KINEMATICS OF PARTICLES
___________________________________
CONTINUOUS AND ERRATIC MOTION
LECTURE #1
TODAY’S LECTURE
We will learn:
 We will be able to find the kinematic quantities of a
particle traveling along a straight path
 Introduce the concepts of position, displacement,
velocity, and acceleration.
 Relations between s(t), v(t),and a(t) for general
rectilinear motion
 Relations between s(t), v(t),and a(t) when acceleration
is
constant
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Introduction
1) Statics
-
Concerned with body at rest.
2) Dynamics
-
F
F
M
x
0
y
0
0
Concerned with body in motion
1. Kinematics: is a study the
geometric aspect of the motion.
2. Kinetics: Analysis of forces that
causing the motion
F
F
M
x
 ma x
y
 ma y
 I
3
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Introduction
 Kinematics: study of the geometry of motion.
Kinematics is used to relate displacement,
velocity, acceleration, and time without reference
to the cause of motion.
 Kinetics: study of the relations existing between
the forces acting on a body, the mass of the body,
and the motion of the body. Kinetics is used to
predict the motion caused by given forces or to
determine the forces required to produce a given
motion.
4
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Introduction
• A Vector quantity has both magnitude and direction.
• A Scalar quantity has magnitude only.
• Scalars (e.g)
• Vectors (e.g.)
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
distance
speed
mass
temperature
pure numbers
time
pressure
area, volume
charge
energy
displacement
velocity
acceleration
force
weight (force)
momentum
5
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Introduction
• Rectilinear :
• Kinematics :
Straight line motion
Study the geometry of the motion
dealing with s, v, a.
• Rectilinear Kinematics : To identify at any given
instant, the particle’s position, velocity, and
acceleration.
(All objects such as rockets, projectiles, or vehicles will
be considered as particles “has negligible size and
shape”
particles : has mass but negligible size and shape
6
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Rectilinear Kinematics: Continuous Motion
• Rectilinear motion: position, velocity, and
acceleration of a particle as it moves along a straight
line.
• Curvilinear motion: position, velocity, and
acceleration of a particle as it moves along a curved
line in two or three dimensions.
7
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Rectilinear Motion
• Particle moving along a straight line is said
to be in rectilinear motion.
• Position coordinate of a particle is defined by
positive or negative distance of particle from
a fixed origin on the line.
• The motion of a particle is known if the
position coordinate for particle is known for
every value of time t. Motion of the particle
may be expressed in the form of a function,
e.g.,
x  6t 2 - t 3
or in the form of a graph x vs. t.
8
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Position and Displacement
A particle travels along a straight-line path
defined by the coordinate axis s.
The position of the particle at any instant,
relative to the origin, O, is defined by the
position vector r, or the scalar s. Scalar s
can be positive or negative. Typical units
for r and s are meters (m) or feet (ft).
The displacement of the particle is
defined as its change in position.
Vector form: Δ r = r’ - r
Scalar form: Δ s = s’ - s
The total distance traveled by the particle, sT, is a positive scalar that
represents the total length of the path over which the particle travels.
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Position, Distance and Displacement
• Displacement : defined as
the change in position.
• r : Displacement ( 3 km )
• s : Distance
( 8 km )
Total length
N
River
City
My Place
X
3km
QUT
8 km
• For straight-line
Distance = Displacement
s
=
r
s =  r
Vector is direction oriented
 r positive (left )
 r negative (right)
10
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
• Distance
• Displacement
– Total length of path
travelled
– Must be greater than (or
equal to) magnitude of
displacement
– Only equal if path is
straight
– Symbol d
– Refers to the change in
particle’s position vector
– Direct distance
– Shortest distance
between two points
– Distance between Start
and End points
– “as the crow flies”
– Can be describe with only
one direction
11
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Speed and Velocity
Velocity is a measure of the rate of change in the position
of a particle. It is a vector quantity (it has both magnitude
and direction).
The magnitude of the velocity is called speed, with units of
m/s or ft/s.
