are chosen to have zero velocity and acceleration are called the reference frame of an observer. Velocities and accelerations of objects can be measured with respect to the reference frame by choosing a coordinate system that is fixed with respect to the reference frame.
To illustrate the situation of two observers using different reference frames, consider two boats \(A\) and \(B\) that move with position vectors \(\overrightarrow{\boldsymbol{r}}_{A}(t)\) and \(\overrightarrow{\boldsymbol{r}}_{B}(t)\) relative to an origin \(O\) that is fixed to the quay reference system (see Figure 6.6). Observer \(O\) stands on the quay and uses an \(x, y, z\) coordinate system that is fixed in the quay reference frame. A second observer \(A\) on boat \(A\) uses coordinate system \(x^{\prime}, y^{\prime}, z^{\prime}\) with origin \(A\), that is fixed to boat \(A\) ’s reference frame. While the position of boat \(B\) in \(O\) ’s reference system is \(\overrightarrow{\boldsymbol{r}}_{B}(t)\), in \(A\) ’s reference system it is given by:
\[\overrightarrow{\boldsymbol{r}}_{B}^{\prime}(t)=\overrightarrow{\boldsymbol{r}}_{B / A}(t)=\overrightarrow{\boldsymbol{r}}_{B}(t)-\overrightarrow{\boldsymbol{r}}_{A}(t) \tag{6.24} \label{6.24}\]
By taking the time derivatives of Equation 6.24, the velocity and acceleration of boat \(B\) in \(A\) ’s reference frame can be determined:
\[\begin{align} \overrightarrow{\boldsymbol{v}}_{B}^{\prime}(t) & =\overrightarrow{\boldsymbol{v}}_{B / A}(t)=\overrightarrow{\boldsymbol{v}}_{B}(t)-\overrightarrow{\boldsymbol{v}}_{A}(t) \tag{6.25} \label{6.25}\\[4pt] \overrightarrow{\boldsymbol{a}}_{B}^{\prime}(t) & =\overrightarrow{\boldsymbol{a}}_{B / A}(t)=\overrightarrow{\boldsymbol{a}}_{B}(t)-\overrightarrow{\boldsymbol{a}}_{A}(t) \tag{6.26} \label{6.26}\end{align}\]
So observer \(O\) will conclude that the acceleration of boat \(B\) is \(\overrightarrow{\boldsymbol{a}}_{B}\), and observer \(A\) will conclude that the acceleration of boat \(B\) is \(\overrightarrow{\boldsymbol{a}}_{B}^{\prime}=\overrightarrow{\boldsymbol{a}}_{B / A}\). When both observers apply Newton’s laws to boat \(B\) observer \(O\) will find that the total force acting on boat \(B\) is \(\overrightarrow{\boldsymbol{F}}_{B}=\overrightarrow{\boldsymbol{a}}_{B} / m_{B}\) and observer \(A\) finds that the force is
\(\overrightarrow{\boldsymbol{F}}_{B}=\overrightarrow{\boldsymbol{a}}_{B}^{\prime} / m_{B}\). Of course the force acting on boat \(B\) cannot depend on the acceleration of boat \(A\). Both observers should therefore find the same value for \(\overrightarrow{\boldsymbol{F}}_{B}\), which is only true if \(\overrightarrow{\boldsymbol{a}}_{B}=\overrightarrow{\boldsymbol{a}}_{B}^{\prime}=\overrightarrow{\boldsymbol{a}}_{B / A}\). This only holds if the acceleration of boat \(A\) in \(O\) ’s reference system is zero: \(\overrightarrow{\boldsymbol{a}}_{A}=\overrightarrow{\mathbf{0}}\).
Inertial reference frames and pseudo forces
From this argumentation we conclude that Newton’s second law cannot be valid in all reference frames, but only in special reference frames that we call inertial reference frames.
An inertial reference frame is a reference frame in which Newton’s second law is valid. It is a reference frame which does not accelerate or rotate substantially (relative to distant stars).
One might ask, how can observer \(A\) check if she is in an IRF? She can do so by testing if Newton’s second law holds in her reference frame, by positioning a point mass at a fixed position in the boat without any forces acting on it. If the mass accelerates in the absence of forces, the observer knows that she is not in an IRF. From the observed acceleration, observer \(A\) might conclude, based on Newton’s second law, that a kind of force is acting on the mass. However, since there are no interactions this force is not a real force, instead it is a fictitious force that is called a pseudo force, since it is only observed because
the experiment is performed in an accelerating reference frame instead of an IRF. Examples of such pseudo forces are the effect pulling you forward when a car brakes and the centrifugal ’force’ that pulls you radially when a car takes a turn. Unless explicitly mentioned, for the rest of this textbook we only deal with dynamics in inertial reference frames (IRFs).