or 6.28 radians. Thus, if a point were to make one complete revolution in one second it would be said to be going at 6.28 radian per second.
There are major differences and similarities between translational forces and rotational forces. Both have work, power, impulse, momentum, and kinetic energy relationships. However, rotational forces have some unique properties.
Angular velocity is generally given in radians per second. It represents the speed of a point which is rotating about a reference or center. The moving point describes a locus (group) of points which describes a circle. A complete revolution is 2 or 6.28 radians. Thus, if a point were to make one complete revolution in one second it would be said to be going at 6.28 radian per second.
An angle in radians is defined as the length of the arc subtended between one point and another, and the radius or distance between the reference point and moving point.
T = 
= s / R
is an angle in radians.
s is the distance of the arc.
R is the distance of the radius.
Angular velocity 
is the ratio of the change in angle by the change in time.
Angular acceleration is the ratio of the change in angular velocity by the change in time.
Angular velocity and angular acceleration are analogous to linear velocity and acceleration.
The angular velocity vector w however, is not quite the same as linear velocity vector. The thing about a vector is it must point in a straight line, but, the moving point is not going in a straight line, rather it is turning. After, much study it was found that the angular velocity vector is really pointing in the direction as the axis of rotation. The right hand rule applies where the thumb is pointing in the direction of the vector and the curl of the fingers is in the direction of rotation. The magnitude of angular velocity vector is the angular velocity.
The angular acceleration vector is the same as the angular velocity vector in that it points in the same direction and that its magnitude is the angular acceleration.
The next interesting aspect of a rotation body is its moment of inertia. This is analogous to mass. The reason it is analogous to mass is because it is derived from linear form of mass. Basically, all the mass points in a rotating body follow the same mass acceleration laws as it would linearly. The development starts by finding the tangential velocity of mass at a distance r from the center of rotation. That particular rotating mass's tangential velocity is going to be affected by forces identically the same as if it were a linear body. Since the tangential velocity is directly proportional to its distance from the center the force required to change the velocity, is also going to be proportional to the distance from the center. The effective mass of the rotating body can be found by summing up all the mass times radius squared points.
Consequently, just like force is equal to mass times acceleration, the required force to change the angular velocity is equal to the moment of inertia times the angular acceleration. This is called torque.
One can relate work, power, impulse, momentum and kinetic energy between translational and rotating bodies too.
The angular momentum vector of any body is the vector product of vector radius through mass times vector tangential velocity.
This follows the 90 degree rule of vector cross products. Thus, the direction of the angular momentum vector is through the axis of rotation which is 90 degrees from the radius and 90 degrees from the tangential velocity.
An extremely interesting property occurs when the angular momentum vector is put under a torque. That is the attempt to twist the angular momentum vector.
The concept of momentum is the ability of a mass to continue in the direction of travel unless acted upon by an external force. In other words, it is the property which resists change.
Angular momentum is no different.
A simple example; get a bike wheel and hook two handles to the axle. Spin the wheel very fast. Move the wheel in and out or up and down. The wheel moves relatively easily. Now attempt to twist the wheel, that is, twist the axis rotation. A resistance to the twisting occurs. One very easily can feel it.
The greatest of all uses for this phenomenon is pitching a mean curve. There is nothing so mean as chucking a baseball directly at the batters face, and having it curve for a strike. Watching the batters eyes is very thrilling. Of course, control is a problem, there is another name for the pitch if the spin isn't right, and that is a bean ball. Bean ball's are also mean, it's just they don't curve correctly.
A base ball with the proper spin will curve all the time exactly the same way. The spin vector reacts with the velocity vectors to cause a third force vector. The third force vector causes the curve. The rate and direction of curving depends upon the angle of spin and angular velocity. To simplify this explanation, our pitcher will spin the ball about the y axis. Almost all hard baseballs have a moment of inertia factor of .005. Say for a better than typical pitcher, the forward velocity is 120 ft/sec, and angular velocity is 40 revolutions per second slightly up and to the side. Thus, at the start of the pitch:
.
Total acceleration a = 
(S X V) - gj
a = (.005) (40)[(3)i-(120)k] - 32 j = 0.6i - 24k - 32j Starting a X0 = 0, Y0 = 5 ft, Z0 = 0.
Integrating twice to get an acceleration in t^2 and adding back the original velocity vector in t and the starting position; the new positional equation is:
P = (120 t + .3t^2)i + ( 5 + 4t - 16t^2)j + (3t - 12t^2)k
Here the k vector component has an acceleration. Although small, the acceleration component is interesting in itself. Make an off center spin and the problem becomes very complex which yields a screw ball because of the gravity vector not being in line with the spin vector. Wind resistance will slow down the spin some, but it will drastically affect the forward speed. Slowing down the forward speed would be a function which will mess up these equations. Furthermore, the ball's stitches grab the air too.
Golfers have the same problem. Many other common devices which kids play with operate with a rotational vector. One such device is the frisbee. The rotational vectors of a frisbee cause the device to fly. If a rotational vector were not present, the frisbee would not work. Try throwing a frisbee without spinning it. A boomerang is another device which works on the same principle of rotational vectors.
Lots of devices work due to spinning. Bicycles require the rotational vector as stability. Have you ever tried to maintain balance on a stationary bike? Bullets from guns operate better when the projectile is spinning. So, rotational vectors are a very important part of our life. This rotational component can be very strong. In the case of gyroscopes it actually will overcome some what the force of gravity. An example of this is a spinning top or gyroscope. If the top or gyroscope is at an angle with respect to its pivotal point one may expect the device to topple. But, it doesn't. It stays lopsided and keeps spinning.