12
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Speed and Velocity
• Velocity : Displacement per unit time
• Average velocity :
V=r /t
• Speed : Distance per unit time
• Average speed :
usp = sT /  t (Always positive scalar )
• Speed refers to the magnitude of velocity
• Average velocity :
uavg =  s /  t
Instantaneous velocity :
V  lim
t 0
For straight-line motion  r =  s
r
t

dr
dt
ds
v
dt
13
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Velocity
The average velocity of a particle during a
time interval Δt is
vavg = Δr/Δt = Δr/Δt i
The instantaneous velocity is the time-derivative of position.
v = dr/dt
Speed is the magnitude of velocity: v = ds/dt
Average speed is the total distance traveled divided by elapsed time:
(vsp)avg = sT/ Δ t
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Acceleration
•
Acceleration : The rate of change in velocity
{(m/s)/s}
V  V  - V
•
Average acceleration :
aavg 
•
V
t
Instantaneous acceleration :
v dv d 2 s
a  lim

 2
t 0 t
dt dt
• If v ‘ > v
• If v ‘ < v
“ Acceleration “
“ Deceleration”
15
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Acceleration
Acceleration is the rate of change in the velocity of a particle. It is a
vector quantity. Typical units are m/s2 or ft/s2.
The instantaneous acceleration is the time
derivative of velocity.
Vector form: a = dv/dt
Scalar form: a = dv/dt = d2s/dt2
Acceleration can be positive (speed
increasing) or negative (speed decreasing).
As the book indicates, the derivative equations for velocity and
acceleration can be manipulated to get
a ds = v dv
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Acceleration
• Consider particle with velocity v at time t and
v’ at t+t,
v
Instantaneous acceleration  a  lim
t 0 t
• Instantaneous acceleration may be:
- positive: increasing positive velocity
or decreasing negative velocity
- negative: decreasing positive velocity
or increasing negative velocity.
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Acceleration
• From the definition of a derivative,
v dv d 2 x
a  lim

 2
dt dt
t 0 t
e.g. v  12t - 3t 2
a
dv
 12 - 6t
dt
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Acceleration
• Consider particle with motion given by
x  6t 2 - t 3
dx
 12t - 3t 2
dt
dv d 2 x
a

 12 - 6t
dt dt 2
v
• at t = 0, x = 0, v = 0, a = 12 m/s2
• at t = 2 s, x = 16 m, v = vmax = 12 m/s, a = 0
• at t = 4 s, x = xmax = 32 m, v = 0, a = -12 m/s2
• at t = 6 s, x = 0, v = -36 m/s, a = 24 m/s2
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Constant Acceleration
The three kinematic equations can be integrated for the special case
when acceleration is constant (a = ac) to obtain very useful equations.
v
t
 dv   a dt
c
vo
o
s
t
 ds   v dt
so
yields
v  vo + act
yields
s  so + v ot + (1/2)a ct 2
o
v
s
vo
so
 v dv   ac ds yields
v2  (vo)2 + 2ac(s - so)
A common example of constant acceleration is gravity; i.e., a body freely
falling toward earth. In this case, ac = g = 9.81 m/s2 = 32.2 ft/s2 downward.
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Velocity as a Function of Time
dv
ac 
dt
dv  ac dt
v
t
 dv   a
vo
c
dt
0
v  v0 + a c t
21
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Position as a Function of Time
ds
v
 v0 + a c t
dt
s
t
 ds   (v
so
0
+ ac t ) dt
0
1 2
s  s 0 + v0 t + a c t
2
22
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Velocity as a Function of Position
v dv  ac ds
v
s
 v dv   a
v0
c
ds
s0
1 2 1 2
v - v0  a c ( s - s 0 )
2
2
v  v + 2 ac ( s - s0 )
2
2
0
23
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
SUMMARY
• Time dependent
acceleration
s (t )
ds
v
dt
2
dv d s
a
 2
dt dt
a ds  v dv
• Constant
acceleration
v  v0 + act
1 2
s  s 0 + v0 t + a c t
2
v  v + 2 ac ( s - s0 )
2
2
0
This applies to a freely falling object:
a  g  9.81 m / s 2  32.2 ft / s 2
24
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Acceleration
• Acceleration given as a function of time, a = f(t):
dv
 a  f t 
dt
dx
 vt 
dt
dv  f t  dt
t
v0
0
 dv   f t  dt
x t 
dx  vt  dt
v t 
vt  - v0   f t  dt
0
t
t
 dx   vt  dt
x0
t
xt  - x0   vt  dt
0
0
• Acceleration given as a function of position, a = f(x):
dx
dx
v
or dt 
dt
v
v dv  f  x dx
v x 
dv
dv
a
or a  v  f  x 
dt
dx
x
 v dv   f  x dx
v0
x0
1 v x 
2
2
- 12 v02

x
 f  x dx
x0
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Acceleration
• Acceleration given as a function of velocity, a = f(v):
dv
 a  f v 
dt
dv
 dt
f v 
v t 
v t 
t
dv
 f v    dt
v
0
0
dv
 f v   t
v
0
dv
v  a  f v 
dx
v t 
v dv
xt  - x0  
v f v 
0
v dv
dx 
f v 
x t 
v t 
v dv
v0 f v 
 dx  
x0
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Determination of the Motion of a Particle
• Recall, motion of a particle is known if position is known for all
time t.
