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
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