Ch7 kinematic

1
Dynamics
a branch of mechanics that deals with
the motion of bodies under the action of
forces.
Two parts

• Kinematics
• Study of motion without reference to the
  forces that cause the motion

2. Kinetics Study of motion under the
action of forces on bodies for resulting motions.
2
Kinematics
Kinetics
study of motion under the action of forces on
bodies about the bodies motions.
study of objects motion without reference
to the forces that cause the motion
kinematics relation is necessary to solve
kinetics problem
How does wAB related with another wCD
(kinematics)
If you want aCD 36.5 rad/s2, how much torques
do you apply to AB
(kinetics)
3
Kinematics
Kinetics
Particles
Before Midterm
After Midterm
Rigid Bodies
4
Kinematics of particles
How to describe the motion?
How to describe the position?

• Frame Ref-Point Coordinate

Motion time-related position
Coordinate
Path of the particle
(x,y) coord
time-related info velocity , acceleration
A'
(r,q) coord

• from A to A' takes ?t second

?s
r
A
q

• Distant traveled
• measured along the path, scalar

O
Reference Point
You are going to learn

• Displacement ( in ?t )

r-q coord
(x,y) coord
n-t coord
relative frame
5
2/2 Rectilinear Motion (1D-motion)
By defining the axis according to the moving
direction
displacement, velocity, acceleration can be
considered as scalar quantity.
Particle moving on straight line path.
reference point
The particle is at point P at time t and point P?
at time t?t with a moving distance of ?s.
P t
P tDt
reference axis
D s
s
average velocity
Positive v is defined in the same direction
as positive s. i.e.) positive v implies
that s is increasing, and negative v implies that
s is deceasing.
instantaneous velocity
6
Rectilinear Motion

• ( similarly as )

instantaneous acceleration
or

• Eliminating dt, we have

or
3 Equations, but only 2 are independent
Positive a is defined in the same direction as
positive v (or s). Ex. positive a implies
that the particle is speeding up (accelerating)
negative a implies that the particle
is slowing down (decelerating).
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Formula Interpretations of a,v,s,t

• (1) Constant acceleration (a constant). Find
  v(t), s(t)

9
5 m
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ag (const)
Time required for 3-m falling.
s2
s1 3
2nd
Thus, the 2nd ball has time to fall
1st
Therefore, the 2nd ball travels
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Formula Interpretations of a,v,s,t

• (2) Acceleration given as a function of time,
• a f(t) . Find v(t), s(t)

Or by solving the differential equation
s(t)
v(t)
13
Formula Interpretations of a,v,s,t

• (3) Acceleration given as a function of velocity,
• a f(v)

Find s(v), t(v)
Find v(t), s(t), a(t)
14
Formula Interpretations of a,v,s,t

• (4) Acceleration as function of displacement
• a f (s) Find v(s), t(s)

Find s(t), v(t), a(t)
15
Graphical Interpretations of a, v, s, t

• The familiar slopes and areas

Area under a-t curve from t1-t2 v(t2) -
v(t1)
s-t curve
a-t curve
v-t curve
Area under v-t curve from t1-t2 s(t2) -
s(t1)
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Graphical Interpretations of a, v, s, t

• Less familiar interpretations

Find a !
v-s curve
a-s curve
Find v !
q
q
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Given the acceleration-displacement plot as
shown. Determine the velocity when x 1.4 m,
assuming that the velocity is 0.8 m/s at x 0
a-s curve
Area under curve 0.16 0.12 0.08 0
0.36
1.4
or -?
ANS
19
The velocity of a particle which moves along the
x-axis is given by
m/s , where t is in second.
Calculate the displacement ?x of the particle
during the interval from t 2 sec to t 4 sec.
Solution
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Case (b)
Case (a)
22
Find v(t), s(t) of the mass if s start at
zero and
or - ?
Altenative Solution Differential equation
s
23
Find h and v when the ball hits the ground.
24.1 m/s
36.5 m
24
SP2/1 Given
x
Find
1) t when v72
How many turns?
2) a when v32
3) total distance traveled from t1 to t4
Correct?
total distance traveled
net displacement
! total distance traveled
25
2/44 The electronic trottle control of a model
train is programmed so that the train speed
varies with position as shown. Determine the time
t required for the train to complete one lap.
Rectilinear motion?
a along the path
not total a
v, a
v, a
s
s
Rectilinear equations can be used for curved
motion if s, v, a are measured along the
curve (more on this soon)
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2/44 The electronic trottle control of a model
train is programmed so that the train speed
varies with position as shown. Determine the time
t required for the train to complete one lap.
Area vds
Slope dv/ds
Slope Constant. C
50.8 s
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Recommended Problem
2/29 2/36 2/46 2/58
slope 0 at both end
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Kinematics of particles
How to describe this particles motion?

