Physical Laws and Frames of Reference
Inertial & Non-Inertial Frames — B.Sc. Physics Notes
1. Frame of Reference
A frame of reference is a coordinate system (origin + three mutually perpendicular axes) together with a clock used to specify the position and time of events.
- Position of a particle is given by the position vector \(\mathbf{r}(t)\) relative to the chosen origin.
- Velocity: \(\mathbf{v} = \dfrac{d\mathbf{r}}{dt}\)
- Acceleration: \(\mathbf{a} = \dfrac{d\mathbf{v}}{dt} = \dfrac{d^{2}\mathbf{r}}{dt^{2}}\)
All mechanical quantities (displacement, velocity, acceleration, force) are measured with respect to a chosen frame. There is no absolute frame of rest in classical mechanics.
2. Inertial Frames of Reference
Definition
An inertial frame is a frame in which a free particle (net external force = 0) moves with constant velocity (Newton’s first law holds). Equivalently, it is a frame that is either at rest or moving with uniform velocity relative to a fixed star (or any other inertial frame).
Properties
- Newton’s laws of motion are valid in their standard form:
\(\mathbf{F} = m\mathbf{a}\) - Acceleration of a free particle is zero.
- If \(S\) is inertial and \(S'\) moves with constant velocity relative to \(S\), then \(S'\) is also inertial.
- All inertial frames are equivalent for the formulation of the laws of classical mechanics (Galilean principle of relativity).
Galilean Transformation
Suppose frame \(S'\) moves with constant velocity \(v\) along the positive \(x\)-axis of inertial frame \(S\), and the origins coincide at \(t = 0\). Then
Differentiating twice with respect to time shows that acceleration is the same in both frames:
Hence \(\mathbf{F} = m\mathbf{a}\) has identical form in all inertial frames. Force and mass are invariant under Galilean transformations.
Examples
- A frame fixed to the distant stars (approximately).
- A laboratory frame on Earth for short-duration experiments (neglecting rotation and orbital motion).
- A spaceship moving with constant velocity in free space.
3. Non-Inertial Frames of Reference
Definition
A non-inertial frame is one that has a linear acceleration or is rotating (or both) with respect to an inertial frame. In such a frame a free particle appears to accelerate.
Linearly Accelerating Frame
Let \(S\) be inertial and \(S'\) have acceleration \(\mathbf{A}\) relative to \(S\). The acceleration of a particle relative to \(S'\) is
Newton’s second law in \(S\) is \(\mathbf{F} = m\mathbf{a}\). Substituting gives
The term \(-m\mathbf{A}\) is called a fictitious (or pseudo) force. It is introduced so that the form \(\mathbf{F}_{\text{eff}} = m\mathbf{a}'\) can still be used in the non-inertial frame.
Rotating Frames
When the frame rotates with angular velocity \(\boldsymbol{\omega}\) relative to an inertial frame, the acceleration transformation is:
(The last term vanishes if \(\boldsymbol{\omega}\) is constant.)
Rearrangement yields the effective force in the rotating frame:
Fictitious Forces
- Centrifugal force
\(\mathbf{F}_{\text{cent}} = -m\,\boldsymbol{\omega}\times(\boldsymbol{\omega}\times\mathbf{r})\)Directed outward from the axis of rotation. Magnitude \(m\omega^{2}\rho\) where \(\rho\) is the perpendicular distance from the axis. - Coriolis force
\(\mathbf{F}_{\text{Cor}} = -2m\,\boldsymbol{\omega}\times\mathbf{v}_{\text{rel}}\)Perpendicular to both \(\boldsymbol{\omega}\) and the relative velocity. It vanishes if the particle is at rest in the rotating frame. - Euler force (if \(\boldsymbol{\omega}\) is time-dependent)
\(\mathbf{F}_{\text{Euler}} = -m\,\dot{\boldsymbol{\omega}}\times\mathbf{r}\)
4. Important Consequences & Applications
- Earth as a non-inertial frame
Because of Earth’s rotation (\(\omega \approx 7.3 \times 10^{-5}\) rad s\(^{-1}\)), two effects appear:- Slight reduction of effective gravity (centrifugal contribution).
- Coriolis deflection of moving objects (projectiles, air masses, ocean currents).
- Effective gravity
On the rotating Earth the measured weight is\(m\mathbf{g}_{\text{eff}} = m\mathbf{g} - m\boldsymbol{\omega}\times(\boldsymbol{\omega}\times\mathbf{R})\)where \(\mathbf{R}\) is the position vector from the Earth’s centre. \(g_{\text{eff}}\) is maximum at the poles and minimum at the equator. - Invariance of physical laws
The fundamental laws of classical mechanics are form-invariant only in inertial frames. In non-inertial frames they retain the same form only after the inclusion of the appropriate fictitious forces.
5. Summary Table
| Aspect | Inertial Frame | Non-Inertial Frame |
|---|---|---|
| Free-particle motion | Uniform rectilinear | Accelerated |
| Validity of \(\mathbf{F}=m\mathbf{a}\) | Yes (standard form) | Only after adding fictitious forces |
| Relative motion of frame | Uniform velocity (or rest) | Accelerated or rotating |
| Transformation | Galilean | More complicated (involves \(\mathbf{A}\) or \(\boldsymbol{\omega}\)) |
| Extra forces | None | Centrifugal, Coriolis, Euler |
| Typical examples | Lab (approx.), deep space | Accelerating lift, rotating Earth, turntable |