Dynamics Lecture

1. MECH-202
DR. MUHAMMAD NAFEES MUMTAZ QADRI
Department of Aerospace Engineering,
College of Aeronautical Engineering, National University of Sciences and Technology,
PAF Academy Asghar Khan
Room AE-12, E-Mail: m.n.mumtaz.qadri@cae.nust.edu.pk
Engineering Dynamics
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2. Lecture 3
KINEMATICS OF PARTICLES-PLANE CURVILINEAR
MOTION, RECTANGULAR COORDINATES & PROJECTILE
MOTION
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3. Plane Curvilinear Motion
 Treat the motion of a particle along a curved path which lies in
a single plane.
 This motion is a special case of the more general 3-D motion
illustrated in the Figure.
 Considering the plane of motion be the x-y plane, then the
coordinates z and ɸ are both zero and R becomes r.
 Vector analysis will be used first to describe the motion,
making it independent of any coordinate system.
 Time derivative of Vector an important concept in dynamics
and will be used frequently in this subject.
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4. Plane Curvilinear Motion
 Consider the continuous motion of a particle along a plane
curve as shown in Figure.
 At time t the particle is at position A, located by position vector
r (from origin O).
 If both magnitude and direction of r are known at time t, then
position of the particle is completely specified.
 At t + Δt, the particle is at position A’, located at position vector
r + Δr.
 This combination is vector addition and displacement of particle
during time Δt is the vector Δr, representing vector change of
position and independent of the choice of origin.
 If the origin was chosen at some other location, position vector
r will change but Δr will be unchanged.
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5. Plane Curvilinear Motion
 Average velocity of particle between A and A’ is defined at
vav = Δr/Δt, which is a vector whose direction is that of Δr.
 Instantaneous velocity v of the particle is defined as the
limiting value of the average velocity as the time interval
approaches zero. Hence,
 The derivative of vector is itself a vector having both a
magnitude and a direction. The magnitude of v is called
speed and is the scalar
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0
lim
t
d
t dt
 
   

Δr r
v v r
ds
v s
dt
  
v

6. Plane Curvilinear Motion
 Careful distinction between magnitude of the derivative and the derivative of the magnitude.
 Example of magnitude of the derivative
It represents the magnitude of the velocity, or the speed, of the particle.
 Example of derivative of the magnitude
 It represents the rate at which the length of the position vector r is changing.
 The two derivatives have different meanings and must be careful to distinguish between them
especially in their notation.
 Recommended symbols for vector quantities:
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d
s v
dt
  
r
r
d dt dr dt r
 
r
ˆ
, ,
v v v

7. Plane Curvilinear Motion
 We denote the velocity of the particle at A by the tangent
vector v and the velocity at A’ by the tangent v’.
 There is a vector change in the velocity during the time
interval Δt.
 The velocity v at A plus (vectorially) the change Δv must
equal the velocity at A’, hence v’-v = Δv.
 From vector diagram it can be seen that Δv depends both
on the change in magnitude (length) of v and on the change
in direction of v.
 Average acceleration of the particle between A and A’ is
defined as Δv/Δt, which is a vector whose direction is that of
Δv.
 The magnitude of this average acceleration is the
magnitude of Δv divided by Δt.
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8. Plane Curvilinear Motion
 Instantaneous acceleration a of the particle is defined as the
limiting value of the average acceleration as the time interval
approaches zero. Hence,
 It is apparent that the direction of the acceleration of a particle
in curvilinear motion is neither tangent to the path nor normal to
the path.
 The acceleration component (due to the change in velocity Δv)
which is normal to the path points toward the center of
curvature of the path.
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0
lim
t
d
t dt
 
   

Δv v
a a v

9. Plane Curvilinear Motion
 Instantaneous acceleration a of the particle is defined
as the limiting value of the average acceleration as the
time interval approaches zero. Hence,
 It is apparent that the direction of the acceleration of a
particle in curvilinear motion is neither tangent to the
path nor normal to the path, rather it acts tangent to the
hodograph
 A hodograph, when constructed, describes the locus of
points for the arrowhead of the velocity vector in the
same manner
 The acceleration component (due to the change in
velocity Δv) which is normal to the path points toward
the center of curvature of the path.
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0
lim
t
d
t dt
 
   

Δv v
a a v

10. Rectangular Coordinates (x-y)
 Particle path shown in Figure along with x and y axes. The
position vector r, the velocity v, and the acceleration a of
the particle shown with their x and y components.
 Scalar values of the components of v and a are merely
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r i j
v = r = i j
a = v = r = i + j
x y
x y
x y
 

,
x y
x x
y y
v x v y
a v x
a v y
 
 
 

11. Rectangular Coordinates (x-y)
 As observed previously, the direction of the velocity is always tangent
to the path, and from the figure it is clear that;
If the angle θ is measured counterclockwise from the x-axis to v for
the configuration of axes shown, then we can see that dy/dx = tan
θ=vy/vx.
 If the coordinates x and y are known independently as functions of
time, x = f1(t) and y = f2(t), then for any value of time we can combine
them to obtain r.
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2 2 2
2 2
tan
x y
x y
y
x
v v v
v v v
v
v

 
 

2 2 2
2 2
x y
x y
a a a
v a a
 
 

12. Rectangular Coordinates (x-y)-Projectile Motion
 Important application of 2-D kinematic theory is the problem of projectile motion. Assumptions:
Neglect aerodynamic drag, curvature and rotation of the earth, and assume that the altitude change
is small enough hence g is constant. Rectangular coordinates will be used .
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0
x
a 
0
( )
x x
v v
 0
( )
y y
v v gt
 
0 0
( )
x
x x v t
 
2
0 0
1
( )
2
y
y y v t gt
  
2 2
0 0
( ) 2 ( )
y y
v v g y y
  
y
a g
 

13. Examples
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Example 1
The velocity of a particle is v = {3i + (6-2t)j} m/s, where t is in seconds. If r = 0, when t = 0,
determine the displacement of the particle during the time interval t =1 s to t =3 s.

14. Examples
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Example 2
The velocity of a particle is given by v = {16t2i + 4t3j +(5t+2)k} m/s, where t is in seconds. If the
particle is at the origin when t = 0, determine the magnitude of the particle’s acceleration when
t = 2 s. Also, what is the x, y, z coordinate position of the particle at this instant?

15. Examples
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Example 3
A rocket is fired from rest at x = 0 and travels along a parabolic trajectory described by y2 =
[120(103)x] m. If the x component of acceleration is ax = (1/4)t2 m/s2, where t is in seconds,
determine the magnitude of the rocket’s velocity and acceleration when t = 10 s.

16. Examples
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Example 4
Determine the minimum initial velocity v0 and the
corresponding angle θ0 at which the ball must be kicked in
order for it to just cross over the 3 m high fence.

17. Examples
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Example 5
Neglecting the size of the ball, determine the magnitude vA of
the basketball’s initial velocity and its velocity when it passes
through the basket.