• Framework for describing the motion

Reference Frame
Path of the particle
(x,y) coord
A'
(r,q) coord

• from A to A' takes ?t second

?s
r
A

• displacement ( in ?t )

q
O
Reference Point

• Distant traveled
• measured along the path, scalar

relative frame
You are going to learn
(x,y) coord
n-t coord
r-q coord
30
2/3 Plane Curvilinear Motion

• Motion of a particle along a curved path in a
  single plane (2D curve)

Reference Frame
Path of the particle
(x,y) coord

• from A to A' takes ?t second

A'

• displacement of the particle during time ?t

?s
(r,q) coord

• Distant traveled ?s, measured along the path,
  scalar

r
A
q
O

• Basic Concept time derivative of a vector

Average Velocity
(Instantaneous) Velocity
31
Speed and Velocity
(Instantaneous) Velocity
Path of the particle
A'
(Instantaneous) Speed
?s
A
O
Path of the particle
A
O
Velocity vector is tangent to the curve path
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Instantaneous Speed
Path of the particle
A'
?s
A
O
Time rate of change of length of the position
vector
speed
(r,q) coord
r
q
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Acceleration
Path of the particle
A'
A
O
The velocity is always tangent to the path of
the particle (frequently used in problems) while
the acceleration is tangent to the hodograph (not
very important)
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Vector Equation and reference frame
Path of the particle
Reference Frame
A'
?s
A
Vector equation is in general form, not
depending on used coordinate.
O

• Rectangular x-y
• Normal-Tangent n-t
• Polar r-?

Reference Frame (coordinate)
Usage will depend on selection. More than one can
be used At the same time.
n-t
rectangular
r-q
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Derivatives of Vectors
Derivatives of Vectors Obey the same rules as
they do for scalars
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Derivatives of Vectors
Derivatives of Vectors Obey the same rules as
they do for scalars
Path of the particle

• The reference frame
• Rectangular, x-y
• Normal-tangent, n-t
• Polar, r-?
• depend on the problem considered

Reference Frame
A'
Vector equation is in general form. No
(detailed) coordinate is need.
?s
A
O
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2/4 Rectangular Coordinates
Reference Frame
O
Path
Both divided particles, are moving in
rectilinear motion
y
O
O
x
Correct?
Basic Agreement Direction of reference axis
x,y do not change on time variation.
0
0
A
Rectilinear Motion in 2 perpendicular
independent axes.
38
Path
y
y
x
O
O
x
rectilinear in 2 dimension, related which other
via time.
If given
can you find
rectilinear in y-axis
rectilinear in x-axis
Velocity is tangent to the path
( slope of curve path)
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Common Cases
Rectangular coordinates are usually good for
problems where x and y variables can be
calculated independently!
From this, you can find path of particles
Ex1) Given ax f1(t) and ay f2(t)
Ex2) Given x f1(t) and y f2(t)
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Projectile motion

• The most common case is when ax 0 and ay -g
  (approximation)
• x and y direction can be calculated independently

