• The previous chapter focused mostly on motion in a straight line. The same
    • principles apply to rotational motion, but end up being reinterpreted in some unusual ways. This chapter provides a short study of the physics of rotational motion.

    [2.1] ROTATIONAL MOTION / ANGULAR VELOCITY
    [2.2] FUNDAMENTAL QUANTITIES / CENTER OF MASS
    [2.3] TORQUE, MOMENT OF INERTIA, & THE SECOND LAW OF MOTION
    [2.4] MOMENT OF INERTIA IN DETAIL
    [2.5] THIRD LAW OF MOTION / WORK & ENERGY
    [2.6] CENTRIPEDAL & CENTRIFUGAL FORCE
    [2.7] CONSERVATION OF ANGULAR MOMENTUM

    [2.1] ROTATIONAL MOTION / ANGULAR VELOCITY

  • To consider rotational motion, let's imagine that Dexter has built a test
    • apparatus, consisting of a mass linked to a rod that is rotating around a pole. The mass is very compact, so its size and shape can be disregarded. The mass of the rod is small and can be ignored.

    The rod spins the mass around the pole in a circle at a rate of, say, six meters per second. For the purposes of this discussion, we can refer to this velocity as the "linear velocity" of the mass, though of course it's not going in a straight line.

    We define this as linear velocity in order to introduce a different concept of velocity applicable to rotational motion, known as "angular velocity". If the mass is positioned on the rod one meter from the pole, then the circular path length of the mass is one meter times 2PI (two times pi), or about 6.28 meters. That means that the number of revolutions the mass makes around the pole in one second is: 

        mass_velocity / ( 2PI * rod_length ) = 0.955 revolutions per second
    This defines the angular velocity of the mass as 0.955 revolutions per second. In general, we can define angular velocity as follows: 
        angular_velocity = linear_velocity / ( 2PI * radius )
    -- with angular velocity given in revolutions per second. Notice the specific difference between linear velocity and angular velocity. Linear velocity gives the number of meters the mass travels in a second, while angular velocity gives the number of revolutions of the mass in a second, regardless of big a circle it travels around.

    Suppose that the mass can be slid forward and back along the rod, and that it Dexter moves it in so that it's only half a meter from the pole. Suppose further that the rod continues to rotate at the same rate. That means that although the angular velocity is the same, 0.955 revolutions per second, the linear velocity has been cut in half, to three meters per second.

    Similarly, suppose Dexter moves the mass out along the rod until it's two meters from the pole, and once again the rod continues to rotate at the same rate. The angular velocity is still the same, but the linear velocity has doubled, to twelve meters per second.

  • Angular velocity is not normally measured in revolutions per second. The
    • popular unit for angular velocity is "revolutions per minute (RPM)", and of course all you have to do is multiply revolutions per second by 60 to to get revolutions per minute. For the example above, the RPM amounts to 60 * 0.955 = 57.3 RPM.

    However, RPM is not a very convenient unit for performing calculations. A somewhat more convenient unit for angular velocity is degrees per second, with 360 degrees in a circle. To get degrees per second, all you have to do is multiply the revolutions per second by 360. For the example above, the mass has an angular velocity of 360 * 0.955 = 344 degrees per second.

    Degrees are a unit of angular measurement, and though we have defined angular velocity as the number of revolutions per second, the term "angular velocity" itself suggests it's more properly defined as the change in angle per second. The two definitions are equivalent, with the first being more intuitive and the second being more useful for calculations.

    In practice, from the point of view of calculations, there is a even more convenient unit for angular measurement than degrees, known as "radians". One radian is just the length of one radius of a circle along the circumference of that circle, and so there are 2PI, or about 6.28, radians in a circle.

