Kinematics of a Particle

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Dynamics

Transcript of Kinematics of a Particle

  • Kinematics of a ParticleKaiser Rex N. Pama

    Engineering Mechanics- Dynamics

  • Rectilinear Kinematics: Continuous Motion

    The Kinematics of a particle is characterised by specifying, at any given instant, the particles

    position, velocity, and acceleration.

  • Position and Displacement

    O ss

    Position

    O s

    Displacement

    s

    ss

    s=s'-s

  • Velocity

    vav =st

    Os

    sv

    v = dsdt+

    (vsp )avg =sT t

  • Accelerationaav =

    v t a =

    dvdt

    Os

    v v'

    a

    Os

    v v'

    aP P

    Deceleration

  • v = dsdt a =dvdt

    dt = dsvdt = dvadsv =

    dva

    ads = vdv

  • Constant accelerationVelocity as a function of time

    Position as a function of time

    Velocity as a function of position

    dvv0

    v

    = ac dt0

    t

    v = v0 + act

    dss0

    s

    = (v0 + act)dt0

    t

    s = s0 + v0t +12 act

    2

    dvv0

    v

    = ac dss0

    s

    v2 = v02 + 2ac(s s0 )

    dvv0

    v

    = ac dt0

    t

    v = v0 + act

    dss0

    s

    = (v0 + act)dt0

    t

    s = s0 + v0t +12 act

    2

    dvv0

    v

    = ac dss0

    s

    v2 = v02 + 2ac(s s0 )

    dvv0

    v

    = ac dt0

    t

    v = v0 + act

    dss0

    s

    = (v0 + act)dt0

    t

    s = s0 + v0t +12 act

    2

    dvv0

    v

    = ac dss0

    s

    v2 = v02 + 2ac(s s0 )

    a = ac

  • Rectilinear Kinematicsv = dxdt a =

    dvdt

  • Curvilinear MotionPosition

    The position is designated by r = r(t).

  • Displacement

    the displacement is r = r - r

  • Velocity

    Average velocity is: vavg = r/ t . Instantaneous velocity is: v = dr/dt . v is always tangent to the path

    What is speed v? s r as t0, then v = ds/dt.

  • Acceleration

    Average acceleration is: aavg = v/t = (v - v)/t

    Instantaneous acceleration is: a = dv/dt = d2r/dt2

  • Position

    The position can be defined as r = x i + y j + z k where x = x(t), y = y(t), and z = z(t) .

    Magnitude is: r = (x2 + y2 + z2)0.5 Direction is defined by the unit vector: ur = (1/r)r

    Curvilinear Motion: Rectangular Components

  • Velocity

  • Acceleration

  • Motion of a Projectile

  • Horizontal Motion

    Vertical Motion

  • Curvilinear Motion: Normal and Tangential Components

    Normal (n) and tangential (t) coordinates are used when a particle moves along a curved path and the path of motion is known

    n and t directions are defined by the unit vectors un and ut, respectively.

    Position

  • Radius of curvature, , is the perpendicular distance from the curve to the center of curvature at that point.

    The position is the distance, s, along the curve from a fixed reference point.

    Radius of

    Curvature

  • The velocity vector is always tangent to the path of motion (t-direction).

    The magnitude is determined by taking the time derivative of the path function, s(t).

    where

    Velocity

  • Acceleration is a = dv/dt = d(vut)/dt = vut + vut

    . .

    Here v represents the change in the magnitude of velocity and ut change in the direction of ut.

    ..

    .

    The acceleration vector can be expressed as:

    where

    or

    Acceleration

  • SPECIAL CASES OF MOTION

    There are some special cases of motion to consider.

    2) The particle moves along a curve at constant speed. at = v = 0 => a = an = v2/

    .

    The normal component represents the time rate of change in the direction of the velocity.

    1) The particle moves along a straight line. => an = v2/ = 0 => a = at = v

    .

    The tangential component represents the time rate of change in the magnitude of the velocity.

  • MOTION RELATIVE TO A FRAME IN TRANSLATION

    relative position of B with respect to A

  • relative velocity of B with respect to A

    relative acceleration of B with respect to A