Emm3104 chapter 1 part2

1
Dr. Azizan As’arry
Semester 1, 2016-2017
7/9/2015
EMM 3104
DYNAMIC

• Business Driven • Technology Oriented • Sustainable Development • Environmental Friendly
Instructor and Textbook Details
• Name: Dr. Azizan As’arry
• Email: zizan@upm.edu.my
• Phone: 03-8946 4377
• HP: 012-3886050
• Office: Room #16, KMP
• Title: Hibbeler, R.C. (2015). Engineering Mechanics:
Dynamics (14th Edition). Singapore: Prentice Hall.
EMM 3104 : Dynamics 2

Course Rules
 Late to class less than 10 minutes
 Be committed to your assignments
 > 80% attendance for lectures.
 If not? Then, you will be banned from the exam hall. Bye-bye
40%.
 Test and assignments will not be repeated without valid excuse.
 Complete all assignments neatly and on time
 Check PutraBLAST frequently to get the latest information about
the course.
 Wear proper shoes (no selipar) and collared t-shirt to the lectures,
lab and KMP office. Follow the faculty rules.

Class Objectives
• Relate concept of rigid body motion, force,
moment, velocity and acceleration (C3 -
Application)
• Solve problems involving rigid body motion (C3 -
Application)
• Explain the free rigid body diagram using
Newton’s law of motion (A3 - Valuing, CS)

Class Evaluation
• Test 1 : 20%
• Test 2 : 20%
• Assignment : 20%
• Final Exam : 40%
• Total : 100%

Introduction to Dynamics
• Static = equilibrium of a body at rest or moving with constant speed
• Dynamic = Accelerated motion of a body
• Dynamic divided into two main parts:
– Kinematic = treat only the geometrical aspects of the motion
– Kinetics = analysis of the forces causing the motion
• Where can you apply these knowledge?
– Structural design of any vehicle; car, planes, train, etc.
– Mechanical devices; motors, pumps, moveable tools, etc.
– Prediction of motion; artificial satellites, projectiles, spacecraft, etc.
• What is important in dynamics?
– Calculus and static basics !

Introduction to Dynamics
• Basic concepts from statics:
– Space
– Time
– Mass
– Force
– Particle
– Rigid Body
– Vector
– Scalar
– Newton’s Law (First, Second and Third)
– SI Units

Method of Attack
1. Formulate the problem
a) State the given data
b) State the desired result
c) State your assumptions and approximations
2. Develop the solution
a) Draw any needed diagrams, and include coordinates which are appropriate for the problem at
hand
b) State the governing principles to be applied to your solution.
c) Make your calculations.
d) Ensure that your calculations are consistent with the accuracy justified by the data.
e) Be sure that you have used consistent units throughout your calculations.
f) Ensure that your answers are reasonable in terms of magnitudes, directions, common sense, etc.
g) Draw conclusions.

Questions?

APPLICATIONS
The motion of large objects,
such as rockets, airplanes, or
cars, can often be analyzed as
if they were particles.
Why?
If we measure the altitude of
this rocket as a function of
time, how can we determine
its velocity and acceleration?

APPLICATIONS
(continued)
A sports car travels along a straight road.
Can we treat the car as a particle?
If the car accelerates at a constant rate, how can we
determine its position and velocity at some instant?

Particle Motion and Choice of
Coordinates:
 Assume P is a particle moving along some
general path in space.
 Position P at any time, t can be describe
through:
 Rectangular coordinates x,y,z
 Cylindrical coordinates r,θ,z
 Spherical coordinates R,θ,ϕ
 Tangent, t and normal, n to the curve

RECTILINEAR KINEMATICS: CONTINUOUS MOTION
A particle travels along a straight-line path
defined by the coordinate axis 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.
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).
The displacement of the particle is defined
as its change in position.
Vector form:  r = r’ - r Scalar form:  s = s’ - s

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.
The average velocity of a particle during a
time interval t is
vavg = r / t
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 / t

ACCELERATION
Acceleration is the rate of change in the velocity of a particle. It is a vector
quantity. Typical units are m/s2.
As the book indicates, the derivative equations for velocity and acceleration
can be manipulated to get a ds = v dv
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) or zero (constant
speed)

SUMMARY OF KINEMATIC RELATIONS:
RECTILINEAR MOTION
• Differentiate position to get velocity and acceleration.
v = ds/dt ; a = dv/dt or a = v dv/ds
• Integrate acceleration for velocity and position.
• Note that so and vo represent the initial position and velocity of
the particle at t = 0.
Velocity:
 =
t
o
v
vo
dtadv  =
s
s
v
v oo
dsadvvor  =
t
o
s
so
dtvds
Position:

Graphical Interpretations:
 From (a), any slope of the line is v = (ds/dt)
 From the slope data, can plot the v versus t graph.
 From (b), any slope of the line is a = (dv/dt)
 From the slope data, can plot the a versus t graph.
 Area under the curve (b), during dt time is v dt. Which is
the displacement of ds.
 Area under the curve (c), during dt time is a dt. It is the
net change in velocity.
 =
t2
t1
s2
s1
dtvds Or s2-s1 = (area under v-t curve)
 =
t2
t1
v2
v1
dtadv Or v2-v1 = (area under a-t curve)

Graphical Interpretations:
 Area under the curve during ds displacement is
a ds. Since a ds = v dv = d (v2/2). So, area
under the curve from s1 to s2 is:
 When v is plotted against s, slope of the curve at
any point A is dv/ds. Having AB normal to the
curve result in CB/v = dv/ds. So, CB = v(dv/ds)
= a.
 All graph should have similar numerical scale.
Example: m or cm for ds.
 =
s2
s1
v2
v 1
dsav dv Or (1/2) (v2
2-v1
2) = (area
under a-s curve)

CONSTANT ACCELERATION
The three kinematic equations can be integrated for the special case when
acceleration is constant (a = ac) to obtain very useful equations. 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 downward. These equations are:
tavv co
+=yields= 
t
o
c
v
v
dtadv
o
2
coo
s
t(1/2) atvss ++=yields= 
t
os
dtvds
o
)s-(s2a)(vv oc
2
o
2 +=yields= 
s
s
c
v
v oo
dsadvv

Example

Example:

EXAMPLE
Plan:Establish the positive coordinate, s, in the direction the
particle is traveling. Since the velocity is given as a function
of time, take a derivative of it to calculate the acceleration.
Conversely, integrate the velocity function to calculate the
position.
Given: A particle travels along a straight line to the right
with a velocity of v = ( 4 t – 3 t2 ) m/s where t is
in seconds. Also, s = 0 when t = 0.
Find: The position and acceleration of the particle
when t = 4 s.

Exercise 1.1
• Problems no (next page):
– 12.4, 12.5, 12.9, 12.10, 12.11, 12.13, 12.15, 12.22.

Emm3104 chapter 1 part2

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