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# Revolving

Revolving of the material particle

Uniform circular motion

The particle that is moving at constant speed on the circular path, performs the uniform circular motion. The particle passes in the same periods of time the same part of the circumference, changing all the time the direction of movement. The speed is so  constant and the direction is changed. According to our earlier definitions such motion is in fact accelerated, the acceleration is present all the time that changes the direction of the moving. The position of the particle in the case of such circular motion can be presented through radius-vector, in other words vector that connects the center of the circle with the position of the particle on the circle, and is directed from the center outside. In the case of uniform circular motion radius-vector in the same periods of time circumscribe the same angles. We can so see that we have to introduce two speeds when describing the revolving:

-         translatory speed v, given with the circular arc that particle passes in the given unit of time

-         angular speed (omega), given with the angle that is circumscribed by the radius-vector of the particle in the given unit of time

While observing the uniform circular motion it is seen that it is in fact the accelerated motion, as the speed although constant, all the time changes the direction. So, there must be an acceleration that changes the direction of the speed and at the same time does not change the speed itself. It is easily seen that the acceleration changes the direction of the motion and does not change the speed itself. It is seen that the given acceleration has the radial direction and it can be expressed like this: a= v2/r=r*w2.

While observing the general case of the rotation, in other words, motion on closed curve, we can conclude that the speed can change it's direction and size. In that case acceleration is divided into two components:

-         radial component in the direction radius - vector

-         tangential component in the direction of the tangent to the direction of the motion

The radial component of acceleration changes the direction, but not the size of the speed. In case of the circular motion it points to the center of the revolving. For any kind of motion along the curve the radial component directs into the center of the curvature. The value of the radial component v2/r, where v is the present translatory speed and r is the present radius of the curvature.

The tangential component of acceleration changes it's value without changing it's direction (speed in any point of path has the direction of tangent). The value of this component will be reached if we observe the temporal change in the value of speed. The total value of acceleration is (page 85).

# The dynamics of revolving

The centripetal force

The force that pulls the material particle towards the center we call the centripetal force. The centripetal force is the consequence of linking the object for the ax or action of gravitation. It is, of course, not any special type of the force. In the inertial rotation system, only centripetal force appears as the force that pulls the body inside. That force is responsible for the revolving as it gives the radial acceleration, which curves the path. It is often talked about the centrifugal force. Centrifugal force does not exist in the inertial reference system, in other words in the system that is still or is an object of uniform motion along the line.

The centrifugal force

Centrifugal force appears only in the uninertial system which rotates. It is an imagined force, which is not present in the still inertial system and which is added to the uninertial system to preserve the value 2. of the Newton Law. F x F = ma, where the inertial force is F=-ma.

Translation and rotation

The rigid body is defined as body with which the distance among the particles can not be changed. In nature do not exist the ideal rigid bodies, as all the bodies can be deformed under the influence of the strong outer forces. But, in practice, the rigid bodies are concerned to be the things made of metal or wood. The describing of the change in the location of those bodies  is in fact far more complicated, from describing the motion of the material particle. All the motions of the rigid bodies can be described as superposition of the translatory and circular motion. The pure translatory motion is the motion at which each material particle has the same momentary speed; the paths of all particles are parallel and congruent curves. The change in position, when translatory motion is concerned is given in three coordinates x,y,z, which show the motion of any material particle of the rigid body.

The pure circular motion is the motion where each material particle makes a circle around one line. That line is called the ax of rotation. In that case all the radius-vectors cover in the same time the same angles from the ax to the each particle. The circular motion is also defined with the tri angles shifted to the axes x, y and z. According to this, each change of the position of the rigid body  is defined with three translatory and tri rotary coordinates, that all together makes six, so called "grades of freedom".

The moment of the force

If the rigid body rotates around the fixed ax and it is under operation of the outer force that changes it's rotation speed. If we decompose this force on the component in the direction of the rotation and the component vertical to it, it is obvious that only vertical component will have influence on the change of the rotation speed, while the component in the direction of the ax of rotation will have no influence on the rotation itself. The change of the angular speed under the influence of the vertical component depends on the moment of force (page 94).The moment of force is always vertical on the force and vector of position. The vertical component of the force is: F=Fsin(delta). The value of moment of the force is : M= r*F*sin (delta).

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