curvilinear motion
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Curvilinear Motion. Lecture III. Topics Covered in Curvilinear Motion. Plane curvilinear motion Coordinates used for describing curvilinear motion Rectangular coords n-t coords Polar coords. Plane curvilinear Motion. - PowerPoint PPT PresentationTRANSCRIPT
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Curvilinear Motion
Lecture III
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Topics Covered in Curvilinear Motion Plane curvilinear motion Coordinates used for describing
curvilinear motion Rectangular coords n-t coords Polar coords
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Plane curvilinear Motion Studying the motion of a particle along a
curved path which lies in a single plane (2D). This is a special case of the more general 3D
motion.
3D
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Plane curvilinear Motion – (Cont.)
If the x-y plane is considered as the plane of motion; from the 3D case, z and are both zero, and R becomes as same as r.
The vast majority of the motion of particles encountered in engineering practice can be represented as plane motion.
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Coordinates Used for Describing the Plane Curvilinear Motion
Rectangular coordinates
Normal-Tangential coordinates
Polar coordinate
s
y
x
P
t
nPA
PB
PC
t
n
t
n
y
x
Pr
r
Path
Path
Path
O O
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Plane Curvilinear Motion – without Specifying any Coordinates
(Displacement)
Note: Since, here, the particle motion is described by two coordinates components, both the magnitude and the direction of the position, the velocity, and the acceleration have to be specified.
P at time t
P at time t+t r
r(t)
r(t+t)
O
or r(t)+r(t)
s
t(
Note: If the origin (O) is changed to some different location, the position r(t) will
be changed, but r(t) will not change.
Actual distance traveled by the particle (it is s scalar)
The vector displacement of the particle
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Plane Curvilinear Motion – without Specifying any Coordinates (Velocity)
Average velocity (vav):
Instantaneous velocity (v): as t approaches zero in the limit,
t
r
vav
Note: vav has the direction of r and its magnitude equal to the magnitude of r divided by t.Note: the average speed of the particle is the scalar s/t. The magnitude of the speed and vav approach one another as t approaches zero.
rrr
v lim0
dt
d
tt
Note: the magnitude of v is called the speed, i.e.
v=|v|=ds/dt= s..
v(t)
v(t)
v(t+t)
v(t+t) v
P
P
Note: the velocity vector v is always tangent to the path.
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Plane Curvilinear Motion – without Specifying any Coordinates
(Acceleration)
Average Acceleration (aav):
Instantaneous Acceleration (a): as t approaches zero in the limit,
t
v
a av
Note: aav has the direction of v and its magnitude is the magnitude of v divided by t.
rvvv
a lim0
dt
d
tt
Note: in general, the acceleration vector a is neither tangent nor normal to the path. However, a is tangent to the hodograph.
C Hodograph
P
P
V1
V2
V1
V2
a1a2
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The description of the Plane Curvilinear Motion
in the Rectangular Coordinates (Cartesian
Coordinates)
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Plane Curvilinear Motion - Rectangular Coordinates
jijirva :on vectoraccelerati The
jijirv :vector velocity The
jir :ectorposition v The
yxaa
yxvv
yx
yx
yx
y
x
PPath
O
j
i
r
v
vy
vx
aay
ax P
22
22
22
:on vectoraccelerati theof magnitude The
:vector velocity theof magnitude The
:ectorposition v theof magnitude The
yx
yx
aaa
vvv
yxr
Note: the time derivatives of the unit vectors are zero because their magnitude and direction remain constant.
Note: if the angle is measured counterclockwise from the x-axis to v for the configuration of the axes shown, then we can also observe that dy/dx = tan = vy/vx.
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Plane Curvilinear Motion - Rectangular Coordinates (Cont.)
The coordinates x and y are known independently as functions of time t; i.e. x = f1(t) and y = f2(t). Then for any value of time we can combine them to obtain r.
Similarly, for the velocity v and for the acceleration a.
If a is given, we integrate to get v and integrate again to get r.
The equation of the curved path can be obtained by eliminating the time between x = f1(t) and y = f2(t).
Hence, the rectangular coordinate representation of curvilinear motion is merely the superposition of the components of two simultaneous rectilinear motions in x- and y- directions.
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Plane Curvilinear Motion - Rectangular Coordinates (Cont.) –
Projectile Motion
)(2
)21(
)( )(
0
22y
2
ooy
oyooxo
oyyoxx
yx
yyg)(vv
gtt)(vyyt )(vxx
gtvvvv
gaa
x
Path
y
vo
(vx)o = vo cos
(vy)o = vo sing
vvy
vx
vvy
vx
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Exercises
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Exercise # 1
2/62: A particle which moves with curvilinear motion has coordinates in millimeters which vary with the time t in seconds according to x = 2t2 - 4t and y = 3t2 – (1/3)t3. Determine the magnitudes of the velocity v and acceleration a and the angles which these vectors make with the x-axis when t = 2 s.
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Exercise # 2A handball player throws a ball from A with a
horizontal velocity (VO ) .Determine :
a) The magnitude of (VO ) for which the ball will strike the ground at point ( D ).b) The magnitude of ( h ) above which the ball will strike the vertical wall at point ( B ).
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Exercise # 3 A helicopter is flying with a constant horizontal
velocity (V) of 144.2 km/h and is directly above point (A) when a loose part begins to fall.The part lands 6.5 s later at point (B) on inclined surface. Determine;
a) The distance (d) between points (A) and (B).b) The initial height (h) .
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Exercise # 4A homeowner uses a snow blower to clear his driveway. Knowing that the snow is discharged at an average angle of 40o with the horizontal, determine the initial speed of the snow .
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Exercise # 52/73: A particle is ejected from the tube at A with a velocity v at angle with the vertical y-axis. A strong horizontal wind gives the particle a constant horizontal acceleration a in the x-direction. If the particle strikes the ground at a point directly under its released position, determine the height h of point A. The downward y-acceleration may be taken as the constant g.