Study of Some Particular Movements: Rectilinear and in-Planar Movements

Rectilinear Movement

A movement is rectilinear if the trajectory[1] followed by the mobile is a straight line. The position M of the mobile is identified by the position vector[2]

  • Position Vector

OM ⃗ = x ( t ) i ⃗ widevec {OM} = x( t ) vec i

If the movement is linear along Ox

  • Instantaneous velocity vector

V ⃗ ( t ) = d OM ⃗ dt = dx dt i ⃗ vec V ( t ) = {d widevec {OM} } over {dt} = {dx} over {dt} vec i

  • Instantaneous acceleration vector

a ⃗ ( t ) = d V ⃗ dt = d 2 x dt 2 i ⃗ vec a ( t ) = {d widevec {V} } over {dt} = { d^{2}x} over { dt^{2}} vec i

Fundamental

We have two types of rectilinear motion: uniform rectilinear motion and uniformly varied rectilinear motion

Uniform rectilinear movement: uniform rectilinear motion is characterized by a constant speed and therefore the acceleration is zero.

a ⃗ = d V ⃗ dt = 0 vec a = {d widevec {V} } over {dt} = 0

Uniformly varied rectilinear movement: uniformly varied rectilinear motion is characterized by constant acceleration.

a ⃗ = d V ⃗ dt = cst vec a = {d widevec {V} } over {dt} = cst

Note

If we have:

a ⃗ . v ⃗ > 0 vec a . vec v > 0 ⇒ the movement is uniformly accelerated

a ⃗ . v ⃗ < 0 vec a . vec v < 0 ⇒ the movement is uniformly delayed

Planar Movement

If the trajectory of the mobile is in planar, we study the movement following the Cartesian coordinates ( O , i ⃗ , j ⃗ O, vec i, vec j ), the polar ( O , U ρ ⃗ , U θ ⃗ O, vec U _{ %rho }, vec U _{ %theta } ) or curvilinear coordinates ( U T ⃗ , U N ⃗ ) ( vec U _{T} , vec U _{N}) , in this section we choose the curvilinear coordinates.

Curvilinear movement is characterized by a curvilinear trajectory which requires knowledge of the radius of curvature R and the center C.

Definition

The curvilinear coordinates is a base connected to the mobile in curvilinear motion. It is defined by the orthonormal base ( U T ⃗ , U N ⃗ ) ( vec U _{T} , vec U _{N}) such as

U T ⃗ vec U _{T} : is a unit vector tangential to the trajectory and in the direction of movement

U N ⃗ vec U _{N} : is perpendicular to the vector U T ⃗ vec U _{T} , and it is directed towards the center of the curvature of the trajectory (the concavity of the trajectory).

Method

  • Position vector

The position of the mobile is determined by the curvilinear abscissa S such that

S ( t ) = R θ ( t ) S ( t )= R %theta ( t )

S is the length of the arc between the two points M(1) and M(2)

  • Instantaneous velocity vector

v ⃗ = v U T ⃗ = dS dt U T ⃗ vec v=v vec U _{T} = {dS} over {dt} vec U _{T}

  • Instantaneous acceleration vector

a ⃗ = dv dt U T ⃗ + v θ ˙ U N ⃗ vec a= {dv} over {dt} vec U _{T}+v dot %theta vec U _{N}

a ⃗ = dv dt U T ⃗ + v 2 R U N ⃗ = a T U T ⃗ + a N U N ⃗ vec a= {dv} over {dt} vec U _{T}+ { v^{2}} over {R} vec U _{N}= a_{T} vec U _{T}+ a_{N} vec U _{N}