    Radians are not as intuitive an angular measurement as degrees, but they make rotational calculations much more consistent and easy to handle. For example, if angular velocity is expressed in radians, then its definition is simplified to: 

        angular_velocity = linear_velocity / radius
    The final result of calculations can be converted into RPM or degrees per second if need be, and the conversion calculations are very simple: 
        RPM  =  60 * radians / 2PI
    degrees per second  =  360 * radians / 2PI
    Angular velocity may seem somewhat arbitrary at first sight, but a moment's thought shows how convenient it is. It is much simpler to describe the rotation of a helicopter's rotor as say, 200 RPM, than it would be to describe it in terms of the linear speed of the rotor tips, as 452 kilometers per hour.

    Of course, if we can define a quantity such as angular velocity, we can also define the related quantities of angular displacement, or the number of radians a revolving object has moved through, and angular acceleration, or the rate of change of angular velocity.

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    [2.2] FUNDAMENTAL QUANTITIES / CENTER OF MASS

  • Defining the concept of angular velocity, as opposed to linear velocity,
    • suggests that other quantities useful in rotational motion can also be defined that are comparable to fundamental quantities in linear motion. In fact, there's a complete and self-consistent system of quantities for rotational motion.

    The basic rules for linear motion are provided by Newton's three laws of motion, and define the basic quantities of force, mass, and acceleration. We can redefine the three laws for rotational motion by defining a rotational equivalent of force called "torque"; a rotational equivalent of mass known as "moment of inertia"; and of course a rotational equivalent of linear momentum known as "angular momentum". The definitions of torque, moment of inertia, and angular momentum are all intertwined, so I'll give all three definitions, and then explain them in detail in the following sections.

    Torque is obtained by multiplying the tangential force that sets an object to spinning times the length of the lever arm, or radius, on which the force acts relative to the center of rotation of the object: 

        torque  =  force * radius
    The moment of inertia, at least for a simple system like the mass on a rod used in the previous section, is given by: 
        moment_of_inertia  =  mass * radius^2
    We'll discuss how this definition is extended for more complicated systems later. For now, just accept that this definition is the truth, just not the whole truth. By the way, moment of inertia is sometimes called "rotational inertia", but we won't use that term in this document.

    The definition of moment of inertia leads to the definition for angular momentum: 

        angular_momentum  =  moment_of_inertia * angular_velocity
    The quantities of energy, work, and power remain the same in rotational motion as they are in linear motion.

  • The "center of rotation" mentioned above is better known as the "center of
    • mass". For any object that is symmetrical around its spin axis, the center of mass is the physical center of the object. However, if the object is not symmetrical, the center of mass is where the object would balance if hung by a thread, using the simple rules of lever balance outlined in the previous chapter to determine the point where the forces from the two masses balance out.

    For example, consider an object that looks like a barbell, with a 15 kilogram mass M1 at one end, a 3 kilogram mass M2 at the other, and with a connecting rod of negligible mass 1.3 meters long between them. Then the balance point X as measured from M1 is given by: 

        M1 * X  = ( 1.3 - X ) * M2
 
    M1 * X  +  M2 * X   =  1.3 * M2
 
    X * ( M1 + M2 )  =  1.3 * M2
 
    X  =  1.3 * M2 / ( M1  +  M2 )  =  1.3 * 3 / ( 15 + 3 )  =  0.22 meters.
    This same "balancing" approach could be extended to an assembly of masses in two or three dimensions, but although the math is not complicated, it can be long-winded, and so will not be discussed further here.

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    [2.3] TORQUE, MOMENT OF INERTIA, & THE SECOND LAW OF MOTION

  • The best way to understand torque, moment of inertia, and angular momentum
    • is to see how Newton's three laws of motion are restated in a rotational context.

    It might seem logical to start with the first law of motion, which is the law of conservation of momentum, expressed very simply as the definition for linear momentum: 

        linear_momentum  =  mass * linear_velocity
    The definition of angular momentum is exactly comparable: 
        angular_momentum  =  moment_of_inertia * angular_velocity
    However, understanding the mechanisms of angular momentum means understanding the concept of moment of inertia as well as a few other subtleties, so an understanding of the first law in a rotational context will have to be put off for bit.

