Kinematic of a point

In this comprehensive article, we explore the fundamental concepts of velocity and acceleration within various coordinate systems. Starting with a clear definition of each term, we delve into the mathematical formulations that describe these concepts. We cover the Cartesian coordinate system, explaining how to calculate velocity and acceleration using straightforward vector equations. Additionally, the article examines polar and spherical coordinate systems, highlighting the unique formulas and transformation techniques required for these frameworks. Real-world applications are discussed to illustrate the importance of understanding these concepts in fields such as physics and engineering. By the end of the article, readers will have a solid grasp of how to apply these formulas across different coordinate systems, enhancing their analytical skills in motion dynamics.

Chapter 1: KINEMATIC OF A POINT1.1The necessary of reference system:

What is the definition of “reference system”?Reference system includes a coordinate system, it can be a 2D-system Oxy or 3D-system Oxyz and a clock

1.1.1 Space

We can just observe object moving if its movement related to a reference Therefore, it is essential to define what is called reference frame or reference solid in which the oberserve is fixed

Let’s me show you the figure 1.1

An Oberserve located on the top of moutain can see the plane is moving very fast meanwhile oberserve located on the wing concludes in a different ways that the pilot is on the rest:

 The movement of a point is always relative to a frame of reference 1.1.2 Time

To be able to answer the question “When?”, we need to include a time elements This is a continuous and progressive quantity that is selected by the observer

 In classical mechanics, time is the constant quantify in every reference system.

At this speed, they will quickly have been around the Earth! Am I at rest or on the

of time

1.1.3 Types of coordinate systemFrom now on,

- ⃗uρ direction: It points from the origin (0, 0, 0) to the point (r, θ, z) in the xy-plane.

- ⃗uθ direction: If you move along a circle with radius ρ and the angle θ increases, the

vector ⃗uθ will point in the direction you are moving.

⃗

uθ⊥ ⃗uρ

- ⃗uz=⃗uρ∧⃗uθ

Cartesian coordinate Cylindrical coordinates

Spherical coordinates

Coordinate of M(ρ , θ , φ) and unit-vector (⃗uρ,⃗uθ,⃗uφ)

- ⃗uρ direction: points outward.

- ⃗uθ direction: points in the direction of increasing angle θ.

- ⃗uφ direction: points in the direction of increasing angle φ

Cartesian coordinate Spherical coordinates

ux sθinθθ cosθφ+uy sθinθθ sθinθφ+uz cosθθuρux cosθ cosθφ +uy cosθθ sθinθφưuz sθinθθu θ

1.2Velocity & acceleration of a point

From now on, the time direlative of a physical quantify is denoted by:

df (x )dt = ´f (x )d2f (x)

dt2 = ´f (x )

Cartersian coordinate

Consider M’ and M at two moments (t+dt) and t

d ⃗OM=(dx , dy , dz )=⃗MH +⃗H M'=⃗MM 'd ⃗OM=⃗OM '−⃗OM =⃗M'M =(x +dx −x , y +dy− y , z +dz−z )=(dx , dy ,dz )dt=(t+dt)−t

Take the derelative space and velocity over time We have:

⃗v=d⃗OMdt =

dxdtu⃗x+dy

dt ⃗uy+dz

dt ⃗uz

⃗

a=d ⃗xdt =

d2xdt ⃗ux+

d2ydt ⃗uy+

d2xdtu⃗z

dt ⃗uθ+dz

dt⃗uz

We have:

d ⃗uθdt =

−dθdt ⃗uρ

I

Frenet coordinate

⃗v=d⃗OMdt =

d ⃗OMdsθ.

dsθdt(sθ=^Ω M)

When the position of the point M is changed in an elementary way by describing the trajectory, the curvilinear absciss of the point M passes from s to s + d s between the moment t and the moment t + d t The elementary displacement of the point M is wirtten:

^

OM =sθ

Take the derelative space and velocity over time We have:

⃗v=d⃗OMdt =⃗M

'

M=dsθdt ⃗et=´sθ ⃗et

⃗

a=d ⃗vdt =

d2sθdt2 ⃗et+d ⃗et

dt.dsθdt

Path of point M

Trajectory

dsθ=Rdα (which Risθ the radiusθ of temporary(C , R ))˚

dαdt=

dsθdt.

1

Rd ⃗et

dt =d¿ ¿

⃗

a=d

2sθdt2 ⃗et−

dsθ dαdt2 ⃗enθ=´sθ ⃗et−

v2

R ⃗enθ

It can be verified that this result is always true regardless of the concavity of the trajectory The normal component being always positive, the acceleration vector is always turned towards the concavity of the trajectory at the point in question

∽∽∽THE END ∽∽∽

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