Unformatted text preview: Chapter II: Planar Rigid Body Kinematics Homework Chapter II Planar Rigid Body Kinematics
Homework II1 II2 Chapter II: Planar Rigid Body Kinematics Homework Planar Kinematics Homework Problems
Chapter II: Planar Rigid Body Kinematics Homework ME 274 II3 Problem II.1
Problem II1
Given: The rectangular BCD ABCD rotates about about a pivot at point If, At the instant shown,
The rectangular plate A plate freely freely rotates a pivot at point A. A. at
the acceleration of point C ispoint C is C = axby ay ˆ.= ax i + ay j , determine the angular
i
shown, the acceleration of given by agiven ˆ + a j
C velocity ! and angular acceleration " of the plate. Find: Determine the angular velocity ω and angular acceleration α of the plate.
y B C
0.25 m A D
0.5 m x Use the following parameters in your analysis: ax = 0 and ay = −20m/sec2 . Use the following: ax = !10 m / sec 2 and ay = 0 . aC II4 Chapter II: Planar Rigid Body Kinematics Homework Planar Kinematics Homework Problems ME 274 Problem II.2 Problem The pulley shown below freely rotates about point C and interacts with two rubber belts
Given: II2
(one horizontal, one vertical). The velocity of point A on the drive belt is measured to be A = vAˆ,
v
i
The pulley shown belowpoint Brotates aboutbelt is C and interacts B = two ,rubber belts do not
with aB ˆ and the belts
and the acceleration of freely on the load point measured to be a
j
(one horizontal, one vertical). If the velocity of point A on the drive belt is measured to
slip. be v A = vA i , the acceleration of point B on the load belt is measured to be a B = aB j , and
Find: do not slip, find the velocity and acceleration of D on the
the belts Determine the velocity and acceleration of point point D. pulley. y A D C
x 0.1 m 0.025 m
B
Usese the following vA = 10 m / secyour analysis:2vm =sec 2 .
U the following: parameters in and aB = ! A / 20m/sec and aB = −5m/sec2 . Chapter II: Planar Rigid Body Kinematics Homework
r Kinematics Homework Problems II5
ME 274 Problem II.3 blem II3 Given: The car lift shown below is controlled by a horizontallymounted hydraulic cylinder attached to point A. The by a horizontally a constant speed v . cylinder
car lift shown below is controlled cylinder extends atmounted hydraulic Assume the upper platform remains
horizontal cylinder extends at a constant speed v, find the vertical
hed to point A. If the and all joints are pinned. ponent of the platform’s velocity for an arbitrary value of !. Assume the upper
Find: Determine the vertical component of the platforms velocity for an arbitrary value of θ.
orm remains horizontal and all joints are pinned. II6 Chapter II: Planar Rigid Body Kinematics Homework
Planar Kinematics Homework Problems ME 274 Problem II.4 Given: The compound wheel assembly shown below is driven by a cable attached to the outer rim
Problem II4
of the assembly at point A. The wheel rolls without slip at point B, and the acceleration at point
A isThe compound xwheel ˆassembly shown below is driven by a cable attached to the outer
given by = a ˆ + ay j
a
i rim of the assembly at point A. Assuming that the wheel rolls without slip about point B,
and the acceleration at point A is given by a A = ax i + ay j , determine the acceleration of Find: Determine the acceleration of the center point C of the pulley. point C. y A !
x C 0.025 m B
0.1 m
Use Use following parameters in your analysis: sec 2 0,ndx a 8m/sec2sec 2 . ay = 12m/sec2 .
the the following: ! = 36.87° , a = 8 m / φ = a a = = 6 m / and
x y Chapter II: Planar Rigid Body Kinematics Homework II7 Planar Kinematics Homework Problems ME 274 Problem II.5 Problem II5 Given: The wheel shown below rolls without slip on cart A. The cart has velocity A = vAˆ and
v
i
ˆ. The center of without slip onvelocity Assuming that the cart has C = aCˆ.
ˆ and acceleration velocity
acceleration A =shown below rolls the wheel has cart A. vC = vC i
a
a
i
The wheel aA i v A = vA i and acceleration a A = aA i , and the center of the wheel has velocity vC = vC i
and acceleration aC = aC i , determine the velocity and acceleration of point B. Find: Determine the velocity and acceleration of point B. y
r B C
x A
aA Use the following parameters in your analysis: r = 0.5m, vA = 5m/sec, vC = −10m/sec, aA =
−8m/sec2 and aC = −6m/sec2 . Use the following: r = 2 ft , vA = 10 ft / sec , vC = !5 ft / sec , aA = !10 ft / sec 2 and
aC = !6 ft / sec 2 . II8 Chapter II: Planar Rigid Body Kinematics Homework Planar Kinematics Homework Problems ME 274 Problem II.6 Problem II6 Given: At the instant shown, the piston has velocity A = vAˆ.
v
i If, at the instant shown, the piston has velocity v A = vA i , determine the resulting angular Find: Determine the resulting angular velocity ωBC of the crankshaft.
velocity ! CB of the crankshaft. B
0.06 m
C y !
