Kinematics of a particle

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CHAPTER.12
Kinematics of a particle
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12.1 Introduction
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12.2 Rectilinear Kinematics Continuous motion

1. Kinematics Analysis of the geometric aspects of
   motion.
2. Particle A particle has a mass but negligible
   size and shape.
3. Rectilinear Kinematics Kinematics of objects
   moving along straight path and characterized by
   objects position, velocity and acceleration.
4. Position(1) position vector r A vector used to
   specify the location of particle P at any instant
   from origin O.

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(2) position coordinate , S An algebraic
scalar used to represent the position coordinate
of particle P
from O to P.

1. DisplacementChange in position of a particle ,
   vector

(1) Displacement
or
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• (2) Distance Total length of path traversed by
  the particle. A positive scalar.
• Velocity(1) Average velocity

(2) Instantaneous Velocity
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1. Acceleration

(1) Average acceleration
(2) (Instantaneous) acceleration

• Relation involving a , s and vvds/dt,
  dtds/vadv/dt, dtdv/a
• so, ds/vdv/a vdvads

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1. Constants acceleration a ac

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10. Analysis Procedure

• Coordinate System
• A. Establish a position coordinate s along the
  path.
• B. Specify the fixed origin and positive
  direction of the coordinate.

(2) Kinematic Equations A. Know the
relationship between any two of the four
variables a, v, a and t. B. Use the
kinematic equations to determine the unknown
varaibles
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12.3 Rectilinear Kinematics Erratic Motion
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Given method Kinematics egn Find
S-t graph Measure slope Vds/dt V-t graph
V-t graph Measure slope Adv/dt a-t graph
A-t graph Area integration v-t graph
v-t graph Area integration s-t graph
A-s graph Area integration v-s graph
v-s graph Measure slope Av(dv/ds) a-s graph
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12-4 General Curvilinear Motion

• 1. Curvilinear motion
• The particle moves along a curved path.

Vector analysis will be used to formulate the
particles position, velocity and acceleration.
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2. Position
s
3. Displacement

change in position of particle form p to p
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4. Velocity
(1) average velocity ??

(2) Instantaneous velocity ??
tangent to the curve at Pt .p tangent to
the path of motion
(3) Speed
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5. Acceleration
(1) Average acceleration
time rate of change of velocity vectors
Hodogragh is a curve of the locus of points for
the arrowhead of velocity vector.
(2) Instantaneous acceleration
which is not tangent to the curve of motion, but
tangent to the hodograph.
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12-5 Curvilinear Motion Rectangular components
xyz fixed rectangular coordinate system
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1. Position vector
Here
magnitude of
unit vector direction of
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2. Velocity
0
0
0
tangent to the path
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3. Acceleration
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12.6 Motion of a projectile
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(No Transcript)
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• 1. Horizontal motion, ax0

Vx (Vx)0 axt (Vx)0
X X0 (Vx)0t
Same as 1st Eq.
One independent eqn
X X0 (Vx)0t
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2. Vertical motion, ay-g constant
Can be derived from above two Eqs.
two independent eqns
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12-7 Curvilinear MotionNormal and Tangential
components.
Path of motion of a particle is known.
1. Planar motion
s
Here t (tangent axis ) axis tangent to
the curve at P and positive in the direction
of increasing S ut unit vector n (normal axis
) axis perpendicular to t axis and directed from
P toward to the center of curvature o un
unit vector o center of curvature r radius
of curvature p origin of coordinate system tn
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(1) Path Function
(known)
(2) Velocity
(3) Acceleration
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at Change in magnitude of velocity an Change in
direction of velocity
If the path in y f ( x )
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12-8 Curvilinear MotionCylindrical Components

1. Polar coordinates

(1) coordinates (r,q)
q
r
p
r
o
Reference line
(2) Position
(3) Velocity
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rate of change of the length of the radial
coordinate.
angular velocity (rad/s)
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(4) Acceleration
angular acceleration
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2. Cylindrical coordinates
Position vector
Velocity
Acceleration
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12.9 Absolute Dependent Motion Analysis of Two
Particles

1. Absolute Dependent Motion The motion of one
   particle depends on the corresponding motion of
   another particle when they are interconnected by
   inextensible cords which are wrapped around
   pulleys.

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2.Analysis procedure
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3. Example
Datum
Datum
B
A

1. position-coordinate equation

(2) Time Derivatives
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12.10 Relative-Motion Analysis of Two Particles

1. Translating frames of referenceA frame of
   reference whose axes do not rotate and are only
   permitted to translate relative to the fixed
   frame.

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1. position vector

3. velocity Vector
VB/A relative velocity observed from the
translating frame.
4. acceleration vector