CHAPTER 12
KINEMATICS OF A PARTICLE
DEFINITION
position
displacement
velocity
acceleration
particle motion along a straight line
graphical presentation
particle motion along a curved path
dependent motion of 2 particles
relative motion of two particles
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 🚩
KINETICS: the forces causing the motion = relationship between the change in motion of a body and the forces that cause this change 🚩
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)
Displacement = the change in its position: a vector quantity ! 🚩 
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
speed = magnitude of velocity (m/s)
AND average speed is :
S T: total distance that a particle travelled
delta t: elapsed time
Acceleration
average acceleration
AND instantaneous acceleration:
a (m^2/s)
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
As a function of time = integration of ac = dv/dt
Position as a Function of Time
❤
Velocity as a Function of Position
(thieu ds):
Deceleration: A particle that is slowing down
derived from a = dv/dt and v=ds/dt by eliminating dt
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
s-t graphs
WITH
v-t graph
WITH
SO
if a-t graph is given, then delta v can be determined. E.G.
WITH: v = v° + delta v
if v-t graph is given, then delta s can be determined. E.G.
if a-s graph is given, then v can be determined:
SO
if v-s graph is given, then a can be determined
IN GENERAL
THEN
KINEMATICS: CURVILINEAR MOTION: a particle moves along a curved path
- 3 D
- vector analysis
position
velocity
average acceleration
with
Hodograph = red line
v is always tangent to
the path and a is always tangent to the hodograph 🚩
:
r: position vector
displacement
during a small time interval delta t:
delta r: displacement = change in the particle’s position
average velocity
instantaneous velocity
speed
instantaneous acceleration
OR
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? ❓
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