How Much Torque Do You Really Need to Shift the Earth?
I’ve been doing physics homework for two weeks – and I’m going to need you all to stop saying, “Fast cars don’t drive down the road – they shift the earth beneath them.”
This hugely exaggerated saying is often tossed around in reference of powerful cars. Of course, Earth is a very heavy thing, and even the torque of the most powerful car ever is grossly insufficient to “shift” it.
Still, I got to wondering how we could quantify this age-old adage, and how much torque would actually be needed to shift the planet. So, I got in touch with Robert Boucher, a physics teacher and the Program Leader for Sciences and Technological Studies at Bishop Alexander Carter Catholic Secondary School in Sudbury, Ontario.
Some 20 years ago, Mr. Boucher (who says I may now call him Rob) was my physics teacher. As a fellow car guy, he made Physics fun by using cars and driving to frame up class exercises on power, force, and acceleration – often using his then-pride-and-joy, an Acura 1.6 EL, in the examples.
Boucher and I had a chat, put our physics hats on, and set to work to determine how much torque you would, in fact, need to shift the earth – or technically, to change the speed of Earth’s rotation. We have a number, which we’ll get to. But first, some things.
For starters, this isn’t a super-straightforward thing to figure out. I’ll skip full explanations of the complex calculations and math we went through, and instead present a summary of our findings with very simplified terminology. If you’re a physics major, stop reading now, or you may get nauseous.
Getting Up to Speed
What is torque?
Torque is a twisting force applied at some distance to something. You exert torque whenever you turn a wrench or push open a door or twist the cap off a bottle of Sprite. Your car’s engine exerts torque on its crankshaft, too. Torque is expressed in many ways – most commonly in pound-feet, or lb-ft, when it comes to cars.
As this exercise deals in causing a rotation of the earth, Boucher reminded me that simple force to push an object in a straight line (multiplied by the distance that object then travels) is called “work”, not torque. “Torque is similar, but it imparts a rotational force, not a force in a straight line,” he said.
That’s Grade 11 physics.
Earth’s mass
Second, nobody knows our planet’s precise mass, but scientists have used fancy formulas related to astrophysics and orbits and gravity to create a generally accepted figure.
That figure suggests that Earth has a mass of about six septillion kilograms. That’s 6 × 1024. If you don’t remember exponents, that number looks like this: 6,000,000,000,000,000,000,000,000. (That’s 24 zeroes!)
As the world turns…
For this exercise, we’re calculating the torque required to accelerate a sphere, equal to the mass and radius of Earth, from stationary (zero rotations per day) up to Earth’s current rotation about its axis (one rotation per day) in a time of one second.
To prevent the contents of my cranium from melting down into a sizzling pool of ooze, we’ve left several details out of the calculations. We assume, for instance, that this sphere is perfectly spherical (Earth is slightly flattened at the poles) and of uniform density (Earth is more like a grilled cheese: hard and crusty on top with a deliciously warm, gooey centre).
We also assume that the earth has no massive bodies of fluid on top of it, though it does. Our calculations also ignore the effects of gravity, Earth wobbling about its axis, the gravitational implications of the sun, and so on.
Translation? This will be a ballpark figure.
Putting the pieces together
So, the fun part: How much torque is needed to bring our six-septillion-kilogram planet from a standstill to a one rotation-per-day rotation around its axis, in the span of one second?
Boucher advised that we’d need figures for the moment of inertia of a solid sphere and the delta of angular momentum.
Then, we needed to solve a simple-looking (it wasn’t actually simple) formula that looks like this:
Torque (T) = change in angular momentum (ΔL) / time (t)
As Boucher explains: “The big picture here is that the torque required is equal to the change in angular momentum, over time.”
If we know the moment of inertia of a solid sphere (we do), and the change in angular momentum we want to apply to it (we do), over a given period of time (one second, in this case) we can then figure out how much torque is needed to make it happen.
Arriving at the Answer
Here we restate the critical numbers for our question, “How much torque do you need to shift the earth?” Or, put another way, “How much torque do you need to bring Earth from a standstill to its current rate of spin?”
Mass of Earth: 5.974 × 1024 kg (six septillion kilograms, rounded)
Radius of Earth: 6,371 kilometres
Rotational speed at start: zero rotations per day
Rotational speed at end: one rotation per day
Duration: one second
Filling in and solving several required formulas, converting rotations to radians, and solving this whole complicated equational mishmash for torque, we arrive at the big number…
Drumroll, please!
You’d need 5.202 × 1033 (or 5.2 decillion) lb-ft of torque to bring our Earth from stationary up to its current rotation, in one second.
That’s 5,202,000,000,000,000,000,000,000,000,000,000 lb-ft of torque. (That number is 34 digits long!)
Coming back down to Earth
But even the most powerful street-legal cars today have, maybe, 800 lb-ft of torque. The fastest top-fuel dragsters have, maybe, 7,000 lb-ft.
So, required torque output is equal to about 8.7 nonillion (8,700,000,000,000,000,000,000,000,000,000) Dodge Vipers, or about 7.4 octillion (743,000,000,000,000,000,000,000,000,000) top-fuel dragsters.
For scale, that’s the combined torque output of about one quintillion (1,000,000,000,000,000,000) top-fuel dragsters for every living person on earth.
Obviously, this is charmingly impossible. According to science and math, unless you’ve got access to an amount of torque that’s thirty-four digits long, you’re not shifting the earth. It’s also safe to say that nothing, or even everything, driving on the earth’s surface has any effect on its rotation.
Sorry, Dodge Demon drivers!
Discussion (and future research?)
Boucher concludes, “Another thing to consider is that all of the cars on Earth are oriented in every possible direction. Due to averages, the forces they apply to the earth’s surface would cancel each other out.
“One can do the math and calculate the forces applied to the Earth’s surface, but considering all of the above, it doesn’t really change anything when you look at the planet as a closed system.”
In summation: you’d need the torque output of an unpronounceable and unfeasibly large number of very, very powerful vehicles to actually shift the earth.
Probably, you’d have traction issues too, but I’m not certain of the math on that.
