equations of motion for a particle
Post on 10-Apr-2018
221 Views
Preview:
TRANSCRIPT
-
8/8/2019 Equations of Motion for a Particle
1/17
Rectilinear Motion: Variable acceleration
If a mathematical relation is established between any two of the four
variables s, v, a, and t, then a third variable can be determined by using one
of the following equations which relates all three variables.
adsvdva
dv
v
dsdt
dt
dva
dt
dsv
!p!!
!
!
IfAcceleration is constant:
2
0 210 tatvss c! tavv c! 0 )(2 0
2
0
2 ssavv c !
Review of Kinematics
-
8/8/2019 Equations of Motion for a Particle
2/17
Cartesian Rectangular x, y z coordinates:
These coordinates are often used when the motion can be resolved into
rectangular components. They are also very useful for studying projectilemotion since the acceleration of the projectile is always downward
za
ya
xa
z
y
x
!
!
!
zv
yv
xv
z
y
x
!
!
!
-
8/8/2019 Equations of Motion for a Particle
3/17
n, t, b Intrinsic coordinates
These coordinates are particularly advantageous for studying
particle acceleration along a known path. This is because the t andn components of a represent the separate changes in magnitude
and direction of the velocity, respectively, and these components
can be readily formulated
V2va
dt
dva
dtdsv
n
t
!
!
!
vdvdsat !
-
-
!
2
2
2/32
1
dx
yd
dx
dy
V)(xfy !When is known
-
8/8/2019 Equations of Motion for a Particle
4/17
zr ,,U Coordinates
These coordinates are used when the data regarding the angular motion ofthe radial coordinate r is given to describe the particles motion. Also, some
paths of the motion can conveniently be described using these coordinates.
zv
rv
rv
z
r
!
!
!
UU
za
rra
rra
z
r
!
!!
UU
U
U2
2
-
8/8/2019 Equations of Motion for a Particle
5/17
!
!
!
zz
yy
xx
maF
maF
maF
!
!
!
0b
nn
tt
a
a
!
!
!
zz
rr
maF
maF
maF
UU
-
8/8/2019 Equations of Motion for a Particle
6/17
!
!
!
zz
yy
xx
a
a
a
Equations of Motion of a particle in RectangularCartesian Coordinate System
za
ya
xa
z
y
x
!
!!
-
8/8/2019 Equations of Motion for a Particle
7/17
!!
!
0b
nn
tt
a
a
V
2va
dt
dv
a
dt
dsv
n
t
!
!
! vdvdsat !
Equations of Motion of a particle in Intrinsic Coordinate Systems
-
8/8/2019 Equations of Motion for a Particle
8/17
!
!
!
zz
rr
maF
maF
maF
UU
zv
rv
rv
z
r
!
!
!
UU
za
rra
rra
z
r
!
!
!
UU
U
U2
2
Equations of Motion of a particle in Cylindrical and PolarCoordinate Systems
In polar coordinate:
0
0
!
!
z
z
a
v
-
8/8/2019 Equations of Motion for a Particle
9/17
Geometric requirements Kinematics (position of the particle s as
function of the spatial variable, velocity, and acceleration)
adsvdva
dv
v
dsdt
dt
dva
dt
dsv
!p!!
!
!
Dynamic Requirements
mvp
madt
dvm
dt
mvd
dt
dpf
!
!!!!)(
Nf
mgf
ksf
mvp
kf
g
s
Q!
!
!
!
Constitutive requirements for system elements and gravitational field
p is linear momentum
Momentum - velocity
Spring force
Gravitational force
Frictional force
Example: Rectilinear Motion with Variable acceleration
Requirements for Solving Dynamic Problems
-
8/8/2019 Equations of Motion for a Particle
10/17
Example1: Consider a simple pendulum consisting of a mass, m, suspended from a
string of length L and negligible mass. We can formulate the problem in polar
coordinates, and noting that r = L (constant), and write r and theta components.
T
mg
U
ru
Uu
x
y
L
o
UUU
U
rra
rrar
2
2
!!
0!!! rrLr
UU
UU
U
mLmgF
mLTmgFr
!!
!!
sin
cos2
Where T is the tension on the string. If we restrict the
motion to small motion, we can approximate,
UU }sin
The equationU becomes 0! UUL
g
max)0( UU !
tL
gCt
L
gCt
! sincos)( 21UIntegrating we obtain the general solution
Where the constants C1 and C2 are determined from the initial conditions. Thus, if
t
L
gt
! cos)(
maxUU 2cos UU mLmgT !and
-
8/8/2019 Equations of Motion for a Particle
11/17
The ball has a mass of 30 kg and speed v = 4 m/s at the instant it
is at its lowest point, Determine the tension in the cord and
the rate at which the balls speed is decreasing at the instant Q20!U
Q
0!U
Neglect the size of the ball
Problem 13-62:
-
8/8/2019 Equations of Motion for a Particle
12/17
Given v0=4 m/s,
determine v and tension T
n
tT
tunu
Umg
ttt
nn
amaF
vTmaF
30sin)81.9(30
430cos)81.9(30
2
!!
!!
U
U
V
2va
dt
dva
dt
dsv
n
t
!
!
!
4!!l
V
vdvdsat !
We can formulate the problem
using n-t coordinates:
vdvdsat ! UUU ddsrs 44 !p!!
!!vv
v
vdvvdvd400
)4(sin81.9 UUU
)1(cos4.3982
14
2
1
2
1cos)4(81.9
222
0!! UU U vv
NT
sma
smv
t
361
/36.3
/357.3
2
!
!
!
At Q20!U
-
8/8/2019 Equations of Motion for a Particle
13/17
-
8/8/2019 Equations of Motion for a Particle
14/17
-
8/8/2019 Equations of Motion for a Particle
15/17
s
1
-
8/8/2019 Equations of Motion for a Particle
16/17
-
8/8/2019 Equations of Motion for a Particle
17/17
top related