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Unformatted text preview: III4 Chapter III: Moving Reference Frame Kinematics Homework Problem III.2
Given: A telescoping arm OP is pinned to a cart at end O. The cart moves along a horizontal
˙
surface with a constant speed of vO . The angle of the arm, θ, is increasing at a constant rate of θ
˙ . The xyz coordinate system is attached to the telescoping arm with
and is extending at a rate of L
its origin at end O of the arm. ing Reference Frame Kinematics Homework Problems ME 274 Find: For this problem: blem III2
• Determine the velocity and acceleration of particle P. Express your answers as vectors in
elescoping arm OP is pinned theircart at end O. The cart moves along a horizontal
terms of to a xy components.
face with a constant speed of vO . The angle of the arm, !, is increasing at a constant
• Make a sketch !
!
L The xyz coordinate system vectors found above.
of ! and is extending at a rate of of .the velocity and accelerationis attached to the
scoping arm with its origin at end O of the arm.
a) Determine the velocity and
acceleration of particle P.
Express your answers as vectors
in terms of their xy
components.
b) Make a sketch of the velocity
and acceleration vectors found
in a) above. P x L
y
O !
vO ˙
Use the following parameters in your analysis: θ = !90◦ , vO = 6m/sec, θ = 5rad/sec, L = 2m,
the following ˙parameters in your analysis: ! = 0 , vO = 10 m / s , ! = "3 rad / s ,
¨ = 3m/sec2 .
L = 0 and L
!!
!
4 m , L = !3 m / s and L = 0 . Chapter III: Moving Reference Frame Kinematics Homework Moving Reference Frame Kinematics Homework Problems ME 274 III11 Problem III.9 Problem Shaft AB rotates about a ﬁxed axis with a constant rotational speed of Ω. A tube is hinged
Given: III9
˙
Shaftshaftrotates about a fixed axis withtheconstant rotational speed of !.constantis
on AB AB with the angle θ between a tube and shaft increasing at a A tube rate of θ. Particle
P moves within the tube at a constant rate the tube and to the tube. The a constant
˙
hinged on shaft AB with the angle " between of xP relativeshaft increasing atXYZ coordinate system
is ﬁxed Particle Xaxis aligned the the ﬁxed rotation axis x the shaft AB. The xyz
!
rate of ! . with theP moves within withtube at a constant rate ofof! P relative to the tube. coordinate
system is attached to the tube with the xaxis aligned with the tube for all time. For the position
The XYZ coordinate system is fixed with the Xaxis aligned with the fixed rotation axis
shown below, the z and Zaxes are aligned.
of the shaft AB. The xyz coordinate system is attached to the tube with the xaxis aligned
with the tube for all time. For the position shown below, the z and Zaxes are aligned.
Find: For the position shown: For th• position shown:
e Determine the angular velocity and angular acceleration of the tube. Write your answers as
vectors in the a of their xyz components.
a) Determine termsngular velocity and angular acceleration of the tube. Write your
answers as vectors in terms of their xyz components.
• Determine the acceleration of particle P. Write your answer as a vector in terms of its xyz b) Determine the acceleration of particle P. Write your answer as a vector in terms
components.
of its xyz components.
x
Y
y xP " P
X O
A !
B U the following parameters in your analysis: ! = = ad / s , ! θ = , ◦! θ = r 4rad/sec
Use se the following parameters in your analysis: Ω 4 r3rad/sec, = 30°60!, =˙ "2 −ad / s , , xP = 2m
!
xPand3xP and 2m/sec. m / sec .
= ˙m = − xP = !4 Chapter III: Moving Reference Frame Kinematics Homework III13 Problem III.11
Given: Block A slides within a straight slot cut in a rotating disk with the position of A given by
x = 2sin (4π t), where x and t are given in inches and seconds, respectively. The angular orientation
of the disk is given by θ = 0.2sin (8π t), where and t are given in radians and seconds, respectively.
