by Tom Rogers
Solving for velocity gives:
However, it’s inconvenient to talk about rotation in terms of tangential velocity. Revolutions per minute (RPM) is a more common way to quantify rotation.
Assume that v is in units of meters per second and r is in units of meters. The object travels the circumference of the circle (2 r) in each revolution. Then the number of revolutions per second (RPS) is as follows:
Substituting for v using equation 15.1 yields:
For Earth g = 9.8 m/s2
For the 400-meter diameter space station in 2001: A Space Odyssey the rotational speed would be calculated as follows:
Next, let’s pull the imaginary giant gravity switch in the sky to the “off” position. Yeah, yeah, it sounds like something in a movie, but doing physics often requires imagination. Our person again feels like he’s weightless. So, it seems that a gravity force is also needed. But wait. When the gravity force was turned off, the normal force also ended. Unfortunately, we still don’t know if we need gravity to have a sensation of weight.To answer the question we have to create an experiment with a normal force but no gravity force. If the person still feels the sensation of weight, then it’s obvious that a normal force is required but a gravity force is not.
To run this experiment, we can have subjects stand against the wall inside an amusement park ride—the kind that is a large hollow cylinder that spins around a vertical axis at its center. When the ride is spinning at full speed and the riders are pressed against the cylindrical wall, the bottom drops out. This results in some screaming, but no one falls out. They feel like they have a large gravity force pressing them against the wall so firmly that they can’t slide out. But there is no gravity force in the horizontal dimension. What they are feeling is the normal force between the cylindrical wall and the back sides of their bodies. This force acts as the centripetal force that makes the subjects speed around in a circular path. A normal force acting without a gravity force can indeed produce the sensation of weight. The conclusion: normal force, not gravity, is responsible for the sensation of weight.
Okay, at this point someone invariably says “but what if a person is hanging from a cable? There is no normal force.” The answer is that a normal force is simply the force a surface exerts on an object—in this case a person. To hold the person up, the surface of the cable or the surfaces of a harness attached to it must be in contact with the person. Add up all the normal forces from the cable or harness, and it will equal the force of gravity but act in exactly the opposite direction. In other words, the normal force does still exist and is indeed the primary force causing the sensation of weight.
Someone will invariably also argue that the amusement ride described above does not turn off the force of gravity. However, the sensation of weight described was in the horizontal dimension corresponding to the normal force exerted by the cylindrical wall. Gravity is in the vertical dimension. Seal the ends of the cylinder, provide it with a life support system, transport it to outer space, spin it up to speed with the people against the wall, and they will still feel a weight-like sensation.
If the cylinder has a large diameter—say 440 yards (400 m) so that it looks disk shaped—and is spinning at exactly the right speed, the people will be able to stand on the cylindrical wall and walk around just like they were standing on Earth. If one of them drops a coin, it will appear to fall to the floor (the cylindrical wall) just like a coin would on Earth. The effect is called artificial gravity. In reality it is an artificially produced normal force. There is no gravity force involved. The normal force is providing a centripetal force in the direction of the center of the cylinder. This centripetal force keeps people rotating around inside the disk. When a coin is dropped, the centripetal force is momentarily removed and the coin falls to the floor, which then reestablishes the centripetal force.
The rotating cylinder needs to have a large diameter for two reasons: it makes the floor’s curvature less noticeable, and it makes the artificial gravity effect on the person more uniform. With artificial gravity the centripetal force provides the artificial normal force needed for the sensation of gravity. Unfortunately, centripetal force depends on one’s distance from the center of rotation, or in this case, the center of the disk-shaped space station. It will be zero at the center and 1.0 g at the floor people stand on. (Keep in mind that the floor is the cylindrical wall of the disk.) If the space station had a diameter of 12 feet (3.7 m) and was rotating fast enough so that objects on the floor would experience 1.0 g of acceleration, six-foot-tall people standing on the rotating floor would experience 1.0 g in their feet and 0 g in their heads. The result: who knows? But nausea, lightheadedness, and disorientation would be good bets, especially if a person bent over, sat down, or stood up.
THE GOOD PORTRAYAL OF ARTIFICIAL GRAVITY
A realistic space station with artificial gravity would look like a gigantic rotating bicycle wheel with large spokes—just like the space station in 2001: A Space Odyssey [GP] (1968).Yes, 2001 has some out-there stuff, such as the monoliths, which have about as much scientific basis as magic wands, not to mention the movie’s colorful but incomprehensible ending (reading the book was required to make any sense of it). But when a scene calls for realistic Newtonian physics, the movie delivers. Its realism created a mood that has never been duplicated.
2001 won an academy award for visual effects, is listed number twenty-two in the American Film Institute’s 100 Greatest Movies list (http://www.afi.com/tvevents/100years/movies.aspx), made $56.7 million (270 million in 2004 dollars), and developed a cult following; and yet it ranks as one of Hollywood’s least copied movies. If influence over other films were measured in height above sea level, 2001 would be in the Marianas Trench. Why? Hollywood has yet to understand its success let alone develop a formula for cloning it. Still, a few aspects of the movie have stuck, such as the vague notion that something has to rotate to produce artificial gravity.