Note a const
a0
v
vy
vo
vx
vx
(vo)y v0 sin?
g
vy
v
?
(vo)x v0 cos?
x-axis
y-axis
vx (vo)x
x xo (vo)xt
( in case of )
41
Determine the minimum horizontal velocity u that
a boy can throw a rock at A to just pass B.
y
x
g
it can be applied in both x and y direction
42
Find x-,y-component of velocity and displacement
as function of time , if the drag on the
projectile results in an acceleration term as
specified. Include the gravitational acceleration.
k const
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Find RR(q)
y
2/95 Find q which maximizes R (in term of v-zero
and a)
x
RR(q)
Find Rmax
a
44
H12-96 A boy throws 2 balls into the air with a
speed v0 at the different angles q1, q2 (q1 gt
q2). If he want the two ball collide in the mid
air, what is the time delay between the 1st throw
and 2nd throw.
The first throw should be q1 or q2?
q1
intersected point
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2/84 Determine the maximum horizontal range R of
the projectile and the corresponding launch angle
q.
this way?
Yes! but why?
h
q45 ?
How do you throw the ball?
h(q)
R
R(h)
Ceils height (5 m) is our limitation!
R(q)
46
2/84 Determine the maximum horizontal range R of
the projectile and the corresponding launch angle
q.
this way?
Yes! but why?
h
q45 ?
How do you throw the ball?
h(q)
R
R(h)
Ceils height (5 m) is our limitation!
R(q)
47
2/84 Determine the maximum horizontal range R of
the projectile and the corresponding launch angle
q.
this way?
48
H12/92 The man throws a ball with a speed v15
m/s. Determine the angle q at which he should
released the ball so that it strikes the wall at
the highest point possible. The room has a
ceiling height of 6m.
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2/5 Normal and Tangential Coordinates (n-t)
Curves can be considered as many tangential
circular arcs

• takes positive t in the direction of increasing
  s

Fixed point on curve
Path known

• and positive n toward the center of the
  curvature of the path

s

• the origin and the axes move (and rotate) along
  with the path of particle

Forward velocity and forward acceleration make
more sense to the driver
The driver is only aware of forward direction (t)
and lateral direction (n).
Brake and acceleration force are often more
convenient to describe relative to the car
(t-direction).
Turning (side) force also easier to describe
relative to the car (n-direction)
51
Normal and Tangential Coordinates (n-t)
generally, not total a
Rectilinear Similarity
Path known
v,a
s
s
s
Consider scalar variables (s) along the path
(t direction)
The reason why we define this coordinate
similar to rectilinear motion
0 why?
s measured along the path
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Velocity
Small curves can be considered as circular arcs
The velocity is always tangent to the path
Path
Speed
A?
d?
? (d?)
A
Fixed point on curve
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Acceleration
Path
t
C
(similarly)
?
db
d?
A?
n
ds ?(d?)
A
d?
d?
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Alternative Proof of
Path
y
t
C
?
?
n
A
O
x
55
Understanding the equation
Path
C
t
?
A?
d?
n
ds ?(d?)
A
at comes from changes in the magnitude of
an comes from changes in the direction of
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What to remember
by definition of n-t axis
generally, not total a
Rectilinear Similarity
v,a
s
s
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Proof
y
r
db
bdb
ds
dy
b
dx
x
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Direction of vt an
an is always plus. Its direction is toward the
center of curvature.
B
B
Speed Increasing
Speed Decreasing
Inflection point
Inflection point
A
A
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n-t coordinates are usually good for problems
where Curvature path is known
1) distance along the curvature path (s) is
concerned
s
2) curvature radius (r) is concerned.
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A rocket is traveling above the atmosphere such
that g 8.43 m/sec2. However because of thrust,
the rocket has an additional acceleration
component a1 8.80 m/sec2 and the velocity v
8333.33 m/sec. Compute the radius of curvature ?
and the rate of change of the speed
Find r
Find
t
a1
30
an
gt ?
60
n
g 8.43 m/sec2
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The driver applies her brakes to produce a
uniform deaccceleration. Her speed is 27.8 m/s at
A and 13.89 m/s at C. She experience a
acceleration of 3 m/s2 at A. Calculate 1) the
radius of curvature at A 2) the acceleration at
the inflection point B 3) the total acceleration
at C
Condition at A
Condition at B
Condition at C
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Circular Motion (special case)
direction n? t?
t
v
at
n
r
an
?
Particle is moving clockwise
with speed increasing.
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2/123 Determine the velocity and the acceleration
of guide C for a given
value of angle q if
C and P shares the vertical velocity,
acceleration.
t
n
q
q
q
q
q
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The motorcycle starts from A with speed 1 m/s,
and increased its speed along the curve at
t
b
Determine its velocity and acceleration at the
instant t 5s .
6.25
velocity is the vector (magnitude direction)!
what is x , when t 5 ?
6.25
3.184
Numerical Method
0
s0, x0
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The motorcycle starts from A with speed 1 m/s,
and increased its speed along the curve at
n
t
b
6.25
b
Determine its velocity and acceleration at the
instant t 5s .
0.1
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2/6 Polar Coordinates ( r - ? )
Radar Coordinate
t r q
0.0 25.1 32.0
0.1 26.2 35.0
0.2 26.1 39.0
0.3 24.8 40.0
0.4 23.2 37.0
0.5 25.2 35.0
Detect
What is the velocity and acceleration of the
plane?

