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CHAPTER 12 KINEMATICS OF A PARTICLE (DEFINITION (MECHANICS: state of rest…
CHAPTER 12
KINEMATICS OF A PARTICLE
DEFINITION
position
displacement
velocity
acceleration
MECHANICS
: state of rest or motion of bodies subjected to the action of forces
STATICS
: the
equilibrium
of a body that is
either
at rest or moves
with
constant velocity
DYNAMICS
=
accelerated motion
of a body
KINEMATICS
: the geometric aspects of the motion :red_flag:
KINEMATICS of a particle along a rectilinear
= straight
line path
RECTILINEAR KINEMATICS
REMINDER
: a particle: mass = negligible size and shape (e.g., with
finite size
At any instant:
position
velocity
acceleration
Position
= vector quantiy that has magnitude and direction
a single coordinate axis s
origin O
distance s (meter)
Position as a Function of Time
:<3:
Displacement
= the change in its position: a vector quantity ! :red_flag:
either + or - values
NOT
the
distance traveled by a particle
= a positive scalar: total length of
path over which the particle travels
Velocity
average velocity
instantaneous velocity
:
OR
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As a function of time = integration of ac = dv/dt
Velocity as a Function of Position
(
thieu ds)
:
Acceleration
average acceleration
AND instantaneous acceleration:
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Constant Acceleration: a = ac
Integration of ac = dv/dt, v = ds/dt, and ac ds = v dv to obtain formulas that relates ac, v, s, and t
Deceleration
: A particle that is slowing down
ERRATIC MOTION
functions of velocity, position, and acceleration
cannot be
a single continuous mathematical function
THEREFORE, graphs should be used such as s-t, v-t, a-t graphs
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KINEMATICS: CURVILINEAR MOTION
: a particle moves along a curved path
3 D
vector analysis
position
:
r
: position vector
velocity
average velocity
instantaneous velocity
speed
average acceleration
with
Hodograph = red line
v is always tangent to
the path and a is always tangent to the hodograph :red_flag:
instantaneous acceleration
OR
displacement
during a small time interval delta t:
delta r: displacement = change in the particle’s position
KINETICS
: the forces causing the motion = relationship between the change in motion of a body and the forces that cause this change :red_flag:
particle motion along a straight line
graphical presentation
particle motion along a curved path
dependent motion of 2 particles
relative motion of two particles
If the particle is at point (x, y, z) on the curved path s, then its location is defined by the position vector
while x = x(t), y = y(t), z = z(t)
so
r
=
r
(t)
with
v
elocity
OR
with
and
=
Acceleration
Motion of a Projectile
Horizontal Motion
Since a x = 0
with
ac . t2
with
with
Vertical Motion
DEPENDENT MOTION OF 2 PARTICLES
position coordinates sA and sB (notice the positive sense of each coordinates) :!:
fixed point O
l CD: length of the cord passing over arc CD
lt: total cord length
sA and sB: the segments of the cord that change in length
SO
OR
what does it mean?
:question:
how do we determine sA and sB?
(1) have their origin at fixed
points or datums
(2) are measured in the direction of motion of each
block
(3) has a positive sense from C to A and D to B
For another fixed point of sB
Procedure for Analysis
use algebraic scalars or
position coordinates of
rectilinear path
changes in magnitudes of velocity and acceleration but not their line of direction
How to do it?
STEP 1
Establish each position coordinate with O point = fixed point or datum
coordinate axis ~ path of motion
what is a datum?
a datum refers to the references from which the measurements are made on the part. This can be a point, a line or a plane or even features.
STEP 2
relate sA, sB to l T or l, l CD (arc segments wrapped over pulleys = do not change length)
STEP 3
Time Derivatives of position coordinates for the required velocity and acceleration equations
