Newton's Laws in Non-Inertial Frames

The Foundation: Inertial vs. Non-Inertial Frames

An inertial frame of reference is one that is not accelerating. In such a frame, Newton's first law holds: an object with zero net force acting on it will have zero acceleration. For most purposes on Earth, the ground is considered an excellent approximation of an inertial frame.

A non-inertial frame of reference is one that is accelerating with respect to an inertial frame. If you are inside an accelerating car, a lift speeding up, or on a spinning merry-go-round, you are in a non-inertial frame.

Why does this matter? Because in a non-inertial frame, Newton's laws appear to be violated. If a bus you're in suddenly accelerates forward, you feel "pushed" backward, even though no physical object is pushing you. To an observer on the ground, your body is simply trying to remain at rest due to inertia while the bus moves forward. To you inside the bus, it feels like a real force.

This following video provides an excellent conceptual introduction to why Newton's laws fail in non-inertial frames and how we "fix" them.

Linearly Accelerating Frames

To make Newton's second law work in a non-inertial frame, we introduce a pseudo force. For a frame that is linearly accelerating with vector acceleration ${\vec{A}}_{\text{frame}}$ relative to an inertial frame, we apply a pseudo force on the object of mass $m$ given by:

This force is added to all the real physical forces (gravity, normal, tension, etc.). With this addition, Newton's second law can be used in its familiar form:

where ${\vec{a}}_{\text{rel}}$ is the acceleration of the object as measured within the non-inertial frame. The great advantage of this method is that we can often choose a frame where the object is at rest, making ${\vec{a}}_{\text{rel}}=0$, which simplifies the problem to a statics problem.

Example 1: The Classic Lift Problem and Effective Gravity

The concept of "apparent weight" in a lift is the quintessential application of linear pseudo force. When a lift accelerates upwards with acceleration $a$, the pseudo force $-ma$ acts downwards. An observer inside the lift feels this in addition to gravity.

The net downward force feels like $mg+ma=m(g+a)$. We call $g_{\text{eff}}=g+a$ the effective gravity.

• Ascending Lift (acceleration $a$ up): ${\vec{F}}_{\text{pseudo}}$ is down. $g_{\text{eff}}=g+a$. You feel heavier.
• Descending Lift (acceleration $a$ down): ${\vec{F}}_{\text{pseudo}}$ is up. $g_{\text{eff}}=g-a$. You feel lighter.
• Free Fall ($a=g$ down): $g_{\text{eff}}=g-g=0$. You feel weightless.

The key problem-solving trick is to replace $g$ with $g_{\text{eff}}$ and solve the problem as if it were happening on the ground.

Example 2: Block on an Accelerating Wedge

A very common JEE Advanced problem archetype involves finding the acceleration of a wedge required to keep a block stationary on its surface.

To solve this, we switch to the frame of the wedge.

1. The wedge accelerates with $\vec{A}$ to the right. It is a non-inertial frame.
2. The block has mass $m$. We apply a pseudo force ${\vec{F}}_{\text{pseudo}}=-m\vec{A}$ to the left on the block.
3. We add the real forces: gravity ($mg$) downwards and the normal force ($N$) perpendicular to the incline.
4. Since the block is stationary relative to the wedge, its relative acceleration is ${\vec{a}}_{\text{rel}}=0$.
5. We then solve the static equilibrium equation: $N+mg+{\vec{F}}_{\text{pseudo}}=0$. This is done by balancing force components parallel and perpendicular to the incline.

The video below demonstrates this exact technique.

Rotating Reference Frames

The situation becomes more intricate when the non-inertial frame is rotating. In this case, two distinct pseudo forces emerge. This is a topic that requires a solid grasp of vector cross products.

For a frame rotating with a constant angular velocity $\vec{\omega }$ relative to an inertial frame, an object with mass $m$, position $\overrightarrow{r^{\prime }}$ and velocity $\overrightarrow{v^{\prime }}$ (both relative to the rotating frame) experiences two pseudo forces:

1. Centrifugal Force: ${\vec{F}}_{\text{cf}}=-m\vec{\omega }\times (\vec{\omega }\times \overrightarrow{r^{\prime }})$

   • This force depends on the object's position.
   • It always points radially outward from the axis of rotation.
   • Its magnitude is $m{\omega }^2{r}_{\perp }$, where $r_{\perp }$ is the perpendicular distance from the axis of rotation.

2. Coriolis Force: ${\vec{F}}_{\text{cor}}=-2m(\vec{\omega }\times \overrightarrow{v^{\prime }})$

   • This force depends on the object's velocity within the rotating frame.
   • It only exists if the object is moving relative to the rotating frame ($\overrightarrow{v^{\prime }}\ne 0$).
   • Its direction is perpendicular to both $\vec{\omega }$ and $\overrightarrow{v^{\prime }}$.

The modified Newton's Second Law becomes:

For JEE Advanced, a full derivation is essential for deep understanding. The following video provides a rigorous derivation of all pseudo force components, including the Euler force which arises from angular acceleration ($\vec{\alpha }$).

To solidify this theory, let's examine the foundational text from MIT OpenCourseWare.

Illustrating Centrifugal and Coriolis Forces

The same video from Physics Sir JEE provides brilliant illustrations to build intuition for these abstract forces.

A Comprehensive Example

Let's conclude with an advanced problem that synthesizes all four components of pseudo force: linear, centrifugal, Coriolis, and Euler. The scenario involves a person on a non-uniformly rotating turntable observing a stationary person on the ground. The goal is to use the full pseudo-force equation to prove, from the observer's perspective, that the stationary person appears to be moving in a non-uniform circle.

Conclusion

Today we've developed a formidable tool for solving mechanics problems. By stepping into an accelerating frame and applying the appropriate pseudo forces, we can often transform a difficult dynamics problem into a much simpler one, sometimes even a statics problem.

Key Takeaways:

• Newton's laws are only valid in inertial (non-accelerating) frames.
• To use Newton's laws in non-inertial frames, we introduce fictitious pseudo forces.
• For linearly accelerating frames (acceleration $\vec{A}$), the pseudo force is ${\vec{F}}_{\text{pseudo}}=-m\vec{A}$.
• For rotating frames (angular velocity $\vec{\omega }$), there are two main pseudo forces:
  • Centrifugal Force: ${\vec{F}}_{\text{cf}}=-m\vec{\omega }\times (\vec{\omega }\times \overrightarrow{r^{\prime }})$, which is position-dependent and points radially outward.
  • Coriolis Force: ${\vec{F}}_{\text{cor}}=-2m(\vec{\omega }\times \overrightarrow{v^{\prime }})$, which is velocity-dependent and acts only on moving objects.
• The general strategy is always: identify the non-inertial frame, apply all real and pseudo forces, and then solve $\sum {\vec{F}}_{\text{total}}=m{\vec{a}}_{\text{rel}}$.

In our next lesson, we will tackle non-uniform circular motion, specifically in vertical planes. This will require us to combine Newton's second law with the work-energy theorem, building directly on the force analysis skills you've honed today.