Find: Determine the acceleration of block A when:
• x = Problems ˙
e Frame Kinematics Homework 0 with x 11 (5/167) >0 • x = 2inches es within a straight slot cut in a rotating
position of A given by x = 2 sin 4! t ,
t are given in inches and seconds,
The angular orientation of the disk is
0.2 sin 8" t , where ! and t are given in
econds, respectively. Determine the
of block A when !
with x > 0 .
inches. ME 274 Chapter III: Moving Reference Frame Kinematics Homework III17 Problem III.15
Given: System shown.
Frame Kinematics Homework Problems 5 (7/25) ME 274 Find: Determine the angular velocity and angular acceleration of shaft B. Assume all rotation
rates to the constant. angular velocity and angular
f shaft B. Assume all rotation rates
t. Use the following parameters in your analysis: Ω = 4rad/sec, ω0 = 3rad/sec, γ = 30◦ and
ing parameters˙ in your analysis:. ! = 4 rad / sec , ! 0 = 3 rad / sec ,
γ = (π /4) rad/sec ! = " / 4 rad / sec . III18 Chapter III: Moving Reference Frame Kinematics Homework Problem III.16
Given: System shown.
K inematics Homework Problems /51) ME 274 Find: Determine the angular acceleration of the rotor of the gyroscope. Assume all rotation rates
to the constant. ular acceleration of the rotor of the
Assume all rotation rates to the constant. Use the following parameters in your analysis: N = 20rev/sec, γ = 30◦ and γ = 4rad/sec.
˙ parameters in your analysis: N = 20 rev / sec , ! = 30° and Chapter III: Moving Reference Frame Kinematics Homework III19 Problem III.17
Moving Reference Frame Kinematics Homework Problems ME 274 ˙
Given: Person A is sitting on a merrygoround that is rotating at a constant rate θ. Person B is
running in a straight line at a constant speed vB along the sidewalk. Problem III17
!
Person A is sitting this problem: round that is rotating at a constant rate ! . Person B is
Find: For on a merrygo
running in a straight line at a constant speed vB along the sidewalk. ! • Determine the velocity and acceleration of person B as seen by the moving observer, person
A, at the instant shown. Use vB = 10m/sec θ = 3rad/sec, θ = 30◦ ,
(a) Determine the velocity and acceleration of person ,B˙ as seen by the movingr = 1.5m, d = 8m, and
h = 4m in A, at the instant
!
observer, person your analysis. shown. Use v = 10 m/s, ! = 2 rad/s, ! = 60°, r
B = 1 • ,Show5thatand h answerin your ﬁrst part is independent of the location of observer A. That is,
m d = m, your = 3 m in the analysis.
(b) Show showythatanswer in the first part is independent of velocity and acceleration of the runner B,
that our any observer on the disk sees the same the location of observer
A. That is, show of the observer’s location.disk sees the same velocity and
regardless that any observer on the
acceleration of the runner B, regardless of the observer's location. !
! ! Moving Reference Frame Kinematics Homework Problems !
III22 ME 274 Chapter III: Moving Reference Frame Kinematics Homework Problem III20!
At the instant shown, the shaft rotates with an angular velocity !P and has an angular
Problem III.20
!
acceleration !P . At the same instant, the disk spins about its axle with an angular
!
velocity !S and has an angular acceleration !S . an angular velocity angular acceleration
Given: At the instant shown, the shaft rotates with Determine the total ωP . At the same instant,
theodisk spins and the accelerationan angular velocity ωS .
f the disk about its axle with of point C at this instant.
Find: Determine the! = 1angular ! = 3 rad/s2, and disk and the ivelocityanalysis. C at this
!P
!
Use !P = 8 rad/s, total 2 rad/s, acceleration of the !S = 2 rad/s2 n your of point
S
instant.
! ! Use the following parameters in your analysis: ωP = 10rad/sec, ωS = 8rad/sec, ωP = −2rad/sec2
˙
and ωS = 3rad/sec2 .