• Typically, conditions of motion are specified by the type of
acceleration experienced by the particle. Determination of
velocity and position requires two successive integrations.
• Three classes of motion may be defined for:
- acceleration given as a function of time, a = f(t)
- acceleration given as a function of position, a = f(x)
- acceleration given as a function of velocity, a = f(v)
27
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Important Points
• Dynamics: Accelerated motion of bodies
• Kinematics: Geometry of motion
• Average speed and average velocity
• Rectilinear kinematics or straight-line motion
• Acceleration is negative when particle is
slowing down
• α ds = v dv; relation of acceleration, velocity,
displacement
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Rectilinear Kinematics: Erratic Motion
• Erratic (discontinuous) motion
• When a particle has erratic or changing motion then
its position, velocity and acceleration can not be
described by a single continuous mathematic
function along its entire path.
29
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Erratic Motion and Graphical Methods
• Graphing provides a good way to handle complex
motions that would be difficult to describe with
formulas.
• Graphs also provide a visual description of motion
and reinforce the calculus concepts of differentiation
and integration as used in dynamics
• The approach builds on the facts that slope and
differentiation are linked and that integration can be
thought of as finding the area under a curve
30
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Erratic Motion and Graphical Methods
• When particle’s motion is erratic, it is described
graphically by using a series of curves
• A graph is used to describe the relationship with any
two of variable: a, v, s, and t
31
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
S-t , v-t, & a-t Graphs
s-t graph →construct v-t
• Plots of position vs. time can be
used to find velocity vs. time
curves. Finding the slope of the
line tangent to the motion curve
at any point is the velocity at that
point
(or v = ds/dt)
• Therefore, the v-t graph can be
constructed by finding the slope at
various points along the s-t graph
32
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
S-t , v-t, & a-t Graphs
v-t graph →construct a-t
Plots of velocity vs. time can be used to
find acceleration vs. time curves. Finding
the slope of the line tangent to the
velocity curve at any point is the
acceleration at that point (or a = dv/dt)
Therefore, the a-t graph can be
constructed by finding the slope at
various points along the v-t graph
Also, the distance moved (displacement)
of the particle is the area under the v-t
graph during time Δt
33
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
S-t , v-t, & a-t Graphs
a-t graph →construct v-t
Given the a-t curve, the
change in velocity (Δ v)
during a time period is
the area under the a-t
curve.
