Circular motion

Bidimensional generic motion[edit | edit source]

Considering a curvilinear trajectory on which we choose an origin and a way of movement:

Definition (Curvilinear abscissa (l))

We define curvilinear abscissa the portion of bend which connects with . If is among the positive , according to the chosen way of movement, the curvilinear abscissa is positive; otherwise if is among the negative , according to the chosen way of movement, the curvilinear abscissa is negative. The velocity of the point defines the concordance of symbol between the chosen way of movement and the way of movement of the point: positive velocity makes the point move according to the chosen way of movement, negative velocity makes the point move in the opposite way.

 
Ascissa curvilinea.jpg


Average velocity

is represented by the vector with the same track of and the same direction of motion. But we notice how this definition of average velocity gives very general information.

We use now the definition of derivative to calculate the instantaneous velocity (or just velocity):

whose vector is tangential to the trajectory in where we find the point at the instant considered.

If we apply a second time the derivative we find the instantaneous acceleration (or just acceleration):

whose vector is parallel to the banding ray in , and perpendicular to the vector .

Generic 3D motion[edit | edit source]

are versor and they are constant all over the space. is the position versor. Breaking it up on the axes x, y e z we find:

From the decomposition of we break up on the three axis :

We also know that:
and we find the equivalence:

We can also break up the acceleration vector :

We also know that:
as we have done for the velocity, we find that:

Circular motion[edit | edit source]

Definition (Circular motion)

We define circular motion the specific curvilinear motion in which the trajectory is a circumference of centre and ray

 
Definition (Curvilinear coordinate)

We define curvilinear coordinate (indicated with ) the oriented length of the arch

 


The equation of motion is of the form

If the circular motion is uniform, the speed is constant: and

As , we find:

we now impose ,

  • Decomposition of motion:

We now have to apply what said on the decomposition of and in the general curvilinear motion, and we find:

where is called centripetal acceleration

Definition (Centripetal acceleration)

We define centripetal acceleration, the acceleration which causes the bending of trajectory without modifying the absolute value of the speed ω

 

→ This is why we talk about of uniform motion even though there is an acceleration


  • Relation between v, ω, a

As we find that
.

As we find that so
We know that and we can find the acceleration as a function of velocity

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