Engineering Dynamics Part 2

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14 Νοε 2013 (πριν από 3 χρόνια και 4 μήνες)

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Engineering Dynamics


Part 2

Projected outcomes for today’s class

-

Reinvigorated with the passion for engineering dynamics

-

Be able to use dynamics terminology correctly

-

Be able to use uniform acceleration rectilinear motion equations

-

Be able to use
instantaneous kinematic relationships

-

Be able to use kinematics in two dimensions (projectile motion)

Definitions




Dynamics
: the study of bodies in motion.


Kinematics:
a study of the description of motion.


Kinetics:
a study of forces and their effe
ct on the motion of bodies.


Rectilinear Motion:
one dimensional motion…motion along a straight line.


Particle Body:
An object which is modeled as a point object. While it has mass, it's volume can
be treated as infinitely small and all concentrated at

a single point.


Position:
A quantity which describes a location of a body with respect to a known reference
system



Displacement:
a quantity which describes the change in position of a body.


Velocity:
the rate of change of position with respect to t
ime.


Acceleration:
the rate of change of velocity with respect to time.

















Equations:


Average velocity
:
Instantaneous velocity
:
Units of velocity
:


ave
x
v
t




0
lim
t
dx x
v
dt t
 

 


ft/s m/s


Average acceleration
:
Instantaneous Accel.
:
Units of acceleration
:


ave
v
a
t




2
2
0
lim
t
dv d x v
a
dt t
dt
 

  


ft/s
2


m/s
2

Workout problems:

Given a function
3 2
2 15 36 10
x t t t
   

where t is in sec and x is in meters.


a) Determine the velocity and acceleration as functions of time.


b) Determine the instantaneous velocity and acceleration at t = 4 s


c) Determine the average velocity and acceleration from 4 to 5 seconds.





























Definitions:

Uniform Rectilinear Motion:

Straight line motion which occurred when velocity is constant.


Uniform Accelerated Rectilinear Motion:

Straight line motion which occurs when acceleration is
constant.



Uniform Rectilinear Motion Equations:



Instantaneous velocity
:


0
x x vt
 



Uniform Accelereated Rec
tilinear Motion Equations:



t x v a


0
v v at
 

√ √ √

2
0 0
1
2
x x v t at
  

√ √ √


0 0
1
( )
2
x x v v t
  

√ √ √


2 2
0 0
2 ( )
v v a x x
  

√ √ √


A truck travels 220 m in 10 s while being decelerated at a constant rate of 0.6 m/s
2
.

Determine


a) its initial velocity,

b) its final velocity

c) the distance traveled during the first 1.5 s.









Instantaneous Kinematic Relationships:



dx
v
dt


dv
a
dt


v dv a dx



Graphical Representation:

The relationships indicated by the instantaneous
kinematic equations above can also be shown
graphically.


Pos
it
ion vs. Time plot
:



Vel
ocity

vs. Time plot
:














position,
x

time,
t

slope =
v

x
-
t
graph

velocity,
v

time,
t

slope =
a
.

Area = Δ
x

v

-
t

graph


Plots Typical of Plots Typical of Uniform


Uniform Motion Accelerated Motion



























t


t

t

acceleration
,
a

x

-
t

graph

time,
t

time,
t

velocity, v

position,
x

v

-
t

graph

a

-
t

graph

a

= constant

v
0


x
0


slope =
a





t


t

t

acceleration,
a

x

-
t

graph

time, t

time,
t

velocity,
v

position,
x

v

-
t
graph

a

-
t

graph

a
=0

v

= constant

x
0

Slope =
v


The velocity
-
time graph for the motion of a train as it moves is given below. Construct the
x
-
t

and the
a
-
t

graphs for the same motion. Assume the train starts from a position of


x

=0 at
t

= 0.
































v

[ft/s]


t

[s]

90

40

30

120

x

[ft]


t

[s]





a

[ft/s
2
]


t

[s]





Projectile Motion

Defined in two dimensions using same kinematic equations that are used for one dimensional
motion

0
x x vt
 

0
v v at
 

2
0 0
1
2
x x v t at
  

0 0
1
( )
2
x x v v t
  

2 2
0 0
2 ( )
v v a x x
  


A stone is thrown horizontally from the top of a 100 ft tall building at 50 ft/s

(g = 32.2ft/


)

a)

A
t what horizontal distance from the point at which it is thrown does it hit the ground?

b)

What is the magnitude of its velocity just before it hits?

c)

With what
angle (with respect to the horizontal) does it hit the ground?