  • That means the best first step is to consider the second law of motion. In
    • the linear context, this is given by: 
        force  =  mass * linear_acceleration
    In the rotational context, the equivalent expression is: 
        torque  =  moment_of_inertia * angular_acceleration
    The two expressions are actually equivalent. The equivalence can be easily seen if you consider the simple system described earlier, with a mass rotating at a certain radius on a rod, and the force causing the torque operating directly on the mass at that radius.

    Then we can restate the rotational version of the second law as follows: 

        torque  =  moment_of_inertia * angular_acceleration
 
    force * radius  =  mass * radius^2 * angular_acceleration
    force           =  mass * radius * angular_acceleration
    force           =  mass * ( radius * angular_acceleration )
    force           =  mass * linear_acceleration
    The underlying physical principles established for rotational motion haven't changed, they've just been restated in a different form.

  • This shows the equivalence between the force and torque, but it doesn't
    • really explain the concepts of torque and moment of inertia.

    The concept of torque is fairly easy to understand. To get something to spin, Dexter has to push it along a direction at a right angle to a radius of rotation. Any force in the direction of that radius might get the entire rotating system to move in a straight line if it's not bolted down, but it won't contribute to rotation.

    Now suppose Dexter has a rotating disk with a hex nut on top, and uses a wrench to get it to rotate. He applies a torque equal to the force he exerts on the wrench times the length of the wrench. Considering the discussion of levers in the previous chapter, obviously the same force will have a greater effect if the wrench is longer. It's just simple mechanical advantage, exerting a smaller force over a longer path to obtain the same amount of work.

    This means that the force actually exerted on the rotating disk, and so its angular acceleration, is proportional both to the length of the lever arm and the force exerted at the end of that lever arm.

    One of the peculiarities of torque is that it's measured in units of mass times distance, or "newton-meters" in metric. Work is also measured in the same units. Does this imply that they're the same thing?

    No, it doesn't. In fact, they're literally sideways to each other. Work is force on an object in the direction of its motion. Torque is exerted at a right angle to a radius, and there's no motion along the radius.

  • Moment of inertia is a bit harder to understand. Let's consider our simple
    • example again of a sliding mass on a rod, but now let's say the rod has an extension a meter long on the other side of the axis, with a handle on it that Dexter can push on to cause rotation.

    Now let's suppose that the mass is positioned on the rod a meter away from the axis. If Dexter pushes on the handle, the same linear force acts on the mass on the other side of the axis, and the mass accelerates just as the second law says it does.

    Now Dexter moves the mass out to two meters and pushes on the handle. Since the mass moves twice as far as the handle, it only receives half the force at any one time, and its linear acceleration is only half as great as it was at one meter. This means he has to push twice as hard to achieve the same linear acceleration.

    However, what we're after here is not linear acceleration, but angular acceleration, a change in RPM. If he doubles the radius to two meters, as noted has to push twice as hard to get the same linear acceleration, but now the circular path along which it moves is twice as long. Even given the same linear velocity in the two positions, the angular acceleration is only half of what it was for the same linear velocity when the radius was one meter.

    This means that to obtain the same angular acceleration when the mass is moved from one meter to two meters, Dexter has to push four times as hard, or proportionally to the square of the radius. This leads back to the original definition of moment of inertia: 

        moment_of_inertia  =  mass * radius^2
    Once again, the same physical laws apply, they're just restated in a different form.

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    [2.4] MOMENT OF INERTIA IN DETAIL