• direction of direction of positive r
  

• direction of direction of positive ?

A
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Polar Coordinates ( r - ? )

• direction of direction of positive r
  

• direction of direction of positive ?

A
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Velocity and Acceleration
r
q
.. the change of the length of the vector
. the rotation of the vector
Physical meaning will be discussed next page
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Understanding the acceleration equation
(in r direction)
(in ? direction)
(in ? direction)
(in -r direction)
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Understanding the acceleration equation

• Magnitude change of

(in r direction)

• Direction change of

(in ? direction)

• Magnitude change of

(in ? direction)

• Direction change of

(in r direction)
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Radar Coordinate
t r q
0.0 25.1 32.0
0.1 26.2 35.0
0.2 26.1 39.0
0.3 24.8 40.0
0.4 23.2 37.0
0.5 25.2 35.0
Detect
What is the velocity and acceleration of the
plane?
Detect
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Circular Motion (Special Case)
r const
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r-? coord
n-t coord

• direction depends on its curvature path.

• r-? coord depend on ref point, ref axis

Usually, Path need to be known
s
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Problem types
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2/145 The angular position of the arm is given
by the shown function, where ? is in radians and
t is in seconds. The slider is at r 1.6 m (t
0) and is drawn inward at the constant rate of
0.2 m/s. Determine the magnitude and direction
(expressed by the angle relative to the x-axis)
of the velocity and acceleration of the slider
when t 4.
At t 4s
v
Ans
y
a
b
q
q
x
Ans
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Motion of the sliding block P in the rotating
radial slot is controlled by the power screw as
shown. For the instant represented,
Also, the screw turns at a constant
speed giving For this
instant, determine the magnitude of the velocity
and acceleration of P
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At the bottom of loop, airplane P has a
horizontal velocity of 600 km/h and no horizontal
acceleration. The radius of curvature of loop is
1200 m. determine the record value of
for this instant.
Use n-t coord to find v,a
a
?
v
?
?
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The piston of the hydraulic cylinder gives pin A
a constant velocity v 1.5 m/s in the direction
shown for an interval of its motion. For the
instant when ? 60, determine
2D vector equation
x-y coord
r-q coord
v
q
60
r-q coord
From viewpoint of
From viewpoint of
v
x-y coord
A
A
150
mm
150
mm
O
O
acceleration 0
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The piston of the hydraulic cylinder gives pin A
a constant velocity v 1.5 m/s in the direction
shown for an interval of its motion. For the
instant when ? 60, determine
2D vector equation
x-y coord
r-q coord
q
60
r-q coord
From viewpoint of
From viewpoint of
v
x-y coord
A
A
150
mm
150
mm
O
O
acceleration 0
84
The piston of the hydraulic cylinder has a
constant velocity v 1.5 m/s For the instant
when ? 60, determine
60
60
x-y coord
r-q coord
mag?
mag?
mag?
mag0
Only max 2 unknown to be solved
x-y coord
0
r-q coord
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velocity
mag?
mag?
mag?
q
q
x-y coord
r-q coord
60
Alternate Solution
?
?
q
Acceleration
mag?
mag?
mag?
x-y coord
r-q coord
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vy
15.401 m
vx
32.495 deg
r
q
x-y coord
At t 0.5 s
12.990 m
27.337 m/s
6.274 m
-0.353 rad/s
25.981 m/s
10.095 m/s
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The slotted link is pinned at O, and as a result
of the constant angular velocity 3 rad/s, it
drives the peg P for a short distance along the
spiral guide r 0.4 q m, where q is in
radians. Determine the velocity and acceleration
of the particle at the instant it leaves the slot
in the link, i.e. when r 0.5 m
Use r-q where reference-origin is at O, and
reference axis is horizontal line.
at r0.5
Constrained motion
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Summary Three Coordinates (Tool)
Velocity
Acceleration
Reference Frame
Reference Frame
Path
Path
x
x
y
r
Observers measuring tool
y
r
O
Observer
Observer
(x,y) coord
(n,t) coord velocity meter
(r,q) coord
r
q
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Choice of Coordinates
Velocity
Acceleration
Reference Frame
Reference Frame
Path
Path
x
x
y
r
Observers measuring tool
y
r
O
Observer
Observer
(x,y) coord
(n,t) coord velocity meter
(r,q) coord
r
q
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Translating Observer
Two observers (moving and not moving) see the
particle moving the same way?
Translating-only Frame will be studied
today
No!
Path
Observers Measuring tool
Which observer sees the true velocity?
Observer B (moving)
(x,y) coord
A
both! Its matter of viewpoint.
This particle path, depends on specific
observers viewpoint
(n,t) coord velocity meter
relative absolute
Observer O (non-moving)
Point if O understand Bs motion, he can
describe the velocity which B sees.
r
(r,q) coord
q
Two observers (rotating and non rotating) see the
particle moving the same way?
No!
translating rotating
Rotating axis will be studied later.
Observer (non-rotating)
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2/8 Relative Motion (Translating axises)

