Non-Inertial Frames and Pseudo-Forces

1. The Simplest Case: Linearly Accelerating Frames

Before tackling rotation, let's consider a reference frame $O^{\prime }$ that is accelerating with a constant linear acceleration $\vec{A}$ with respect to an inertial frame $O$.

Let $\vec{r}$ be the position of a particle in the inertial frame $O$ and $\overrightarrow{r^{\prime }}$ be its position in the non-inertial frame $O^{\prime }$. The position of the origin of $O^{\prime }$ with respect to $O$ is $\vec{R}$.

The relationship between the positions is:
$\vec{r}(t)=\overrightarrow{r^{\prime }}(t)+\vec{R}(t)$

Differentiating twice with respect to time gives the relationship between the accelerations:
$\vec{a}=\overrightarrow{a^{\prime }}+\vec{A}$
where $\vec{a}$ is the acceleration in the inertial frame, $\overrightarrow{a^{\prime }}$ is the acceleration in the non-inertial frame, and $\vec{A}$ is the acceleration of the frame itself.

Newton's Second Law, ${\vec{F}}_{real}=m\vec{a}$, is only valid in the inertial frame. Substituting our acceleration relation into it, we get:
${\vec{F}}_{real}=m(\overrightarrow{a^{\prime }}+\vec{A})$
Rearranging to solve for the motion within the non-inertial frame, we get:
$m\overrightarrow{a^{\prime }}={\vec{F}}_{real}-m\vec{A}$

This is a profound result. It looks like Newton's Second Law, but with an extra term. To an observer in the accelerating frame $O^{\prime }$, it appears as though there is an additional force acting on the particle, ${\vec{F}}_{fictitious}=-m\vec{A}$. This is our first pseudo-force. It's not a real interaction, but a consequence of the frame's acceleration.

You can explore this concept in the following reading.

2. The Kinematics of Rotation: A General Time Derivative

Rotation is more complex because the direction of the coordinate axes themselves are changing. Our goal is to find a general formula that relates the time derivative of any vector as seen from an inertial frame to its time derivative as seen from a rotating frame. This is the mathematical heart of the lesson.

The following video provides a rigorous, step-by-step derivation of this crucial relationship and then uses it to derive the full acceleration equation. Because of your interest in formal derivations, I recommend you watch it carefully, perhaps with pen and paper to follow the steps.

Step 1: The Fundamental Operator

First, let's derive the relationship between the time derivative of a vector in the inertial ("fixed") frame and the rotating frame.

The key result from this segment is what I'll call the master equation for derivatives. For any vector $\vec{Q}$, its rate of change in the inertial (fixed) frame is related to its rate of change in the rotating frame by:
${\left(\frac{d\vec{Q}}{dt}\right)}_{fixed}={\left(\frac{d\vec{Q}}{dt}\right)}_{rot}+\vec{\Omega }\times \vec{Q}$
Here, $\vec{\Omega }$ is the angular velocity vector of the rotating frame. This equation is incredibly powerful. We will now apply it twice, starting with the position vector.

Step 2: From Position to Velocity to Acceleration

Now, we apply the master equation to the position vector $\vec{r}$ to find the velocity transformation, and then apply it again to the velocity vector to find the acceleration transformation. This second step is where the Coriolis and centrifugal terms naturally appear.

After that detailed derivation, we arrive at the complete transformation for acceleration between a translating, rotating frame and an inertial frame. The result, presented clearly by Professor Lewin in a separate lecture, is:
${\vec{a}}_{in}={\vec{A}}_{frame}+{\vec{a}}_{rot}+\vec{\alpha }\times \vec{r}+2(\vec{\Omega }\times {\vec{v}}_{rot})+\vec{\Omega }\times (\vec{\Omega }\times \vec{r})$

Let's break this down term by term:

• ${\vec{a}}_{in}$: Acceleration of the object in the inertial frame.
• ${\vec{A}}_{frame}$: Acceleration of the origin of the rotating frame (the translational part).
• ${\vec{a}}_{rot}$: Acceleration of the object as seen by an observer in the rotating frame.
• $\vec{\alpha }\times \vec{r}$: Euler acceleration, present if the frame's rotation speed is changing ($\vec{\alpha }=d\vec{\Omega }/dt$).
• $2(\vec{\Omega }\times {\vec{v}}_{rot})$: Coriolis acceleration, which depends on the object's velocity in the rotating frame.
• $\vec{\Omega }\times (\vec{\Omega }\times \vec{r})$: Centripetal acceleration, which depends on the object's position in the rotating frame. It always points toward the axis of rotation.

3. Dynamics: The Emergence of Pseudo-Forces

Now we are ready to find the equations of motion. We start with Newton's law in the inertial frame, ${\vec{F}}_{real}=m{\vec{a}}_{in}$, and substitute our monster expression for ${\vec{a}}_{in}$. Then we rearrange it to see what an observer in the rotating frame would conclude.

The modified Newton's Second Law for an observer in a non-inertial frame is:
$m{\vec{a}}_{rot}={\vec{F}}_{real}+{\vec{F}}_{fictitious}$
where the total fictitious force is the sum of four terms:

1. Translational Force: ${\vec{F}}_{trans}=-m{\vec{A}}_{frame}$
2. Euler Force: ${\vec{F}}_{Euler}=-m(\vec{\alpha }\times \vec{r})$
3. Coriolis Force: ${\vec{F}}_{Cor}=-2m(\vec{\Omega }\times {\vec{v}}_{rot})$
4. Centrifugal Force: ${\vec{F}}_{cf}=-m(\vec{\Omega }\times (\vec{\Omega }\times \vec{r}))$

The centrifugal force is perhaps the most familiar. The vector identity $\vec{A}\times (\vec{B}\times \vec{C})=\vec{B}(\vec{A}\cdot \vec{C})-\vec{C}(\vec{A}\cdot \vec{B})$ can be used to show that for an object rotating in a circle, this force simplifies to $m{\Omega }^{2}r$ pointing radially outward, as you would expect.

4. Application: An Ant on a Turntable

Let's solidify this theory with a challenging example that incorporates all the forces we've derived.

This example is excellent because it requires you to consider:

• An accelerating origin? No, so ${\vec{F}}_{trans}=0$.
• A changing angular velocity? Yes, so there is an Euler force.
• A position in the rotating frame? Yes, so there is a centrifugal force.
• A velocity in the rotating frame? Yes, so there is a Coriolis force.

The real force of friction must counteract all these pseudo-forces to provide the ant's observed acceleration in the rotating frame.

Conclusion

Today, we've accomplished a significant goal in classical mechanics. We have rigorously derived the full equations of motion for a particle in a general non-inertial reference frame. This powerful result allows us to solve problems from a more convenient, albeit non-inertial, viewpoint.

Key Takeaways:

• Newton's Second Law can be applied in non-inertial frames if we introduce pseudo-forces.
• These forces are not real interactions but mathematical terms that account for the frame's acceleration.
• For a frame with translational acceleration $\vec{A}$ and angular velocity $\vec{\Omega }$, the key pseudo-forces are:
  • Translational: $-m\vec{A}$
  • Centrifugal: $-m(\vec{\Omega }\times (\vec{\Omega }\times \vec{r}))$
  • Coriolis: $-2m(\vec{\Omega }\times {\vec{v}}_{rot})$
  • Euler: $-m(\vec{\alpha }\times \vec{r})$
• The Coriolis force is particularly interesting as it acts only on objects moving within the rotating frame and is responsible for many large-scale phenomena on Earth.

Preview of the Next Lesson:
We have now established a comprehensive framework for the dynamics of a single particle. In the next lesson, we will shift our focus from single particles to systems of many particles. We will introduce the concept of the center of mass and derive a powerful theorem for how the system as a whole moves, which will allow us to analyze the motion of complex, extended objects.