A
x 0.03 m
Use the following parameters in your analysis: φ = 30◦ and vA = −50m/sec. Use the following: ! = 53.13° and vA = !20 m / sec . v Chapter II: Planar Rigid Body Kinematics Homework II9 Problem II.7
Consider the four mechanisms shown below. Respond to the questions posed. Feel free to draw
directly on these sheets and submit for your homework solution. Use a straight edge when making
ics Homework Problems
ME 274
your drawings. 7
MECHANISM NO. 1
four mechanisms shown below. Respond to the questions posed. Feel free to
Link OA is rotating in the clockwise sense. Determine the
y on these sheets and submit for your homework solution. Use a straight location of the instant center (IC) of
link AB.
aking your drawings. From the location of this IC, determine the sense of rotation for links AB and BC. Justify
your answers in words. #1
rotating in the clockwise sense.
he location of the instant center (IC) of
m the location of this IC, determine the
tion for links AB and BC. Justify your
ords. #2
he location of the instant center
AD. From the location of this IC,
e sense of rotation for links AD
tify your answers in words. II10 Chapter II: Planar Rigid Body Kinematics Homework MECHANISM NO. 2
Determine the location of the instant center (IC) of link AD. From the location of this IC, determine
the sense of rotation for links AD and DB. Justify your answers in words. #2
he location of the instant center
AD. From the location of this IC,
he sense of rotation for links AD
stify your answers in words. Chapter II: Planar Rigid Body Kinematics Homework II11 MECHANISM NO. 3
Link OA is rotating in the clockwise sense. Determine the location of the instant center (IC) of
link
cs Homework Problems AB. From the location of this IC, determine the sense E 274
M of rotation for link AB and the sense of
translation for the piston. Justify your answers in words. #3
otating in the clockwise sense.
e location of the instant center (IC)
rom the location of this IC,
e sense of rotation for link AB and
translation for the piston. Justify your
ords. #4
B are moving in the directions
he speeds provided. Assume that the
t slip on the racks. Determine the
e instant center (IC) of the gear.
ation of this IC, determine the sense
r the gear. Justify your answers with
words. vA = 0.8 m/s vB = 0.6 m/s II12 Chapter II: Planar Rigid Body Kinematics Homework MECHANISM NO. 4
Racks A and B are moving in the directions shown with the speeds provided. Assume that the gear
does not slip on the racks. Determine the location of the instant center (IC) of the gear. From the
location of this IC, determine the sense of rotation for the gear. Justify your answers with numbers
and words. are moving in the directions
speeds provided. Assume that the
ip on the racks. Determine the
nstant center (IC) of the gear.
n of this IC, determine the sense
he gear. Justify your answers with
rds. vA = 0.8 m/s vB = 0.6 m/s Chapter II: Planar Rigid Body Kinematics Homework II13 lanar Kinematics Homework Problems ME 274 Problem II.8 roblem II8 Given: The mechanism below is driven by a motor at point A, which produces an angular velocity
of ωAB in the direction shown.
he mechanism below is driven by a motor at point A, which produces an angular
elocity of ! ABFind: Using the shown. of instantaneous centers, ﬁnd the velocity of find C.
in the direction method Using the method of instantaneous centers, point
e velocity of point C.
!
AB B 0.2 m
C 0.1 m
A " # Use the following parameters in your analysis: φ = 0, θ = 60◦ and ωAB = 5rad/sec. se the following: BC is horizontal, ! = 0 , ! = 30° and ! AB = 8 rad / sec . II14 Chapter II: Planar Rigid Body Kinematics Homework
Planar Kinematics Homework Problems ME 274 Problem II.9 P The sliding
Given: roblem II9 collar moves horizontally along the shaft producing an oscillation in AB. At
the instant shown, AB and BC are horizontal and vertical, respectively, and the velocity of C is
The sliding collar moves C moves to the right.
not changing (vC = constant) ashorizontally along the shaft producing an oscillation in AB. If,
at the instant shown, AB and BC are horizontal and vertical, respectively, and the velocity
f C is not changing ( vC = cons ant )
Find: oDetermine the angular acceleration tof AB.as C moves to the right, find the angular
acceleration of AB. A r B L C vC
Use the following: L = 0.6 m , r = 0.8 m and v = 25 m / sec . Use the following parameters in your analysis: L = 2fC, r = 1.5f t and vC = 40f t/sec.