• Sometimes it is convenient to describe motions
  of a particle relative to a moving reference
  frame (reference observer B)

• If motions of the reference axis is known, then
  absolute motion of the particle can also be
  found.

• A a particle to be studied

• B a (moving) observer

Reference frame O
Reference frame B

• Motions of A measured by the observer at B is
  called the relative motions of A with respect to
  B

• Motions of A measured using framework O is
  called the absolute motions

• For most engineering problems, O attached to the
  earth surface may be assumed fixed i.e.
  non-moving.

frame work O is considered as fixed
(non-moving)
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Relative position

• If the observer at B use the x-y coordinate
  system to describe the position vector of A we
  have

Y

• Here we will consider only the case where the x-y
  axis is not rotating (translate only)

X
other coordinates systems can be used e.g.
n-t.
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Relative Motion (Translating Only)

• x-y frame is not rotating (translate only)

y
Y
Direction of frames unit vectors do not change
x
0
X
Notation using when B is a translating frame.
Note Any 3 coords can be applied to Both 2
frames.
0
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Understanding the equation
Path
Translation-only Frame!
Observer B
A
O B has a relative translation-only motion
This particle path, depends on specific
observers viewpoint
Observer O
reference framework O
reference frame work B
Observer B (translation-only Relative velocity
with O)
Observer O
Observer O
This is an equation of adding vectors of
different viewpoint (world) !!!
100
The passenger aircraft B is flying with a linear
motion to theeast with velocity vB 800 km/h. A
jet is traveling south with velocity vA 1200
km/h. What velocity does A appear to a passenger
in B ?
101
Translational-only relative velocity
You can find v and a of B
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Is observer B a translating-only observer
B
relative with O
Yes
Yes
O
Yes
No
?
104
To increase his speed, the water skier A cuts
across the wake of the tow boat B, which has
velocity of 60 km/h. At the instant when
? 30, the actual path of the skier makes an
angle ? 50 with the tow rope. For this
position determine the velocity vA of the skier
and the value of
Relative Motion (Cicular Motion)
Consider at point A and B as r-? coordinate
system
30
A
A
o
o
B
B
30
D
?
?
O.K.
M
Point Most 2 unknowns can be solved with 1
vector (2D) equation.
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2/206 A skydriver B has reached a terminal speed
. The airplane has the
constant speed and is
just beginning to follow the circular path shown
of curvature radius 2000 m Determine (a) the
vel. and acc. of the airplane relative to
skydriver. (b) the time rate of change of the
speed of the airplane and the radius of
curvature of its path, both observed by
the nonrotating skydriver.
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(b) the time rate of change of the speed
of the airplane and the radius of curvature
of its path, both observed by the nonrotating
skydriver.
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