So we can construct a v-t
graph from an a-t graph
if we know the initial
velocity of the particle
34
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
S-t , v-t, & a-t Graphs
v-t graph →construct s-t
We begin with initial
position S0 and add
algebraically increments Δs
determined from the
v-t graph
Equations described by vt
graphs may be
integrated in order to
yield equations that
describe segments of the
s-t graph
35
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
S-t , v-t, & a-t Graphs
36
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
a-s & v-s Graphs
• a-s graph → construct v-s graph
• A more complex case is presented by the a-s graph. The area
under the acceleration versus position curve represents the
change in velocity
• This equation can be solved for v1, allowing you to solve for
the velocity at a point. By doing this repeatedly, you can
create a plot of velocity versus distance
37
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
a-s & v-s Graphs
• v-s graph → construct a-s graph
Another complex case is presented by the v-s graph. By reading the
velocity v at a point on the curve and multiplying it by the slope of the
curve (dv/ds) at this same point, we can obtain the acceleration at
that point. a = v (dv/ds)
• Thus, we can obtain a plot of a vs. s from the v-s curve
38
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Analysing problems in dynamics
Coordinate system
• Establish a position coordinate S along the path and specify its
fixed origin and positive direction
• Motion is along a straight line and therefore s, v and α can be
represented as algebraic scalars
• Use an arrow alongside each kinematic equation in order to
indicate positive sense of each scalar
Kinematic equations
• If any two of α, v, s and t are related, then a third variable can be
obtained using one of the kinematic equations
• When performing integration, position and velocity must be
known at a given instant (…so the constants or limits can be
evaluated)
• Some equations must be used only when a is constant
KINEMATICS OF PARTICLES
Continuous and Erratic Motion
Problem Solving
1. Read the problem carefully (and read it again)
2. Physical situation and theory link
3. Draw diagrams and tabulate problem data
4. Coordinate system!!!
5. Solve equations and be careful with units
6. Be critical. A mass of an aeroplane can not be 50 g
7. Read the problem carefully
Explanation of Example 12.7 (A)
Explanation of Example 12.7 (A)
Explanation of Example 12.7 (B)
Groups think about this problem please
Given: The v-t graph shown
Find: The a-t graph, average
speed, and distance
traveled for the 30 s
interval
Hint
Find slopes of the curves and draw the a-t graph.
Find the area under the curve--that is the distance traveled.
Finally, calculate average speed (using basic definitions!)
Solution to the problem (A)
For 0 ≤ t ≤ 10
a = dv/dt = 0.8 t ft/s²
For 10 ≤ t ≤ 30
a = dv/dt = 1 ft/s²
a(ft/s²)
8
1
10
30
t(s)
Solution to the problem (B)
s0-10 =  v dt = (1/3) (0.4)(10)3 = 400/3 ft
s10-30 =  v dt = (0.5)(30)2 + 30(30) – 0.5(10)2 – 30(10)
= 1000 ft
s0-30 = 1000 + 400/3 = 1133.3 ft
vavg(0-30) = total distance / time
= 1133.3/30
= 37.78 ft/s
Try at home please (I)
What we have learned today?
 Concepts
such as position, displacement, velocity and
acceleration are introduced
 Study the motion of particles along a straight line.
Graphical representation
 Investigation of a particle motion along a curved path.
 Use of different coordinate systems
 Analysis of dependent motion of two particles
 Principles of relative motion of two particles.
 Use of translating axis
Next Lecture




- General curvilinear motion
- Curvilinear motion:
Rectangular components
(Cartesian coordinate)
QUESTIONS
THANK YOU
FOR YOUR
INTEREST
KINEMATICS OF PARTICLES
___________________________________
GENERAL CURVILINEAR MOTION &
RECTANGULAR COMPONENTS
LECTURE #1
TODAY’S LECTURE
Students will able to understand:
 The
motion of a particle traveling along a curved
path.
 Kinematic quantities in terms of the rectangular
components of the vectors.
 Particle motion along a curved path using
rectangular coordinate system
Kinematics of Particles
General Curvilinear Motion
Curvilinear Motion
• Path is described in three dimensions
• Position, velocity, and acceleration are vectors
56
Kinematics of Particles
General Curvilinear Motion
Applications
The path of motion of each plane in
this formation can be tracked with
radar and their x, y, and z coordinates
(relative to a point on earth) recorded
as a function of time.
How can we determine the velocity
or acceleration of each aircraft at any
instant?
Should they be the same for each
aircraft?
Kinematics of Particles
General Curvilinear Motion
Applications
A roller coaster car travels down
a fixed, helical path at a constant
speed.
How can we determine its
position or acceleration at any
instant?
If you are designing the track, why is it important to be
able to predict the acceleration of the car?
Kinematics of Particles
General Curvilinear Motion
Position and Displacement
A particle moving along a curved path undergoes curvilinear motion. Since
the motion is often three-dimensional, vectors are used to describe the
motion.
A particle moves along a curve defined by the
path function, s.
