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.
1. Inertial Frames
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).
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.
2. 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:
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 |