  • As noted earlier, the initial definition of moment of inertia:
    •     moment_of_inertia  =  mass * radius^2
    -- was only for the simple mass on a rod, and was not exactly true for other cases. Let's consider a more complicated example, where the object whose moment of inertia we want to calculate has several different masses of different size, each at a different distance from the axis of rotation of the object. This leads to a more general definition for moment of inertia: 
        moment_of_inertia  =  SUM( mass * radius^2 )
    In simple terms, the moment of inertia of the entire object is the sum of the moments of inertia of its components. If the object had three different masses, each at a different distance from the axis of rotation, then the moment of inertia of the object is: 
        moment_of_inertia  =   mass1 * radius1^2
                            + mass2 * radius2^2 
 		              + mass3 * radius3^2
    Let's consider more specific examples, such as a barbell with weights connected by a rod. Ignoring little details like the weight of the rod, if each weight has a mass Mi and the rod has a length L = 2*R, then its moment of inertia for rotation around the midpoint of the rod is: 
        2 * Mi * ( L/2 )^2  =  2 * Mi * R^2
    If we designate the sum of the mass of the weights as Ms, then this becomes 
        Ms * R^2
    If we have a pair of rods of length L = 2 * R joined in a cross, with weights with mass Mi at each end of the rods, then the moment of inertia for rotation around the center of the cross is: 
        4 * Mi * R^2
    -- or once again: 
        Ms * R^2
    In fact, this moment of inertia also applies to any thin ring or hollow thin cylinder rotating around its axis. They're exactly the same as the barbell in this respect because we still have a mass at a particular radius, and it makes no difference if it's two individual masses or a continuous ring.

    Calculating moments of inertia for other objects generally requires more math than is appropriate here, but a short list of the moments of inertia of a few different objects of mass M will help give a better feel for the concept.

  • The concept of moment of inertia makes a certain amount of intuitive sense.
    • For example, it's much easier to rotate a barbell if the weights are close together on the bar instead of at the ends.

    Similarly, a tightrope walker may often carry a long pole held tightly in her hands to provide balance. The longer the pole, the greater its moment of inertia, and more it prevents the acrobat from tipping.

    In other cases, it's not as intuitive. Suppose Dexter has two cylindrical masses, one in the form of a hollow cylinder and the other in the form of a solid cylinder. They could have the same or different diameters, or the same or different masses, it doesn't matter.

    If Dexter drops drop these two cylinders from the same height, as emphasized in the last chapter, ignoring drag they'll hit the ground at the same time. If he were to put these two cylinders on end on top of a ramp and let them slide down, if the ramp is very slick, once again they'll reach the bottom at the same time.

    Now if Dexter rolls these two objects down a plane, the solid cylinder will always beat the hollow cylinder. This is because the hollow cylinder has a greater moment of inertia, and its angular acceleration is lower for a given force. The relative sizes of the two cylinders doesn't matter any more than it does for dropping them, because the force required to get them to move is proportional to the mass, but the hollow cylinder just can't get rolling as fast as the solid cylinder.

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    [2.5] THIRD LAW OF MOTION / WORK & ENERGY

  • After considering the second law of motion in a rotational context, it is
    • almost too easy to consider how the third law fits in. Obviously, if Dexter generates a torque on a rotating system, the same torque is produced on him in the opposite direction.

    This is most obvious in the case of a helicopter. A helicopter spins a rotor at high speed in one direction, which means that it tends to spin in the other direction. This is why most helicopters have a little rotor attached on the side of the tail, just to stop it from spinning. The British call this a "penny-farthing" configuration.

    There are other approaches to the perform the same trick. Some large cargo helicopters have rotors on both ends, spinning in different directions. Yet another scheme is the "eggbeater" configuration, with twin rotors meshing in a side by side configuration, or the "coaxial" configuration, with one rotor sitting on top of the other but rotating in opposite directions.

    There is a way around the problem entirely. Some helicopters have been designed with little jet engines on the end of the rotor blades, with fuel passed up from the helicopter through the rotor blades. This causes the rotor to turn itself, and eliminates the need to cancel torque. However, for various reasons, it seems particularly noise, no "tipjet" helicopter has ever been produced in quantity.

    Coaxial propellers were also used on some of the very last propeller-driven fighter aircraft. These aircraft had such powerful engines that they generated an enormous amount of torque that made the aircraft very hard to control, and so they were often fitted with "contra-rotating" propellers, with two sets of blades turning in opposite directions, driven by the same engine using a wheels-within-wheels "planetary gear" transmission.

