Transcript Document
Objective
• To investigate particle motion along a curved path “Curvilinear
Motion” using three coordinate systems
– Rectangular Components
• Position vector r = x i + y j + z k
• Velocity
v = vx i + vy j + vz k
• Acceleration a = ax i + ay j +az k
(tangent to path)
(tangent to hodograph)
– Normal and Tangential Components
• Position (particle itself)
• Velocity
v = u ut
u2
a u u t u n
• Acceleration
– Polar & Cylindrical Components
1 (dy / dx)
2 3/ 2
d 2 y / dx 2
(tangent to path)
(normal & tangent)
Curvilinear Motion: Cylindrical
Components
• Section 12.8
• Observed and/or guided from origin or from
the center
r , , and z
• Cylindrical component
r and
• Polar component “plane motion”
Application: Circular motion but observed
and/or controlled from the center
Polar Coordinates
•
•
•
•
•
•
Radial coordinate r
Transverse coordinate
and r are perpendicular
Theta in radians
1 rad = 180o/p
Direction ur and u
Position
• Position vector
• r = r ur
Velocity
• Instantaneous velocity
= time derivative of r
r r ur
v r r u r r u r
u u
r
v r u r r u
v ur u r u u
• Where
ur r and u r
v r radial velocity
v transvers e velocity
Velocity (con.)
• Magnitude of velocity
u (r) 2 (r) 2
• Angular velocity
• Tangent to the path
• Angle = d
1 u
d tan ( )
ur
d
Acceleration
• Instantaneous acceleration = time derivative of v
v r u r r u
a v r u r r u r r u r u r u
u r u
u u r
a (r r 2 )u r (r 2 r )u
ar r r 2
a r 2 r
a ar u r a u
Acceleration (con.)
•
Angular acceleration
d 2 / dt 2 d / dt (d / dt )
•
Magnitude
a (r r 2 ) 2 (r 2r) 2
•
•
Direction “Not tangent”
Angle f
a
f tan ( )
ar
1
f
Cylindrical Coordinates
• For spiral motion
cylindrical coordinates
is used r, , and z.
• Position
rp r u r z u z
• Velocity
v r u r r u z u z
• Acceleration
a (r r 2 ) u r (r 2r) u z u z
Time Derivative to evaluate r, r, , and
• If r = r(t) and (t)
r 4t
2
(8t 6)
3
r 8t
24 t 2
r 8
48 t
• If r = f() use chain rule
r 5
2
r 10
r 10[ () () ]
r 10 2 10
Problem
• The slotted fork is rotating about O at a constant
rate of 3 rad/s. Determine the radial and
transverse components of velocity and
acceleration of the pin A at the instant = 360o.
The path is defined by the spiral groove r =
(5+/p) in., where is in radians.
360o 2p rad
r 5
p
vr r
7 in
2p
3
p
3 rad/s
3
r in/s
p p
0.955 in/s
v r 7(3) 21 in/s
0 rad/s 2
r p 0 in/s 2
ar r r 2 0 7(3) 2 63 in/s 2
3
a r 2r 0 2( )(3) 5.73 in/s 2
p
Example 12-20
r 0.5(1 cos ) ft
v 4 ft/s
a 30 ft/s 2
find and
at 180o
r 0.5(1 cos )
r 0.5( sin )
r 0.5(cos )() 0.5( sin )
at 180o
r 1 ft
r 0
r -0.5θ 2
u (r) 2 (r) 2 (0) 2 (1) 2 4
4 rad/s
a (r r 2 ) 2 (r 2r) 2 [0.5(4) 2 1(4) 2 ]2 [1 2(0)( 4)]2 30
18 rad/s 2
Problem
A collar slides along the smooth vertical spiral rod, r = (2)
m, where is in radians. If its angular rate of rotation is
constant and equal 4 rad/s, at the instant = 90o. Determine
- The collar radial and transverse component of velocity
- The collar radial and transverse component of acceleration.
- The magnitude of velocity and acceleration
p
2
rad
r 2
r 2*
p
2
p m
4 rad / s
r 2
r 2
r 2 * 4 8 m / s
vr r 8 m/s
v r p (4) 12.56 m/s
u (8) 2 (12.56) 2
0 rad / s 2
r 0
ar r r 2 0 p (4) 2 50.24 m/s 2
a r 2r 0 2(8)( 4) 64 m/s 2
a (50.24) 2 (64) 2