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Uniform circular motion

a) uniform circular motion

DEF: The motion of a body that is a circle (a circle) with speed (in form) is said constant uniform circular motion. When we refer only to the intensity of the speed we are talking about speed climb.

ATTENTION: The speed is a vector, it is characterized by intensity, direction and orientation.
in uniform circular motion is the intensity of speed to be constant, the direction and to change the time!
NOTE: For each type of trajectory, the velocity vector is always tangent to the trajectory.
Since the form of speed is constant, one might be tempted to consider a motion is not accelerated. But we must remember the definition of acceleration (it is also a carrier!) And note that the difference of two vectors with the same form is not 0. The fact that the speed changes of direction, even if it does not change in intensity, uniform circular motion is therefore an accelerated motion.
For the second principle of dynamics, if the motion is accelerated, then this is a force.

Some important parameters relating to the uniform circular motion are:

1)

PERIOD The period is the time it takes to make a full circle. It is measured in SI (International System) in seconds. It is usually indicated by a capital letter T.
We observe that the concept of time also applies to any motion that they be periodic but the characteristic of, or to "wipe" to the same point after a certain time.

2) FREQUENCY

The frequency indicates the number of revolutions made per unit time. In SI, the frequency is measured in hertz (Hz) and the number of revolutions per second. It is usually denoted by the lowercase Greek letter f or ν. The frequency features in general a periodic phenomenon qualunque.Fra the period and frequency there is a mathematical relationship important:

f = 1 / T;

ie the frequency is the inverse of the period.

3) SPEED 'SCALAR

The velocity scale of uniform circular motion is, as with all speed, measured by the ratio space / Time. If the radius of the circle is R, whereas the entire circumference measure 2 π R and that the total time to tour is the period T, then you will have:
v = s / t = 2 π R / T.
This is the formula of the scalar speed of rectilinear motion. It can also be expressed as a function of frequency taking into account that f = 1 / T. Then you get: v = 2 π R
f.
speed climb, of course, is measured in SI in m / s.

b) centripetal acceleration

The rectilinear motion with uniform acceleration because the direction of its velocity changes point by point. Let's see how this acceleration is calculated and its characteristics.
consider velocity vectors at points A and B respectively, and call them v1 and v2:
means for accelerating the change in velocity per unit time. Dv call with the change in velocity between points A and B for which we have:

v2 = v1 + dv

since the speed at point B is the speed at point A plus the change in speed (all three are vectors!).
For convenience, we report the carrier at point A by a parallel shift. We get :

Remember that the intensity of v1 and v2 are the same and to make the sum of two vectors, you must use the rule of the parallelogram.
We have obtained the vector change in velocity dv that is directed toward the center of the circle along which the motion takes place.
If we divide this vector for the time Dt in which the point is from A to B, we finally obtain the sought acceleration which is itself a carrier who has the same direction and orientation (since the time that we share is a positive number ) of the vector change in velocity dv.
The acceleration is then:

a = dv / dt

. . Please note that we have indicated the acceleration with the "subscript" c. This means that the acceleration "point" toward the center, and this is called centripetal acceleration .
NOTE: This acceleration, at a given point on the circumference, is exactly pointing to the center though, looking at the graph, this would seem true only approximately. In the chart, we took two points (A and B) "somewhat" distant for reasons of simplicity. If we take them "very close" (infinitely close), you would see that dv is directed exactly toward the center and there would then the instantaneous change in speed.

How much is the intensity of centripetal acceleration? Need to derive some basic knowledge of differential calculus, for which we give directly the result. The strength of the centripetal acceleration is: a

c = v ^ 2 / R

where v is the velocity scale of the motion and R the radius of the circle. Also note that here, speed and acceleration are staggered arrangements.
Note that the centripetal acceleration is directly proportional to the square of the velocity and inversely proportional to the radius. This means that if the speed is doubled, the acceleration quadruples and so on. If the radius doubles, the acceleration halved, if the radius by half, the acceleration doubles etc..

c) centripetal force

If a body moves with accelerated motion, it is because it suffers the action of a force (resultant). For the second law of dynamics, the relationship between force and acceleration is given by the formula: F = ma

m is the mass of a scalar, the force and acceleration vectors.
in uniform circular motion, then a force acts, the so-called centripetal force, which is due to the fact that the body along a circular path. If your body does not act force (resultant), the body moves in rectilinear motion (first law of motion ).
The centripetal force is then: F

c = ma c

and will be targeted as the centripetal acceleration, m is the mass of a positive number (multiplying a vector by a positive number, direction and orientation the carrier that you get do not change).

The intensity of the centripetal force will be:

the centripetal force for the same considerations of direct and inverse proportionality that we have done for the centripetal acceleration.