    In addition, twin-engine propeller-driven aircraft often had their two propellers turning in opposite directions to cancel out the effects of torque. This wasn't always done, because it meant that the aircraft required two different types of engines, one "left handed" and the other "right handed", which complicated maintenance.

  • It is also easy to see that the expression for work in the rotational
    • context is similar to the expression for work in the linear context, and is in fact equivalent: 
        work  =  torque * angular_distance
          =  force * radius * angular_distance
 	 =  force * linear_distance
    If you recollect the example of a mass on a rod, rotated around a pole by a one-meter extension with a handle on the other side, then pushing the handle over an arc a meter long with a force of a newton does a joule of work. It doesn't matter how far out along the rod the mass is, the work done on the mass is the same as the work done on the handle.

    Calculation of kinetic energy demonstrates the same sort of equivalence: 

        kinetic_energy  =  (1/2) * moment_of_inertia * angular_velocity^2
                    =  (1/2) * mass * radius^2 * angular_velocity^2
                    =  (1/2) * mass * ( radius * angular_velocity )^2
 		   =  (1/2) * mass * linear_velocity^2
    Kinetic energy is due to mass moving at a certain linear velocity. It doesn't matter if the velocity in a straight line or in a circle, the kinetic energy is the same.

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    [2.6] CENTRIPEDAL & CENTRIFUGAL FORCE

  • We now have a good grasp of the second and third laws of motion in the
    • rotational context, and so it might seem time to go back and nail down the first law. However, we end up having to take yet another side trip before we can do that, and consider the forces on a rotating object.

    As explained in the previous chapter, an object moving along a straight line at a certain velocity will continue to do so unless a force acts on it. A revolving in a circle is always changing direction, which means there is a force on it causing it to do so. If terms of our simple mass-on-rod example, the rod exerts a force on the mass, directed towards the pole.

    This is known as "centripedal force", and it is always directed towards the center of rotation.

    The concept of centripedal force sounds very much like the more popular notion of centrifugal force, which most people recognize as the force that squeezes the water out our clothes in a washing machine by spinning them very rapidly.

    In fact, they're really the same thing, just seen differently. Centrifugal force is perceived as the force of a rotating object trying to pull outward, which is reality the mirror reflection of the centripedal force on a rotating object trying to pull it inward. During the washing machine's spin dry cycle, the water just keeps going out the holes in the basket in a straight line, while the basket pulls the clothes themselves into a spinning curve at a rapid rate.

    From a casual point of view the distinction is small, but physicists get nitpicky about it, since using centrifugal force in the mathematical analysis of a rotating system tends to be awkward as best. However, most people don't understand the term centripedal force, and that leads to a simple rule: use the term "centrifugal force" in casual conversation and "centripedal force" when you're being technical.

  • One application of centripedal force is to provide artificial gravity in
    • spacecraft. Back in the 1950s and 1960s, space stations were generally seen as big rotating wheels, spinning to create artificial gravity, as in Stanley Kubrick's movie 2001: A SPACE ODYSSEY.

    2001 has come and gone and no such thing ever happened, but the idea remains valid. A similar but less ambitious concept is to link a pair of spacecraft by a long cable or "tether" and set them to spinning around each other. This may need to be done on a long Mars mission or other expedition where people would be in weightless condition for long periods of time.

    Suppose the tether is 100 meters long, giving each spacecraft a distance from the center of their mutual rotation of 50 meters, and the spin rate is 3 RPM, or one rotation every 20 seconds. Giving the formula for centripedal acceleration without derivation: 

        centripedal_acceleration  =   radius * angular_velocity^2
    Plugging in the quantities: 
        50 * ( 2PI / 20 )^2  =  4.93 meters per second
    This is about half of Earth gravity. By the way, the maximum rotation rate that human beings can tolerate without suffering from vertigo or nausea seems to be about 2 or 3 RPM, so this is a plausible scheme for a Mars mission.