THE BAD PORTRAYAL OF ARTIFICIAL GRAVITY
To give itself credibility, Mission to Mars [RP] (2000) had its spaceship rotate to produce artificial gravity and its inhabitants correctly walk around on the inside of its outer wall. The ship, however, was only about 24 feet in diameter (7.3 m). When standing on the floor in this ship, the feet of people six feet tall would experience 1 g while their heads would experience 1/2 g. If they sat down, the acceleration of their heads would increase by 50 percent.When they climbed the ladder leading to the center of the ship, their centripetal acceleration would have decreased to near zero, yet none of these changes produced even the slightest nausea.
THE UGLY PORTRAYAL OF ARTIFICIAL GRAVITY
Armageddon [RP] (1998) took the foolishness even further. They had an astronaut press a button in a Mir-type Russian space station (mass = 124,340 kg) and spin it up to full artificial gravity level in a matter of seconds. The stresses from the high acceleration required to spin up so quickly would have torn the space station apart. Inside the space station the distance from the axis of rotation varied from 39.4 feet (12 m) to as little as 6.6 feet (1 m), yet the gravity level was always the same in any part of the station. Both the station inhabitants and its visitors experienced no discomfort when their heads were located at 0 g and their feet at 1.0 g. The artificial gravity was so remarkable that it always acted in the correct direction even when the visitors were climbing through tubes pointed directly away from the axis of rotation.
Red Planet [RP] (2000) takes the award for most muddled understanding of artificial gravity. The spaceship in it looks like an oversized shoebox with large counterrotating rings on each end that seem like they belong on a carnival ride with neon lights. While it’s not completely clear, the main crew quarters seem to be located in the nonrotating section of the shoebox. So it’s a mystery how the crew can walk around inside it just like they were on Earth.
The box-like shape of the central ship also makes no sense— assuming it’s pressurized for human habitation. The internal pressure inside it would tend
to make the sides bulge outward. A sphere or a cylinder with spherical ends would be a far better shape for containing internal pressure.
Having the rings counterrotate with a stationary shoebox in between means the rings must have large-sized bearings and seals to prevent air loss. Thanks to seal friction, the rings would have to constantly be driven by electric motors—a power drain on long space voyages. Compare this to the elegance of the 2001 space station in which the entire space station rotates. Once it’s turning, it will rotate indefinitely with no further power input. There is no air resistance or friction in space to slow it down, nor are there any bearings or seals required.
When a solar flare disables the Red Planet spaceship, the counterrotating rings stop moving, causing the artificial gravity to fail. The rings on the ship would have a huge amount of rotational inertia, and yet they are brought to a halt in seconds. The accelerations required to do this would likely tear the rings apart. A large spacecraft would likely not be designed for unexpectedly high stresses, since every pound of material used to build it would cost thousands of dollars just to lift it off Earth and send it into space for assembly.
As the rings stop, loose objects aboard the ship begin to float about. But why? They would still be moving in the direction of rotation. Quickly stopping the rings should send objects tumbling into the nearest wall in the direction of their motion—that is, if the crew quarters were correctly set up in the rotating rings.
When the rings are again restarted, artificial gravity is instantly restored and all the randomly floating objects immediately drop to the floor. If they really were gently floating inside a stationary ring, the objects would also be stationary. When the ring started rotating, the wall opposite of the one they had previously tumbled into would appear to move forward and bump into all the floating objects. As the rings accelerated the objects would slide toward the floor and pile up on top of each other.
Artificial gravity on the starship Enterprise in the Star Trek movie series is beyond the categories of good, bad, and ugly because no attempt is made to use centripetal force as the source. Star Trek proposes that humanity is so advanced, a spaceship can be built that manipulates gravity in a similar way as it manipulates other phenomena such as electricity—once considered esoteric. The series offers no mumbo-jumbo explanations. If the Enterprise couldn’t generate its own gravity, the entire movie series would have to be rethought. The stretch is forgivable but in a begrudging manner. Had the writers used reliable Newtonian physics to generate artificial gravity, the Enterprise would have looked different and the stories been altered, but it still would have had the Star Trek flavor.
At first glance, a rotating ring like the huge 2001 space station flying through space sounds weird, but why not? As long as the vehicle is neither required to land nor take off, there’s no reason for making it aerodynamic. It will encounter no air resistance. The ship will have no fuel-guzzling liftoffs, and once it’s up to cruising speed it will need no energy to keep it there. Hence, small size is not critical. Such a ship would still have a great deal of screen presence.
Robert Zubrin’s highly creative proposal for a Mars ship would break all cinematic paradigms in spaceship design. He suggests that a section of used-up booster could be tethered several hundred meters from the living quarters of the Mars ship and the two rotated around a point in the middle. In a military version both sections could hold living quarters. Imagine the spectacle of an invading army in hundreds of tethered ships spinning as they silently traveled through space. Realistic artificial gravity does impose design constraints on cinematic spaceships, but even in the world of reality there’s still plenty of room for creativity and gee-whiz effects.