Review what we have re
-
learned today

-

Reinvigorated with the passion for engineering dynamics

-

Be able to use dynamics terminology correctly

o

Dynamics

o

Kinematics

o

Kinetics

o

Rectilinear Motion

o

Particle Body

o

Position

o

Displacement

o

Velocity

o

Acceleration

o

Uniform Rectilinear Motion

o

Uniform Accelerated Rectilinear Motion:

-

Be able to use uniform acceleration rectilinear motion equations

o

0
x x vt
 

o

0
v v at
 

o

2
0 0
1
2
x x v t at
  

o

0 0
1
( )
2
x x v v t
  

o

2 2
0 0
2 ( )
v v a x x
  


-

Be able to use instantaneous kinematic relationships

o

dx
v
dt


dv
a
dt


v dv a dx


-

Be able to use kinematics in tw
o dimensions (projectile motion)

o

Constant velocity in the x
-

direction

o

Constant acceleration in the y
-

direction






A particle moves in a straight line with a velocity shown in the figure. Knowing that
x
=
-
540 ft at
t

= 0,
a) construct the
a
-
t

and the
x
-
t

curves for 0 <
t
< 50 s, and determine

b) the total distance traveled b the particle when
t

= 50 s, and determin
e

c
) the total distance traveled by the particle when
t
= 50 s,

d
) the two times at which
x

= 0.
















A stone is thrown vertically upward from a point on a bridge located 40 m above the water.
Knowing that the stone strikes the water 4 seconds after being released, determine


a) the speed with which the stone was thrown upward,

b) how high the stone rises above the bridge,

c) the speed with which the stone strikes the water.




















The motion of a particle is defined by the relation
3 2
12 18 2 5
x t t t
   
where
x

and
t
are
expressed in meters and second, respectively.

Determine


a)

The
position when the acceleration o
f the particle is equal to zero

b)

The
velocity when the
acceleration o
f the particle is equal to zero









































A commuter train traveling at 40 mi/h is 3 mi from a station. The train then decelerates so that its speed is
20 mi/h when it is 0.5 mi from the station. Knowing that the train arrives at the station 7.5 min after
beginning to decelerate and assuming cons
tant decelerations, determine

a) the time required for the train to travel the first 2.5 mi,

b) the speed of the train as it arrives a the station,

c) the final constant deceleration of the train.

















Example 3:

An automobile at rest is passed by a truck traveling at a constant speed of 54 km/hr

(15m/s). The
automobile starts and accelerates for 10 s at a constant rate until it reaches a speed of 90 km/hr
(25 m/s). If the automobile then maintains a constant speed of 90 km/hr, determine when and
where it will overtake the truck, assuming that t
he automobile starts 3 seconds after the truck has
passed it.

















v

[m/s]


t

[s]

x

[m]


t

[s]





a

[m/s
2
]


t

[s]



A Bus is accelerated at a rate of 0.75 m/s
2

as it travels from A to B. The speed of the bus was
18km/
hr as it passed A. Determine the time required to for the bus to reach B, and the
corresponding speed as it passes B.



















B

A

150 m

A body's acceleration is given by the function:
2
25 3
a x
 


The body starts from rest at
x
=0. when time
t

is 0.

a) Find the velocity when
x

= 2

b) Find the position when body’s velocity returns to 0.

c) Find the position where the velocity is a maximum.















A sprinter in a 100 m race accelerates
uniformly for the first 35 m and then runs with a constant velocity.
If the sprinter’s t
ime for the first 35 m is 5.4 s

D
etermine

a) his acceleration

b) his final velocity

c) his time for the r
ace