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    [2.7] CONSERVATION OF ANGULAR MOMENTUM

  • The physics of rotational motion also introduces a number of elaborations
    • to the physics we learned for linear motion. The basic physical rules stay the same, but new games are played.

    For example, let's return to our moving mass on a rod, rotating around a pole. In the previous example, Dexter moved the mass in and out, keeping it at the same angular velocity. In reality, the mass wouldn't stay at the same angular velocity as it moved in or out, unless the rod was being driven from the pole or its tip.

    Let's see what actually happens if Dexter moves the mass in or out in or out, and the mass is not being driven. If the mass is moving at a linear velocity of six meters per second when it is one meter from the pole, and it is moved to a half meter from the pole, assuming no friction or other forces on the rod or mass, the law of conservation of momentum ensures that the mass will continue to move at six meters per second.

    Since the radius of the circle has been cut in half, so has its circumference, and the mass now revolves around the pole at twice the angular rate that it did when it was a meter from the pole. Moving the mass inward causes the angular velocity to increase.

    Similarly, if the mass moves out along the rod until it is two meters from the pole, in the absence of any external forces it is still moving at six meters per second, but the circle is twice as big. The angular velocity is now cut in half. Moving the mass outward causes the angular velocity to decrease.

  • This is an example of the principle of the "conservation of angular
    • momentum", or Newton's first law of motion applied to rotational motion. There's no new fundamental principle involved, it's just the implications of the conservation of linear momentum applied to a rotational system, but using the same basic rules in a different games gives new results.

    There are many practical examples of the law of conservation of angular momentum. The most popular is to consider Didi on skates going into a spin. As she goes into the spin, she leaves her arms spread out wide, and the spin is slow, but as she draws her arms in to herself, she spins faster and faster. When she spreads her arms out wide again, the spin slows down.

    A more cosmic example is the formation of a "neutron star". A neutron star is a very small but dense star, the size of a mountain but as massive as a star like the Sun. It is created as a "cinder" left over from a "supernova", an explosive collapse of a star much bigger than the Sun. This explosion blasts off the outer layers of the star, while the star's interior falls inward to create the neutron star.

    As the interior falls inward, conservation of angular momentum causes the rate of rotation to increase, and after the explosion the neutron star is spinning very fast, much faster than the oversized star that gave it birth. Young neutron stars often have "hot spots" that generate radio noise, and so they emit radio pulses as they spin, bringing the hot spot in and out of view. Such a "pulsing" neutron star is known as a "pulsar".

    A spinning neutron star loses angular momentum through indirect forces over time, slowing down its spin. Astronomers can roughly estimate the age of a pulsar by its spin.

  • Another example of the conservation of angular momentum is known as the
    • "Coriolis effect". If Dexter stands at the North Pole, properly dressed for the weather, and points a finger straight out in front of him, his fingertip will trace out a full circle in 24 hours. Now he goes to the equator, discards his winter gear, and hold his finger straight up over his head, it will also trace out a full circle in 24 hours.

    However, at the Pole, the actual distance traveled by his finger in 24 hours is a little over six meters, while at the equator it's about: 

        PI * 12,750  =  40,055 kilometers
    The angular velocity is the same in both places, but the actual linear speed along the circular path is very much greater at the equator, since the distance between Dexter and the Earth's axis of rotation is very much greater as well.

    Similarly, suppose Dexter fires a cannon at a high northern latitude pointed due south, so that the cannonball arcs high through the sky towards the equator.

    The cannon is moving from west to east along with the rotation of the Earth, and of course the cannonball has the same motion once it's fired. This isn't noticeable at short range, since everything in the vicinity is moving from west to east at the same rate. Farther south, however, the distance of the cannonball from the Earth's axis of rotation will increase, and the linear velocity of the Earth' surface underneath the cannonball will increase accordingly.

    The cannonball will start to lag behind the Earth's rotation, curving to the west. This is the Coriolis effect. It has to be calculated for missile and spacecraft launches, and is particularly important in weather studies, as it has a definite effect on air or cloud masses moving north or south. One of the interesting subtle results of the Coriolis force is that large cyclonic storms rotate counterclockwise in the northern hemisphere and clockwise in the southern hemisphere.

