Animated Solution for Physics - Laws of Motion: A reference frame attached to the earth (1986)

Have you ever sat perfectly still in your room, reading a book, and felt completely at rest? It is a very convincing illusion. In reality, you are aboard a massive rocky spaceship hurtling through the cosmos at mind-boggling speeds. This brings us to a profound and classic question in physics: Is the Earth a valid inertial reference frame? To answer this, we must strip away our human perception and look at the cold, hard kinematics of our planet.

Inertial vs. Non-Inertial Frames

Before we judge the Earth, we need to define our terms. In Newtonian mechanics, an inertial frame of reference is the gold standard. It is a frame that is either completely stationary or moving in a perfectly straight line with a constant speed. Mathematically, this means the net acceleration of the frame is exactly zero:

In an inertial frame, Newton's laws of motion work flawlessly. If no net force acts on an object, it stays at rest or keeps moving at a constant velocity.

Conversely, a non-inertial frame is any frame of reference that is accelerating. It could be speeding up, slowing down, or changing its direction of motion. Because the frame itself is accelerating, objects inside it appear to move without any physical force acting on them. To make Newton's laws work in such a rebellious frame, physicists have to invent mathematical corrections known as pseudo forces (like the centrifugal force).

Earth's Revolution

Now, let's put the Earth on trial. Does it move in a straight line with constant speed? Absolutely not. The Earth's most prominent motion is its grand orbit around the Sun. It travels in a massive, nearly circular path, completing one revolution every year.

Even though the Earth's orbital speed is relatively constant, the direction of its velocity vector is changing every single second to keep it on that curved path. In physics, a change in direction is a change in velocity, and a change in velocity means there is an acceleration. Specifically, the Earth experiences a centripetal acceleration directed towards the Sun, given by the formula:

Where $v$ is the orbital speed and $R$ is the distance to the Sun. Because this acceleration exists, the Earth immediately fails the test for being an inertial frame.

Earth's Rotation

But the story doesn't end there. The Earth is a multi-tasker; it isn't just revolving, it is also spinning. Every 24 hours, the Earth completes one full rotation about its polar axis.

Imagine you are standing on the equator. As the Earth spins, you are being carried along a giant circular path. Just like the orbit around the Sun, this circular motion requires a centripetal acceleration, this time directed inwards towards the Earth's axis of rotation. The magnitude of this acceleration is:

Where $\omega$ is the angular velocity of the Earth and $r$ is your distance from the axis. This means that every point on the Earth's surface (except exactly at the poles) is constantly accelerating.

Why We Need Pseudo Forces

Because the Earth is accelerating due to both its revolution around the Sun and its rotation about its own axis, it is strictly a non-inertial frame.

You might wonder, "If it's non-inertial, why do we use Newton's laws in our high school physics problems without worrying about it?" The answer is that for small-scale, short-duration events—like a block sliding down a ramp or a ball being thrown—the Earth's acceleration is so incredibly tiny that its effects are negligible. We approximate the Earth as an inertial frame for convenience.

However, for large-scale phenomena, the non-inertial nature of Earth becomes glaringly obvious. The rotation of the Earth gives rise to the Coriolis force, a pseudo force that dictates the swirling direction of hurricanes and ocean currents. Without acknowledging Earth as a non-inertial frame, we couldn't explain global weather patterns!

Let's look at the choices provided in the problem: - (a) is an inertial frame by definition: This is false. We just proved it is accelerating. - (b) cannot be an inertial frame because the earth is revolving round the sun: This is a true statement. The centripetal acceleration of revolution makes it non-inertial. - (c) is an inertial frame because Newton's laws are applicable in this frame: This is false. Newton's laws are only strictly applicable if we introduce pseudo forces, proving it is non-inertial. - (d) cannot be an inertial frame because the earth is rotating about its own axis: This is also a true statement. The centripetal acceleration of rotation makes it non-inertial.

Therefore, the correct answers are (b) and (d).