Frame of Reference (FOR) : Inertial and Non-Inertial (NFOR)
1. Frame of Reference (FOR)
A frame of reference is a coordinate system along with a clock, relative to which the position, velocity, and acceleration of a particle are measured.
Position vector of a particle:
\[ \vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k} \]
Without a frame of reference, the concepts of rest and motion have no meaning. Hence, motion is always relative.
2. Relative Motion (Important)
Relative position:
\[ \vec{r}_{AB} = \vec{r}_A - \vec{r}_B \]
Relative velocity:
\[ \vec{v}_{AB} = \vec{v}_A - \vec{v}_B \]
Relative acceleration:
\[ \vec{a}_{AB} = \vec{a}_A - \vec{a}_B \]
Velocity and position depend on the frame of reference, while acceleration is invariant between inertial frames.
3. Inertial Frame of Reference (IFOR)
An inertial frame of reference is one in which a body remains at rest or moves with uniform velocity in a straight line unless acted upon by an external force.
Condition for inertial frame:
\[ \sum \vec{F} = 0 \Rightarrow \vec{a} = 0 \]
- Newton’s laws are valid directly
- No pseudo force is required
- Frame is non-accelerating
Examples: Deep space frame, train moving with constant velocity
4. Non-Inertial Frame of Reference (NFOR)
A non-inertial frame of reference is a frame which is accelerating or rotating with respect to an inertial frame.
- Newton’s laws do not hold in original form
- Pseudo (fictitious) forces must be introduced
Examples: Accelerating lift, rotating Earth, turning car
5. Pseudo Force
Pseudo force is an apparent force introduced in a non-inertial frame to apply Newton’s laws.
If the frame has acceleration \( \vec{a}_0 \), pseudo force is:
\[ \vec{F}_{pseudo} = -m\vec{a}_0 \]
- Acts opposite to frame acceleration
- Proportional to mass
- No physical interaction
- Exists only in non-inertial frames
6. Earth as a Frame of Reference
Earth rotates about its axis and revolves around the Sun. Hence, Earth is strictly a non-inertial frame.
For short-duration and small-scale experiments, Earth can be treated as an approximately inertial frame.
7. Experimental Evidence of Non-Inertial Nature of Earth
(a) Foucault Pendulum
Angular velocity of rotation of the oscillation plane at latitude \( \lambda \):
\[ \Omega = \omega \sin \lambda \]
(b) Coriolis Force
\[ \vec{F}_{cor} = 2m(\vec{v} \times \vec{\omega}) \]
Deflection is to the right in Northern Hemisphere and to the left in Southern Hemisphere.
(c) Variation of g with Latitude
Effective gravity:
\[ g' = g - \omega^2 R \cos^2 \lambda \]
8. Centrifugal Force (Explicit)
Centrifugal force acting on a particle in a rotating frame:
\[ \vec{F}_{cf} = m\omega^2 r \]
It acts radially outward and exists only in a rotating (non-inertial) frame.
9. Acceleration Transformation (Rotating Frame)
Relation between acceleration in inertial and rotating frame:
$\vec{a}_{i} = \vec{a}_{r} + 2(\vec{\omega} \times \vec{v}) + \vec{\omega} \times (\vec{\omega} \times \vec{r})$
Second term → Coriolis acceleration
Third term → Centrifugal acceleration
10. Galilean Transformation (Classical Mechanics)
Between two inertial frames moving with relative velocity \( v \):
\[ x' = x - vt,\quad y' = y,\quad z' = z \]
\[ t' = t \]
Newton’s laws are invariant under Galilean transformation.
11. Comparison: IFOR vs NFOR
| Inertial Frame | Non-Inertial Frame |
|---|---|
| Newton’s laws valid directly | Pseudo forces required |
| No frame acceleration | Frame accelerating/rotating |
| No fictitious forces | Coriolis & centrifugal forces present |
12. One-Line Exam Facts (MCQs)
- Motion is always relative
- Acceleration is invariant in inertial frames
- Pseudo forces exist only in non-inertial frames
- Coriolis force depends on velocity
- Earth is non-inertial but approximately inertial
- The non inertial character of the earth is evident from the fact that a falling object does not fall straight down but slightly deflects to the east.
Final Conclusion
Frame of reference is fundamental to the description of motion. Newton’s laws are valid only in inertial frames. Earth is a rotating, hence non-inertial frame, but it can be treated as approximately inertial for most laboratory experiments.