  • The discussion of rotational motion so far has been as simplified as
    • possible. While it's about as much as anyone needs to know if they're not going on to engineering studies, it's worthwhile to point out some of the complications to prevent things from appearing simpler than they really are.

    One of the nonintuitive aspects of angular momentum is that is it a vector quantity, just as linear momentum is. The tricky question is: what is the direction of the angular momentum vector? If a mass is spinning on a cord, then its direction of its motion changes continuously, covering every possible direction in its plane of revolution.

    The only direction left is the axis of rotation. There's the second question of whether the direction of angular momentum of a object spinning in the horizontal plane is up or down, but the convention followed is a "right hand rule". This means that Dexter grasps a rotating shaft with his right hand so that his fingers point in the direction of the rotation, then sticks his thumb out, his thumb gives the direction of the angular momentum vector. The direction of the angular momentum vector of a platter spinning counterclockwise on a table is up. Spin the platter clockwise, and the direction is down.

    One of the most popular demonstrations of the vector nature of conservation of angular momentum is an experiment often performed in basic physics classes. Didi stands on a platform that is free to rotate and holds a bicycle tire with handgrips on an axle in front of her, with the tire in the vertical plane. Dexter uses a power tool of some sort to set the tire spinning rapidly.

    As long as Didi holds the tire vertical, nothing happens. If she tries to tip it to one side, she starts to spin on the platform in the opposite direction to the rotation of the tire. If she manages to get the tire completely horizontal, she and the platform spin at an angular velocity proportional to the ratio of the moment of inertia of the tire and the moment of inertia of Didi and the platform.

  • The vector conservation of angular momentum is also illustrated by a
    • spinning top or gyroscope, which tends to remain upright no matter where it is moved. Gyroscopes spinning on various sorts of low-friction bearings have long been used for autopilots and "inertial navigation" systems, since they provided a reference to establish the motion of an aircraft or other vehicle.

    Gyroscopes have been largely replaced in such applications by more sophisticated systems using light beams focused in a loop, or sets of micro-machined acceleration sensors. Details of such technologies is a topic for another document.

    Gyroscopic action is often used as a stabilization technique. Firearms and artillery generally have "rifling", or spiral grooves in the barrel, to set a projectile spinning when fired. This also helps average out the unbalanced effects of wind resistance due to imperfections in the shape of the projectile.

    Similarly, spacecraft generally have to maintain a specific orientation in space to perform their mission, and so them are often set to spinning to stabilize them. Such "spin stabilized" spacecraft will have a communications dish antenna mounted on the spin axis and point the spin axis back to Earth so that constant communications can be maintained.

    However, they also may need to keep some of their equipment, such as telescopes or communications dishes, pointed at a specific target, and so such gear is mounted on a section of the spacecraft that is mechanically spun to cancel out the effects of the spacecraft's spin. Electrically connecting this confusingly-named "despun" section to the rest of the spacecraft is a tricky matter, requiring devices known as "rotary couplers", but that's another story.

    Most modern spacecraft use what is known as "three axis stabilization" to maintain orientation without spinning. Although three axis stabilized spacecraft use little rocket thrusters to maintain their orientation, they may also use a set of "reaction flywheels" set in the X, Y, and Z directions to maintain orientation, adjusting the speed of a specific flywheel to transfer angular momentum and change the spacecraft's orientation slightly.

  • Tops and gyroscopes illustrate one of the most counterintuitive features of
    • vector conservation of angular momentum. Any force that tends to tip the top or gyro to one side sets up a torque that actually moves the top or gyro at a right angle.

    This is why a top starts going in lazy circles as it burns off its rotational energy through friction. Gravity is trying to pull it over, but the result is to cause the top to "precess" in circles that get wider and wider until the top falls over. A detailed analysis of precession is way beyond the scope of this document.

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    From Vectorsite.net.

    by Greg Goebel