13.1 newton’s law of motion 1.newton’s 2 nd law of motion (1) a particle subjected to an...
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13.1 Newton’s law of motion1.Newton’s 2nd law of motion
(1) A particle subjected to an unbalanced force F
experiences an acceleration a
having the same
direction as F
and a magnitude that is directly
proportional to the force.
F
=m a
m = mass of a particle
=a quantitative measure of the resistance of the
particle to a change in its velocity.
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(2) The unbalanced force F
acting on the particle is
proportional to the time rate of change of the particle’slinear momentum.
F
am
vdt
dm
dt
vdm
vmdt
d
)(
(if m=constant)
2. Newton’s Law of Gravitational Attraction
r
m1 m2221
r
mmGF
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G = universal constant of gravitation = 2312 skg/m10x73.66
r = distance between centers of two particles
Weight of a particle with mass m1 = m
22
r
mmGF
)r
Gm(m
22
=mg
m2 : mass of the earth
r = distance between the earth center and the particle
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22
r
Gmg=
= acceleration due to gravity
=9.81 2sm measured at a point on the surface of the
earth at sea level and at a latitude of 045
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1F
2F
a
p
13-2 The equations of motion
1.Equations of motion of a particle subjected to more than one force.
amFFR
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2F
1F
FFR p
amKinetic diagram of particle p.
p
amFFR
………...equation of motion
Free body diagram of particle p.
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D’A lembert principle
:
0
am
amFR
inertia force vector
Dynamic equilibrium diagram
RF
amp
( 慣性力 )
0am若 則此狀態為靜平衡
amF
0amF
0F
R
R
+
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)0(
/
dt
vdaa
aaa
oop
opop
(1) Inertial frame
pathp
ap
x
y
o
0v
2. Inertial frame of reference (newtonian)A coordinate system is either fixed or translates in a given direction with a constant velocity.
(2) Noninertial frame
a
pathp
p
o
y
x
0a
0/ aaa pop
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irif
iFz
y
x
i
Equation of motion of particle i.Dynamic equilibrium diagram of particle i.
iii fF
iiam
13-3 Equation of motion for a system of particle
xyz: Inertial Coordinate System
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iiii amfF
i
i
f
F
resultant external force
resultant internal force
n
ijj
ijf1
Equation of motion of a system of particles.
............
......)(......)()(
2211
2211
ii
ii
amamam
fFfFfF
iii
i
iiii
amF
f
amfF
0
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By definition of the center of mass for a systemof particles.
iiG rmrm
i
G
mm
r
Position vector of the center of mass G.
Total mass of all particles.
)()(2
2
2
2
iiG rmdt
drm
dt
d
Assume that no mass is entering or leavingthe system.
iiG
ii
G
amam
dt
rdm
dt
rdm
2
2
2
2
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Hence:Gi amF
This equation justifies the application of the equation of motion to a body that is representedas a single particle.
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amF
RectangularCoordinatesystem.
z
y
x
zF
path
xF yF
Equation of motion of particle P.
In rectangular components
kmajmaimakFjFiF zyxzyx
13-4 Equations of motion : Rectangular
Coordinate
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zz
yy
xx
maF
maF
maF
scalar eqns.
Analysis procedure1. Free Body Diagram. (1) Select the proper inertial coordinate system. (2) Draw the particle’s F.B.D.2. Equation of motion (1) Apply the equations of motion in scalar form or vector form.
amF
or
zz
yy
xx
maF
maF
maF
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(2) Friction force
NF kf (3) Spring force
ksFs
3. Equations of kinematics Apply vdvads
dt
dsv
dt
dva 、、 for the solutions
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Curve path of motion of a particle is known.
tu
nu
bu
=Tangential unit vector
=Normal unit vector
=Binormal unit vector
= tn uu
13.5 Equation of Motion:Normal and Tangential
Coordinates
P
nb
t
Curve
tunu
bu
F
path
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amF
Or scalar form
tF
uF
bF
tam
nam
=
=
= 0
ntbbnntt amamuFuFuF
Equation of motion
dtdvat /
2van
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1. Free body diagram
Identify the unknowns in the problem.
2. Equation of motion
Apply the equations of motion using normal and tangential coordinates.
3. Kinematics
Formulate the tangential and normal components of acceleration.
Analysis procedure
ds/vdvdt/dva t
2van
3/2
22
2
/
/1
dxyd
dxdy
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r
z
zzuF uF
rruF
amF
13.6 Equation of Motion :Cylindrical coordinate
Equation of motion in cylindrical coordinates
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zrzzrramamamuFuFuF
zz
rr
maF
maF
maF
za
rra
rra
z
r
2and
Cylindrical or polar coordinates are suitable for a problem for which Data regarding the angular motion of the radial line r are given, or in Cases where the path can be conveniently expressed in terms of these coordinates.
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Normal and Tangential forceIf the particle’s accelerated motion is not completely specified, then information regarding the directions or magnitudes of the forces acting on the particle must be known or computed. Now, consider the case in which the force P causes the particle to move along the path r=f() as shown in the following figure.
r=f() :path of motion of particle
P:External force on the particle
F:Friction force along the tangent
N:Normal force perpendicular to tangent of path
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Direction of F & N
dr :radial componentrd :transverse componentds:distance
The directions of F and N can be specified relative to the radial coordinate r by computing the angle . Angle is defined between the extended radial line and the tangent to the path.
dr
rd
ddr
r
dr
rdtan
+ positive direction of
- negative direction of