At this point there is another force applied to the top or gyroscope and that is the force of gravity. This force is attempting to twist the angular momentum axis and creates the condition of another vector product.
Thus, another torque is developed. This torque gives the system a secondary velocity which is called precession.
Vector 
L is the change in angular momentum produced in time t by the moment â of 's force. t is a small interval of time. In this small interval of time L = t. The net result is a torque in the direction of L. This torque has the same affect as the weight only in a different direction. In turn, a new torque is produced which affects the angle of with respect to L. This causes an up and down oscillating motion.
Additional rotational forces are created which really makes the system very complex. As the system processes another cross product is introduced. In a top, this product of the system attempts to seek equilibrium, because as the force of gravity pulls against the original momentum vector, the angle changes. The system wants to recover so, an upward force is created. This pushes the system up until overshoot occurs and the force reverses. Thus, the system actually oscillates up and down as it processes. The term for this is nutation. The Earth is under these forces. The period of precession caused by the force of the Sun and Moon on the Earth is said to be about 26,000 years. This is an interperted measured value. However, measured percession is erratic.
The liquid or plastic center of the Earth has angular momentum. The Earth crust has angular momentum. However, these angular momentums are not quite the same. The crust is solid and the inner Earth is not. This starts to have meaning when we go back to the egg theory. That is the Earth is analogous to an egg. It is just the opposite of the hard boiled egg and soft boiled experiment seen in most physics classes. The hard boiled experiment: Simply tell which egg is hard boiled and which is not. To perform the experiment, spin both eggs on a flat surface. One spins for a long time the other stops spinning very fast. The reason is the liquid mass is at rest while only the outer mass spins causing the egg to come to rest very fast. The hard boiled egg is entirely spinning and angular momentum of the hard mass keeps the egg spinning.
With the Earth the liquid-plastic interior is moving while the solid crust spins about the interior. The vector forces are different for liquids and solids. And, for the Earth, a force which affects the crust may not affect the interior. However, a force which affects the interior will eventually affect the crust.
Another change which could affect the spin velocity is the change in radius of a rotating mass. Ice skater knows what this is. As a mass contracts, the conservation of energy forces the velocity to increase and visa versa. As the skater brings in their limbs the mass radius decrease and the spinning velocity increases.
This concept can be applied to the Earth too. The change in radius to account for a one half millisecond change in a day can be calculated. Some really oversimplified assumptions must be made. They are that the Earth's mass is uniform through out the body and that expansion is uniform through out the body. The net angular momentum of the Earth before expansion or contraction must be equal to the net angular momentum after expansion or contraction. Total mass drops out as it is the same in both situations.
This is an increase of approximately 10 meters per 0.5 millisecond through the day. The Earth is not expanding that much regularly in a day, a 10 meter increase in radius would raise or lower the sea level by about the same distance, and this doesn't happen. A 60 millisecond change in Earth's radius would be very noticeable in the yearly change of the sea level. Thus, this can not be the reason the period of the day changes. This would become really noticeable over a 60 millisecond change which occurs every year.
Obviously, there is some effect on the Earth's rotational period due to cooling and expansion, but it is minimal compared to the large time change the Earth undergoes.
Further we can describe the energy of rotation, the kinetic energy, the change in kinetic energy and a minimum rotational energy.
The energy of rotation can be given in a quantum system as:
Applying a vibrational or wave energy to rotational energy we can determine the lowest rotational energy levels to maintain the angular frequency for most bodies.
In chemistry, we can determine the lowest rotational energy to maintain molecular consistency given a binding force. We can also find that there becomes a maximum rotational energy which a given molecule will withstand before it separates. Such is the case for the states of water.
Water has three very discrete states solid, liquid and gas. The state change requires fairly large quantum energy steps. For example; water will remain at 32 degrees as more energy is supplied to ice-liquid system, until all the ice is converted to liquid. In other words, the liquid does not increase in temperature independently of the solid. The same is true for liquid to gas. This concept can be explained by the rotational energy restricted by the binding energy
In the solar system there are several rotating systems which the Earth is associated. The spin of the Earth, the Earth's movement about the Sun, the Sun's rotation, the Sun rotating in the galaxy. Even the galaxy is rotating in the universe. These rotating forces are all susceptible to vector forces.
Work in a rotational system is important because it is what is required to speed up or slow down a rotating body. Work is given by:
This work formula is the one used to determine how much energy it requires to increase the Earth's rotation 0.5 milliseconds per day or 60 milliseconds per year. Similarly, it requires energy to be given up to slow the Earth thus work is negative .
Power is found at the point of application and is:
NOTES: Two easily determined errors in current idealogy can be shown mathematically.
The power stored within the Earth's mass-spin can not supply the energy required to drag the terawatt aberration of the solar wind. Since momentum would not account for the drag, inertia spin must. The fact is neither are sufficient.
Precession formulas, as stated, do not calculate out to 26 thousand years either. However, when one actually analyses the forces which are said to cause the Earth's precession via rotation, one finds that they don't exist. The common development for precession was by analyzing a spinning top. A top has a single fixed point on a gravitational plain. The Earth-Sun arrangement does not have a fixed point at one end. The only Torque variance in the Earth's spinning mass would be from the difference spinning mass pointed away from the Sun and the mass pointed toward the Sun. At ninety degree points it would be totally equal. Furthermore, the variance reverses itself every year thus, the total difference would be from the difference between the inertia of the two hemispheres relative to the difference in distance between perihelion and aphelion. When these two considerations are taken into account the precession time takes trillions of years. But, that's not all! Centripetal force is as real force as gravety. When considering the Earth's centripetal force negates the Sun's gravity, precession force using Newtonian physics and rotation becomes none existent.
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