Pseudo Forces in Non-Inertial Frames

Introduction

When you sit in a car that suddenly takes a sharp turn, you might feel as though a mysterious force is pushing you outward, pressing you against the door. Similarly, when you ride a merry-go-round, you sense a strange pull trying to fling you away from the center.

Where does this “force” come from? Why does it seem to act only in certain situations? And why do physicists sometimes call it a pseudo force or fictitious force?

The answer lies in understanding frames of reference—inertial and non-inertial—and how our perception of motion changes depending on where we observe from. The “force” you feel on a rotating ride is not a real interaction like gravity or friction but rather a pseudo force known as the centrifugal force.

In this article, we will dive deeply into pseudo forces, with special emphasis on centrifugal force. We’ll derive its equations, explore its applications, and explain why it matters in physics, engineering, and everyday life.

Frames of Reference

To understand pseudo forces, we must first discuss frames of reference.

What is a Frame of Reference?

A frame of reference is simply the perspective or coordinate system from which an observer measures motion.

Inertial Frame of Reference:
- A frame in which Newton’s laws hold true without modification.
- It moves with constant velocity or is at rest.
- Example: A stationary observer watching a ball fall freely under gravity.
Non-Inertial Frame of Reference:
- A frame that is accelerating or rotating.
- In such frames, Newton’s laws do not hold unless we introduce extra forces—pseudo forces.
- Example: An observer inside a car that suddenly brakes or turns.

Why Do Pseudo Forces Appear?

Newton’s second law states: F=maF = maF=ma

This is valid only in inertial frames.

But if you’re inside an accelerating frame, objects appear to accelerate without any visible force acting on them. To account for this, we invent pseudo forces so that Newton’s laws remain usable.

Thus, pseudo forces are not caused by real physical interactions but by the acceleration of the observer’s reference frame.

Common Types of Pseudo Forces

Centrifugal Force – experienced in rotating frames, directed outward from the center of rotation.
Coriolis Force – affects moving objects in a rotating frame, responsible for large-scale weather patterns on Earth.
Translational Pseudo Force – felt in linearly accelerating frames (e.g., a car suddenly moving forward or stopping).

Our focus here is centrifugal force.

Centrifugal Force – The Outward “Pull”

Everyday Observation

On a merry-go-round, you feel pushed outward.
In a turning car, you feel pushed against the door.
In a washing machine’s spin cycle, clothes are pressed against the drum wall.

All these are examples of centrifugal force, but remember—it’s a pseudo force.

Definition

Centrifugal force is the pseudo force experienced in a rotating frame of reference, directed radially outward from the axis of rotation.

It balances the centripetal force in the rotating frame, making Newton’s laws appear valid.

Centripetal vs. Centrifugal Force

Centripetal Force:
- Real force.
- Acts toward the center of the circular path.
- Required to keep an object moving in a circle.
Centrifugal Force:
- Pseudo force.
- Experienced only by observers in the rotating frame.
- Acts outward, opposite to centripetal force.

Thus, centrifugal force is essentially a “frame-dependent illusion.”

Mathematical Derivation of Centrifugal Force

Consider a particle moving in a circular path of radius rrr with angular velocity ω\omegaω.

From an inertial frame:
- The particle experiences centripetal acceleration: ac=ω2ra_c = \omega^2 rac=ω2r
- Required force: F_c = m \omega^2 r \] (toward the center).
From a rotating (non-inertial) frame:
- The particle appears at rest.
- To explain why it does not move, we introduce a pseudo force outward: Fcf=−mω2rF_{cf} = -m \omega^2 rFcf=−mω2r

This outward pseudo force is the centrifugal force.

Real-Life Examples of Centrifugal Force

Turning Car:
Passengers feel pushed outward because the car provides centripetal force to change direction, while their bodies resist change in motion.
Merry-Go-Round or Rotating Ride:
Riders feel pressed against the wall, as the wall provides centripetal force, while centrifugal force is perceived outward.
Earth’s Rotation:
- Due to Earth’s spin, we feel a tiny outward centrifugal force.
- This reduces effective gravity slightly at the equator.
- Actual gravity at the poles is greater than at the equator because of this.
Banked Roads and Circular Tracks:
Engineers design curves by considering centrifugal effects to ensure vehicles don’t skid.
Centrifuges in Laboratories:
Machines use high-speed rotation to separate substances of different densities (e.g., blood plasma separation).

Centrifugal Force on Earth

The Earth rotates once every 24 hours. This causes a centrifugal effect that:

Reduces effective weight at the equator.
Shapes Earth into an oblate spheroid (flattened at poles, bulging at equator).
Slightly influences ocean tides along with the Moon’s gravity.

Mathematically, effective gravity at latitude θ\thetaθ: geff=g−ω2rcos⁡2θg_{eff} = g – \omega^2 r \cos^2\thetageff=g−ω2rcos2θ

Coriolis Force vs. Centrifugal Force

Though both are pseudo forces:

Centrifugal Force: affects stationary objects in rotating frames.
Coriolis Force: affects moving objects in rotating frames, causing deflection (e.g., trade winds, cyclones).

Both are essential for geophysics and meteorology.

Misconceptions

“Centrifugal force is real.”
- Not true; it only exists in the rotating frame.
“Passengers are pushed outward in turns.”
- Actually, their inertia keeps them moving straight while the car turns inward beneath them.
“Gravity is weaker only because of Earth’s shape.”
- The centrifugal effect of rotation also contributes.

Historical Context

The idea of centrifugal force dates back to early studies of circular motion by Huygens (17th century).
Newton distinguished between real forces (centripetal) and apparent ones (centrifugal).
Later, d’Alembert’s principle formalized the role of pseudo forces in dynamics.

Applications in Engineering and Science

Amusement Rides – Designed to maximize thrill while keeping centripetal and centrifugal effects safe.
Banked Curves in Highways – Designed to counteract outward centrifugal force.
Aerospace Engineering – Simulated artificial gravity using rotating space stations.
Centrifuges – Widely used in labs, industries, and medicine.

Practice Problems

A car turns in a circle of radius 50 m at speed 20 m/s. Find the centrifugal force on a passenger of mass 70 kg.
Earth’s angular speed is 7.27×10−57.27 \times 10^{-5}7.27×10−5 rad/s. Calculate the reduction in apparent gravity at the equator due to centrifugal force.
A merry-go-round rotates at 0.5 rad/s. What centrifugal acceleration is felt at 3 m from the center?

Conceptual Questions

Why do we call centrifugal force a pseudo force?
How does centrifugal force affect a person at the poles compared to the equator?
What role does centrifugal force play in artificial gravity?