The position of the particle at any instant is
designated by the vector r = r(t). Both the
magnitude and direction of r may vary with time.
If the particle moves a distance s along the curve
during time interval t, the displacement is
determined by vector subtraction:  r = r’ - r
Kinematics of Particles
General Curvilinear Motion
Velocity
Velocity represents the rate of change in the position of a particle.
The average velocity of the particle during the
time increment t is
vavg = r/ t
The instantaneous velocity is the time-derivative
of position
v = dr/dt
The velocity vector, v, is always tangent to the
path of motion.
The magnitude of v is called the speed. Since the arc length  s
approaches the magnitude of r as t→0, the speed can be obtained by
differentiating the path function (v = ds/dt). Note that this is not a vector!
60
Kinematics of Particles
General Curvilinear Motion
Acceleration
Acceleration represents the rate of change in the velocity
of a particle.
If a particle’s velocity changes from v to v’ over a time
increment  t, the average acceleration during that
increment is:
aavg = v/ t = (v - v’)/  t
The instantaneous acceleration is the time-derivative of
velocity:
a = dv/dt = d2r/dt2
A plot of the locus of points defined by the arrowhead of the
velocity vector is called a hodograph.
The acceleration vector is tangent to the hodograph, but
not, in general, tangent to the path function.
61
Kinematics of Particles
General Curvilinear Motion
Acceleration
• Average acceleration:
aavg 
v
t
• Hodograph curve “velocity
arrowhead points”
v dv d 2 r
• Instantaneous acceleration: a  lim   2
dt dt
t 0 t
•
•
a acts tangent to the hodograph
a is not tangent to the path of
motion
• a directed toward the inside or
concave side
62
Kinematics of Particles
General Curvilinear Motion
Rectangular Components
Rectangular : x, y, z frame
63
Kinematics of Particles
General Curvilinear Motion
Rectangular Components - Position
It is often convenient to describe the motion of a particle in terms of its x, y, z or
rectangular components, relative to a fixed frame of reference.
The position of the particle can be
defined at any instant by the position
vector
r=xi+yj+zk .
The x, y, z components may all be
functions of time, i.e.,
x = x(t), y = y(t), and z = z(t) .
The magnitude of the position vector is:
r = (x2 + y2 + z2)0.5
The direction of r is defined by the unit
vector: ur = (1/r)r
64
Kinematics of Particles
General Curvilinear Motion
Rectangular Components - Velocity
The velocity vector is the time derivative of the position vector:
v=
dr/dt = d(xi)/dt + d(yj)/dt + d(zk)/dt
Since the unit vectors i, j, k are constant in magnitude and direction, this equation
reduces to
v = vxi + vyj + vzk
Where; vx = dx/dt, vy = dy/dt, vz = dz/dt
The magnitude of the velocity vector is
v = [(vx)2 + (vy)2 + (vz)2]0.5
The direction of v is tangent to the path of
motion.
65
Kinematics of Particles
General Curvilinear Motion
Rectangular Components - Acceleration
The acceleration vector is the time derivative of the
velocity vector (second derivative of the position vector):
a = dv/dt = d2r/dt2 = axi + ayj + azk
where
ax = dvx /dt, ay = dvy /dt, az =dvz /dt
The magnitude of the acceleration
vector is
a = [(ax)2 + (ay)2 + (az)2 ]0.5
The direction of a is usually not
tangent to the path of the particle.
66
Kinematics of Particles
General Curvilinear Motion
1.
Appendix C will help
you with vectors
2.
Kinematic equations used because rectilinear motion
occurs along each coordinate axis
Magnitudes of v and a for x,y,z vector components can
be found using Pythagorean theorem
Curvilinear motion can cause
changes in both magnitude and
direction of the position, velocity and
acceleration vectors
Use rectangular
coordinate system to
solve problems
By considering the component
motions, the direction of
motion of the particle is
automatically taken into
account
When using rectangular
coordinates, the components
along each of the axes do not
change direction.