˙ Moving Reference Frame Kinematics Homework Problems ME 274 !
Chapter III: Moving Reference Frame Kinematics Homework
Problem III21! III23 At the instant shown, the helicopter is moving upwards with speed vH and acceleration
aH . At the same instant, the frame H, not the horizontal blade, rotates about a vertical Problem III.21 axis At a constant shown, the helicopter . moving upwards rotates with a and acceleration
Given:with the instant angular velocity of !H is If the tail blade Bwith speed vH constant
aH angular velocity !B / H ,the frame H, (not the H, determine the velocityabout a vertical axis with
. At the same instant, measured relative to horizontal blade) rotates and acceleration of
a constant angular velocity of ωH . The tail blade B rotates with a constant angular velocity ωB/H ,
point P, located on the end of the blade, at the instant the blade is in the vertical position.
measured relative to H.
Use v = 4 m/s, a = 2 m/s2, ! = 0.6 rad/s, and ! = 200 rad/s in your analysis. H
H
B
Find: Determine the velocity andHacceleration of point/ HP, located on the end of the blade, at the
instant the blade is in the vertical position. Use the following parameters in your analysis: vH = 6m/sec, aH = 3m/sec2 , ωH = 0.5rad/sec and
ωB/H = 250rad/sec. Moving Reference Frame Kinematics Homework Problems ME 274 Chapter III: Moving Reference Frame Kinematics Homework III27 Problem III25
Given: The mechanism shown below is driven by a motor attached at B. End A of
Problem III.25
link AB is free to slide within a straight slot within link OC. The angular
elocity of !AB i a constant 2 is driven by a motor attached
Given: vThe mechanism sshown below rad/s in the direction shown. at B. End A of link AB is
free to slide within a straight slot within link OC. The angular velocity of ωAB is in the direction
shown. For the position shown below, determine
Find:
Find: For a) tposition shown, determine:
the he angular velocity of OC. b) the velocity of A relative to an observer positioned on OC and rotating
with OC. • the angular velocity of OC. • the velocity of A relative to an observer positioned on OC and rotating with OC.
c) the angular acceleration of OC.
• the angular acceleration of OC. elative to an observer positioned on OC and
d) the acceleration of A r rotating with OC. • the acceleration of A relative to an observer positioned on OC and rotating with OC. Express your answers as vectors in terms of either their xyz or their XYZ components.
Y B !AB
30 ° y # L 0.17 m x
C " 60 °
O A
0.1 m X Use the following parameters in your analysis: ωAB = 2rad/sec, L = 0.17m, θ = 30◦ and φ = 60◦ . a) find the angular velocity vector of the boom OP.
b) find the angular acceleration vector of the boom OP. ( ) ( ) c) evaluate the Frame aO , v P / O
, a P/O
Chapter III: Moving Reference vectorsKinematics Homework rel and r P / O in the
rel III31 following equation for the acceleration of point P:
a P = aO + a P / O
+ ! " r P / O + 2# " v P / O
+ # " # " r P/O
Problem III.29
rel
rel ( ) ( ) ( ) Note that you are NOT asked to evaluate the right hand side of this Given: A crane is moving to the right with a constant speed of vC and is rotating about a vertical
equation for a P .
˙
˙
axis with a constant rate of Ω. The boom is being raised at a rate of θ with θ changing at a rate
¨.this problem, use an observer attached to the boom and with a moving set of xyz
of θ
For coordinate axes also attached to the boom, as shown in the figure below. XYZ are fixed
Find:
axes. Determine the acceleration of end P of the boom.
Y ! P x d
vC L
y
"
O X Use the following parameters in your analysis: vC = 0.5f t/sec, Ω = 1.5rad/sec, θ = 36.87◦ ,
˙
¨
θ = 0.6rad/sec, θ = 1.2rad/sec2 , L = 20f t and d = 4f t. ...
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 Fall '09

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