Summary of Movie Physics Rating Rubrics
The following is a summary of the key points discussed in this chapter that affect a movie’s physics quality rating. These are ranked according to the seriousness of the problem. Minuses [–] rank from 1 to 3, 3 being the worst. However, when a movie gets something right that sets it apart, it gets the equivalent of a get-out-of-jail-free card. These are ranked with pluses [+] from 1 to 3, 3 being the best.
[–] [–] Depictions of spacecraft in which rotating parts produce artificial gravity (AG) but the part of the craft with the gravity does not rotate.
[–] [–] Depictions of AG in which the direction of the AG does not match with the direction of centripetal force. (Note: centripetal force always points at the center of rotation.)
[–] [–] Depictions of AG in which peoples’ heads are subjected to totally different AG conditions than their feet.
[–] [–] Starting or stopping rotation in massive spacecraft parts in a matter of seconds.
[–] Objects falling straight to the floor when interrupted AG produced by rotation is restored.
[+] Astronauts becoming dizzy and possibly vomiting when they climb ladders taking them from high rotationally produced AG to low AG.
[+] [+] Using a Zubrin design for producing AG.
CHAPTER 16
THE MOVIE MERRY-GO-ROUND:
How Filmmakers Create Ridiculous Spin
MARTIAL ARTS
It’s possibly the most famous movie kick ever.Trinity leaps in the air, arms extended like the wings of a white crane about to fly. For a moment time stops. When it resumes, she falls toward Earth slamming her foot forward and slightly downward into a fat cop’s chest. He flies horizontally backward across the room and crashes into a wall (The Matrix [RP]).The action does take place inside a computer simulation, and Trinity has supposedly mastered altering the simulation’s physics, but what about the fat cop? Surely, he hasn’t. He’s part of the simulation and should have rotated backward and crashed into the ground a short horizontal distance from where he got kicked.
Increasing the power of the kick would cause the hapless cop to slam into the ground harder, but would have little effect on the distance he flies backward. In fact, at some point the extra power will mostly go into crushing the rib cage and collapsing the lungs rather than increasing the victim’s kinetic energy.
The only way a victim can be sent straight backward is to apply a horizontal force through or, in other words, lined up with the victim’s center of mass. Even so, the force must have a slightly upward direction to knock the victim’s feet off the ground, thereby eliminating the friction force between the victim’s feet and the floor. Since Trinity’s kick does not act through the center of mass, the friction force would otherwise cause backward rotation.
The force from Trinity’s kick obviously did not match the conditions required for translation (moving in a linear fashion), so why did the cop fly such a long horizontal distance? Run the scene in slow motion, and it’s possible to see the glint of a horizontal wire pulling the hapless cop backward. Indeed, the laws of movie physics always require the hapless victim to fly backward—if he or she is going to be thrown horizontally at all— because slamming a stunt person front wise, even into a fake wall, tends to be hard on the nose, not to mention that pulling people forward tends to bend their backs harmfully backward.
The Basics of Torque
Torque is a twisting action that causes an object to rotate similar to the way a force causes an object to translate (move in a linear fashion). A force F applied at a ninety-degree angle to a lever arm or moment arm r creates a torque as follows:
τ = F (r)
The Rules of Rotation/Translation
An object’s motion can be categorized as translation, rotation, or some combination of the two. Translation is motion along a smooth path. Rotation is a spinning motion around a pivot point. If the object can move freely, the pivot point will be its center of mass. The center of mass is like a balance point. For example, a child’s seesaw will balance if the pivot is located at or under the center of mass.
Like forces, torques can counteract each other. For example, a torque that causes a twisting action in a clockwise direction can be counteracted by one causing a twisting action in the counterclock
wise direction. When all the torques are added up, the result is called the net torque; and if the net torque is not zero, the object will have rotational acceleration. To understand what torques and forces do, let’s see what happens when we apply them to a stationary object that could otherwise move freely. Like many things in physics, the resulting motion can be reduced to a few simple rules:
A net force (the result of adding up all the forces) acting on an object will always cause translation in the direction of the force regardless of whether the object rotates or not.
A net force that does not act through the object’s center of mass or pivot point will cause translation and will create a torque that causes the object to rotate. (Assuming no other torques counteract it. See A.)
An object will translate without rotation (see B) if the net force on it acts through the center of mass.
An object will rotate without translation (see C) if the net force acting on it is zero but the net torque is not.
Concepts such as center of mass, rotation, and translation are key to understanding many martial arts. Practitioners of Aikido (considered the primary martial art of Steven Segal) actually spend time meditating on their centers of mass in order to remain balanced while sending their opponents flying. Attackers foolish enough to charge headlong at an Aikido master will likely find their linear velocity tuned into head over heels rotation. Up to a point, the same mastery of applied physics can be simulated with clever camera and wire tricks that make even moderately trained actors look like martial artists. If overdone, however, the wire and camera stunts merely make the actors look like cartoon parodies.