Velocity vector is always
directed tangent to the path
In general the acceleration vector is not
tangent to the path, but rather, to the
hodograph
Only magnitude and algebraic
sign will change
67
KINEMATICS OF PARTICLES
___________________________________
GENERAL CURVILINEAR MOTION &
NORMAL & TANGENTIAL COMPONENTS
AND CYLINDRICAL COMPONENTS
LECTURE #1
Kinematics of Particles
General Curvilinear Motion
OBJECTIVE
Students should be able to:
1. Determine the normal and tangential components of
velocity and acceleration of a particle traveling along a
curved path.
2. Determine velocity and acceleration components using
cylindrical coordinates
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Kinematics of Particles
General Curvilinear Motion
Normal and Tangential Components I
When a particle moves along a curved path, it is sometimes convenient
to describe its motion using coordinates other than Cartesian. When
the path of motion is known, normal (n) and tangential (t) coordinates
are often used
In the n-t coordinate system, the origin is
located on the particle (the origin moves
with the particle)
The t-axis is tangent to the path (curve) at the instant considered,
positive in the direction of the particle’s motion
The n-axis is perpendicular to the t-axis with the positive direction
toward the center of curvature of the curve
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Kinematics of Particles
General Curvilinear Motion
Normal and Tangential Components II
The positive n and t directions are
defined by the unit vectors un and ut,
respectively
The center of curvature, O’, always lies
on the concave side of the curve.
The radius of curvature, r, is defined
as the perpendicular distance from the
curve to the center of curvature at that
point
The position of the particle at any instant is defined by the
distance, s, along the curve from a fixed reference point.
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Kinematics of Particles
General Curvilinear Motion
Velocity in the n-t coordinate system
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)
v = vut
where
v = ds/dt
Here v defines the magnitude of the velocity (speed) and
ut defines the direction of the velocity vector.
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Kinematics of Particles
General Curvilinear Motion
Acceleration in the n-t coordinate system I
Acceleration is the time rate of change of velocity:
·
a = dv/dt = d(vu )/dt = vu
t
.
+ v ut
.
Here v represents the change in
.
the magnitude of velocity and ut
represents the rate of change in
the direction of ut.
t
After mathematical manipulation,
the acceleration vector can be
expressed as:
.
a = vut + (v2/r)un = atut + anun
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Kinematics of Particles
General Curvilinear Motion
Acceleration in the n-t coordinate system II
There are two components to
the acceleration vector:
a = at ut + an un
The tangential component is tangent to the curve and in the direction of
increasing or decreasing velocity.
.
at = v
or
at ds = v dv
The normal or centripetal component is always directed toward the center of
curvature of the curve. an = v2/r
The magnitude of the acceleration vector is
a = [(at)2 + (an)2]0.5
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Kinematics of Particles
General Curvilinear Motion
Special cases of motion I
There are some special cases of motion to consider
1)
The particle moves along a straight line.
r

=>
an =
v2/r
=0
=>
.
a = at = v
The tangential component represents the time rate of change in the
magnitude of the velocity.
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Kinematics of Particles
General Curvilinear Motion
Special cases of motion II
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/r
The normal component represents the time rate of change in the
direction of the velocity.
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Kinematics of Particles
General Curvilinear Motion
Special cases of motion III
There are some special cases of motion to consider
3) The tangential component of acceleration is constant, at = (at)c.
In this case,
s = so + vot + (1/2)(at)ct2
v = vo + (at)ct
v2 = (vo)2 + 2(at)c(s – so)
As before, so and vo are the initial position and velocity of the
particle at t = 0
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Kinematics of Particles
General Curvilinear Motion
Special cases of motion IV
There are some special cases of motion to consider
4) The particle moves along a path expressed as y = f(x).
The radius of curvature, r, at any point on the path can be calculated
from
[ 1 + (dy/dx)2 ]3/2
r = ________________
d2y/dx 2
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What we have learned today?
 Concept of Curvilinear Motion
 Position, Displacement, Velocity and Acceleration in curvilinear motion of
a particle
 Rectangular components of the vectors
 Rectangular Components
 Normal and Tangential Components
 Polar and Cylindrical Components
Next Lecture



Motion of a projectile
Normal and Tangential Components
Cylindrical Components
ASSIGNMENT
DEADLINE
NEXT LECTURE
STUDY ALL THE EXAMPLES
QUESTIONS
THANK YOU
FOR YOUR
INTEREST