The Day The Earth Stood Still


What would it take to make the earth stop spinning?  This scenario is not unheard of in B-movies and bad sci-fi shows.  It isn’t uncommon to have plots involving the Earth’s core slowing down or aliens from a different galaxy stopping the Earth’s rotation.  A lot of these plots have the Earth stop spinning either instantaneously or within a very short period of time.  Intuitively, we know that spinning bodies have energy.  The Earth is a pretty massive spinning body.  How much energy would the Earth have to shed to stop rotating?  How would that energy affect us worldly inhabitants?

Here, I will discuss the physics behind rotation and rotational energy.  We shall use simple facts about the Earth’s rotation to calculate what would happen to it were it to stop spinning.

The Solution

The Earth holds as much energy in its rotation as a trillion of the most powerful hydrogen fusion bombs that man has ever produced.  This is quite an unimaginable amount of energy and would need some very creative solutions to deal with.

Rotational Mechanics

The rotational motion of bodies can be described in a manner analogous to linear or translational motion.  In linear motion, we deal with mass (m), distance (s), velocity (v), acceleration (a) and force (F).  A body’s mass determines how much it will accelerate when a force acts on it.  A body’s velocity is the distance  it travels in a unit of time.  When studying rotational motion, we deal with quantities such as moment of inertia (I), angle (θ), angular velocity (ω), angular acceleration (α) and torque (τ).  The quantities are analogues to the linear motion quantities.

\begin{array}{|l|l|} \hline\text{Linear quantity (unit)} & \text{Rotational quantity (unit)}\\\hline\text{mass} \;m\; (\text{kg}) & \text{moment of inertia} \;I\; (\text{kgm}^2)\\\text{distance} \;s\; (\text{m}) & \text{angle} \;\theta\; (\text{radians})\\\text{velocity} \;v\; (\text{ms}^{-1}) & \text{angular velocity} \;\omega\; (\text{s}^{-1})\\\text{acceleration} \;a\; (\text{ms}^{-2}) & \text{angular acceleration} \;\alpha\; (\text{s}^{-2})\\\text{force} \;F\; (\text{kgms}^{-2}) & \text{torque} \;\tau\; (\text{kgm}^2\text{s}^{-2})\\\hline\end{array}

A body’s moment of inertia about a particular axis determines how quickly it will rotationally accelerate about that axis when a torque is applied to it.  The moment of inertia of a body is analogous to its mass, but also dependent on the geometry and mass distribution of the body.  The larger the moment of inertia, the slower a body accelerates for a given torque.  Torque is the rotational analogue of a force.  You can think of it as a twisting force.  It is dependent on the force applied and how far away from the axis of rotation it is applied. It is easier to rotate a couch by pushing it near the edge that it is closer to the center.  The distance from the axis of rotation to the point of application of force is known as the moment arm. The stronger the force, the higher to torque.  The longer the moment arm, the higher the torque.

A body’s angular velocity is the angle of rotation it covers in a unit of time.  We can describe rotational motion in terms of the angle swept by the body per unit time.  The rate at which the object spins (sweeps an angle) is known as angular velocity.  The rate of change of this angular velocity is known as angular acceleration.

Kinetic Energy

We have been discussing rotational mechanics but haven’t related any of it to energy.  With linear motion, if a body is moving, it posses energy proportional to its velocity known as kinetic energy.  A body of mass m moving with velocity v has kinetic energy equal to mv2/2.  Similarly, for rotational motion, a body spinning with angular velocity ω about an axis with moment of inertia I has rotational kinetic energy equal to 2/2.  There is a law of nature known as the conservation of energy.  It states that energy can neither be created nor destroyed[1].  Instead, it can be converted from one form to another.  Appealing to this law, if the Earth were to suddenly stop spinning, all of its rotational kinetic energy would have to go somewhere.  Hence, if we calculate the rotational kinetic energy of Earth, we would know how much energy Earth would have to shed to stop rotating.

Number Soup

And now we get to actually figuring out the Earth’s kinetic energy.  Note that we would need to know the moment of inertia of the earth and it angular velocity.  Calculating the angular velocity is easiest; it is the angle swept divided by the time taken to do it.  The Earth spins once in 24 hours.  One full rotation is an angle of 2π in radians.  Hence the Earth’s angular velocity is:

\omega = \frac{2\pi}{24 \text{hours}} = \frac{2\pi}{24 \times 3600 \text{s}} = \frac{2\pi}{86400 \text{s}}

How about the moment of inertia?  This is a harder question.  One can estimate the moment of inertia of the Earth along its axis of rotation given its mass and radius and making some simplifying assumptions, but I will leave that exercise for another post (or to the intrepid reader).  Instead, we use a value for the moment of inertia from scientific literature:

I = 8\times 10^{37} \text{kg}\text{m}^2

This is a massively, stupendously, insanely large value.  Let’s calculate the total rotational kinetic energy:

E = \frac{1}{2}I\omega^2 = \frac{1}{2}(8\times 10^{37}\text{kg}\text{m}^2)\times\left(\frac{2\pi}{86400 \text{s}}\right)^2 \approx 2 \times 10^{29} J

The J above stands for Joules which is a unit of energy.  A 100-Watt bulb uses 100 joules every second.  A stove can use a couple thousand joules a second.  But none of these household items come close to consuming anywhere near the magnitude of energy that the earth commands due to its rotation.  Let’s try to visualize this quantity – 2 * 10^29, in terms of other, better known occurrences.  The energy released by detonating a ton of TNT is about 4 * 10^9 joules (or 4 gigajoules).  The energy released by the atom bomb dropped on Hiroshima is about 6 * 10^13 joules.  One of the most powerful hydrogen bombs ever tested (the Tsar Bomba) put out about 2 * 10^17 joules.  This bomb registered on seismometers all over the world – it quite literally shook the entire world.  But even this is still 10^12 times lower than the energy held by the Earth owing to its rotational kinetic energy.  The Earth holds as much energy as a trillion Tsar Bombas, solely due to its rotation.  Let’s try even harder to imagine this much energy.  The world’s population consumes about 4*10^20 joules of energy every year.  The entire Earth receives about 5*10^24 joules of energy from the Sun every year.  In other words, the amount of energy derived for the Earth’s rotation is 5000 times the total energy received by the Earth from the Sun in a year.  This is a stupendous number and deserves some respect.  If it were to be unleashed by some aliens in a B-movie and they didn’t want to instantly vaporize all known life on earth and reduce every recognizable land mark to dust, they are going to have to try very hard to handle that energy.  On the other hand, the sheer magnitude of this number should give us comfort – we really aren’t slowing the earth’s rotation by any appreciable amount by pushing against it with rockets or running around it.  Phew!

Further Reading

Moment of Inertia

Rotational Analogues to Linear Motion

Rotational Kinetic Energy

Orders of Magnitude of Energy

[1] This is technically not entirely true as phrased. It works as a reasonable approximation for bodies that are neither too small, too large, going too fast, or acting weirdly. Earth, in this case, reasonably qualifies as a body following Newtonian physics and hence, following the traditional conservation of energy theorem. I call nuclear fusion/fission as bodies acting weirdly. Don’t even think about virtual particles.

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