Frames of reference

In the realm of physics and engineering, the concept of a "frame of reference" is foundational, providing a perspective or viewpoint from which we observe, measure, and describe the position and motion of objects. Simply put, a frame of reference is a set of criteria or stated conditions relative to which measurements or judgments can be made. It's the backdrop that determines how we perceive that govern them. This blog delves into the intricacies of frames of reference, highlighting their significance and exploring specific types such as the Earth-Centered Earth-Fixed (ECEF) frame as a prime example.

What is a Frame of Reference?

At its core, a frame of reference is essentially a coordinate system and a set of physical reference points that help us quantify the position and orientation of objects in space. It allows us to answer questions like: "Where is it?" and "How fast is it moving?" But the answers to these questions can vary dramatically depending on the chosen frame of reference. This is because motion is relative: an object might be moving in one frame of reference and stationary in another.

Inertial Frames of Reference

An inertial frame of reference is one in which an object not subjected to external forces moves in a straight line at a constant speed. According to Newton's first law of motion, these frames are either at rest or move at a constant velocity. They provide a 'non-accelerating' backdrop for observing physical phenomena, allowing Newton's laws to apply in their simplest form.

In an inertial frame of reference, Newton's first law (also known as the law of inertia) holds true. This law states that an object will remain at rest or move at a constant velocity unless acted upon by a net external force. Mathematically, this can be written as:

F_net = ma

where:

F_net is the net external force acting on the object, m is the mass of the object, a is the acceleration of the object.

For an inertial frame of reference:

a = 0 when F_net = 0

Now, let's consider the image, assuming it represents an overhead view where a car is moving with a constant velocity v to the left. From the car’s point of view (in the car’s frame of reference):

1.  If the car is moving at a constant velocity and is not accelerating (no net forces acting on it), then:
   a = 0

2.  Any object inside the car will also not accelerate relative to the car:
   a_relative = 0

3. If the car turns or brakes (accelerates in any way), the frame of reference becomes non-inertial, and objects inside the car will experience apparent forces.

In the inertial frame of reference, which could be represented by the stationary stick figure on the road, the car's velocity v is constant, and the car's acceleration :
v = v0
a = 0

where v0 is the constant velocity vector of the car.

If we were to describe the car's position over time in the stick figure's frame of reference, we could use the kinematic equation:

r(t)=r0 +vt

where:

1.  r(t) is the position vector of the car at time 

2.  r0 is the initial position vector of the car when t = 0.

3. v is the constant velocity vector of the car.

4. t is time

Non-Inertial Frames of Reference

In a non-inertial frame, Newton's second law must be modified to account for these fictitious forces. The modified equation of motion for an object of mass m is given by:
ma' = F - ma_frame

where, 

1.  m is the mass of the object

2.  a is the acceleration of the object as measured in the non-inertial frame.

3.  F is the sum of the real forces acting on the object.

4.  ma_frame represents the fictitious force term, where a_frame is the acceleration of the non-inertial frame itself.

This equation states that the apparent acceleration of an object in a non-inertial frame (a') is due to the actual forces acting on the object minus the product of the mass of the object and the acceleration of the frame.

Examples 1: 

Rotating Frame

Consider a car that is turning to the left with a constant angular velocity. In this case, the non-inertial effects become significant. The car is the non-inertial frame of reference.

Centrifugal Force

In a turning car, an object inside will experience an outward force known as the centrifugal force, which is a fictitious force. This force can be described by:

 F_centrifugal = mω×(ω×r)

ω is the angular velocity vector of the car's turn.

r is the position vector of the object from the axis of rotation.

Coriolis Force

If the object is moving inside the car, it will also experience the Coriolis force, which acts perpendicular to the object's velocity relative to the Earth and the angular velocity of the car. The Coriolis force is given by:

F_coriolis =  -2m(v_relative×ω)

Examples 2:

Earth's Frame

The ECEF frame is a geocentric coordinate system that rotates with the Earth. As such, it is a non-inertial frame of reference because it is in constant rotational motion due to the Earth's spin. The ECEF has its origin at the center of the Earth and is fixed relative to the surface of the Earth. This means that any point on the Earth's surface has constant coordinates in the ECEF frame despite Earth's rotation.

In the ECEF frame, the position of a point on the Earth's surface can be represented by a three-dimensional vector r with coordinates (x, y, z). As the Earth rotates, this point maintains its position in the ECEF frame, but in an Earth-Centered Inertial (ECI) frame (which does not rotate with the Earth), its position is constantly changing.

Let's consider an object stationary on the Earth's surface at the equator. For simplicity, we will ignore Earth's ellipticity and assume a spherical shape. The only motion the object experiences is due to Earth's rotation. The velocity v of the object due to Earth's rotation can be expressed as:

v=ω×r

where:

1. ω is the angular velocity vector of the Earth's rotation (∣ω∣=2π/T, with T being the period of rotation, approximately 24 hours).

2.  r is the position vector of the object relative to Earth's center.

At the equator, the radius of the Earth (R) is maximal, so the linear velocity is maximal due to the rotation. If we assume ω points along the Earth's axis from the South Pole to the North Pole and the object is on the equator, the velocity vector v would point due east, and its magnitude would be:

v=ωR

A 2D look to complicated 3D diagram :

The Earth rotates about its axis, which is closely aligned with the Z_ecef axis in the ECEF coordinate system. An object at rest on the Earth's surface, such as a building or a tower, has fixed ECEF coordinates because the ECEF frame rotates with the Earth. The object's position relative to the Earth's center remains constant even as the planet spins.

From the image and the given ECEF coordinates, the position vector for a point P is given by the matrix as shown in the figure. R is the Earth's radius. h is the height above the Earth's surface. ϕ is the latitude.

Velocity Due to Earth's Rotation

An object at the surface of the Earth at the equator will have the greatest linear velocity due to Earth's rotation because it is at the maximum radius from the axis of rotation. This velocity v can be calculated by:

v = ω × r

where:

1. ω is the angular velocity vector of Earth's rotation.

2. r is the position vector of the object from Earth's center.

At the equator (ϕ=0) and sea level (h=0), the velocity magnitude is maximized, and the equation simplifies to:

v=ωR

where:

1. v is the tangential velocity at the equator.

2. ω is the angular velocity of Earth (ω= 2π/T, with T being the period of one rotation, approximately 86400 seconds).