t Chapter II: PlanarHomework Problems
Planar Kinematics Rigid Body Kinematics Homework II15
ME 274 Problem II.1010
Problem II
Given: The mechanism below is drivenby a motor at point A,A, which produces angular velocity
The mechanism below is driven by a motor at point which produces an an angular
of ωAB and angular acceleration of αAB . BCE and CD are horizontal and vertical, respectively, at
velocity of ! A and angular acceleration of ! AB . Provided BCE and CD are horizontal
the instant shown. B and vertical, respectively, at the instant shown, find the velocity and acceleration vectors
of Determine the velocity and acceleration vectors of point E.
Find: point E. 0.3 m Use the following parameters in your analysis: ωAB = 3rad/sec, αAB = 0 and φ = 36.87◦ . Use the following: ! AB = 8 rad / sec , ! AB = "5 rad / sec 2 and ! = 53.13° . II16 Chapter II: Planar Rigid Body Kinematics Homework Problem II.11
Given: Bar OP rotates about a shaft passing through end O of the bar. At the instant shown,
the angular velocity and angular acceleration of OP are given by ω and α.
atics Homework Problems
ME 274 II11 (5/12) Find: Determine the velocity and acceleration of end P. Write your answers as vectors.
tates about a shaft passing
d O of the bar. At the instant
e angular velocity and angular
n of OP are given by ! and ! .
the velocity and acceleration of
ite your answers as vectors. y
P rotation of
bar h
O
b x Use the following parameters in your analysis: ω = 2rad/sec (CCW), αAB = 3rad/sec2 (CW),
b ) ! mm and h c 2 ( CW )
rad / sec ( CCW= ,500= 3 rad / se= 200mm,. b = 500 mm and h = 200 mm in sis. Chapter II: Planar Rigid Body Kinematics Homework II17 Problem II.12
Given: Point A of plate ABC moves to the right with a speed of vA as the plate rotates with an
angular speed of ω .
Find: For the instant shown, determine the velocity of point C on the plate. Write your answer
ME 274
as a vector. omework Problems (5/61)
ABC moves to the right with a
the plate rotates with an angular
r the instant shown, determine the
t C on the plate. Write your answer Use the following parameters in your analysis: vA = 3f t/sec and ω = 5rad/sec (CCW). sec and ! = 5 rad / sec ( CCW ) in your analysis. II18 Chapter II: Planar Rigid Body Kinematics Homework Problem II.13
Given: Link OA of the mechanism shown rotates with an angular speed of ω0 . At the position
shown, link BC is vertical, and the coordinates of pin A are given by (xA , yA ).
omework ProblemsFind: (5/88) ME of
For this position, determine the angular velocities 274 links AB and BC. Write your answers
as vectors. mechanism shown rotates with an
f ! 0 . At the position shown, link BC
the coordinates of pin A are given by
is position, determine the angular
ks AB and BC. Write your answers as Use the following parameters in your analysis: ω0 = 10rad/sec and (xA , yA ) = (−60, 80) mm. d / sec and ( x A , yB ) = ( !60, 80 ) mm in your analysis. Chapter II: Planar Rigid Body Kinematics Homework II19 Problem II.14
mework Problems Given: The vertical control rod of the mechanism shown moves downward with a speed of v.
ME 274 5/108) Find: Use the method of instant centers to determine the velocity of end C of link ABC. rol rod of the mechanism shown moves
speed of v. Use the method of instant
ine the velocity of end C of link ABC. our analysis. Use the following parameter in your analysis: θ = 60◦ . II20 Chapter II: Planar Rigid Body Kinematics Homework Problem II.15
Given: Link OA rotates at a constant rate of ω . At the instant shown, link OA is aligned with
the horizontal line of OC and link AB is vertical.
Find: For this position, determine the angular accelerations of links AB and BC. Write your
ME 274
answers as vectors. tics Homework Problems 15 (5/137)
tates at a constant rate of ! . At the
n, link OA is aligned with the
ine of OC and link AB is vertical.
ition, determine the angular
s of links AB and BC. Write your
ectors. Chapter II: Planar Rigid Body Kinematics Homework II21 Problem II.16
Given: The orientation angle θ for the circular disk of radius R is given by: θ (t) = π sinπ t (with
3
θ in radians and t in seconds).
Find: Determine the velocity and acceleration vectors for point A on the outer edge of the disk
for:
• t=0
mework Problems • t = 1/3seconds ME 274 Make sketches of these vectors for the above two instants in time. ngle ! for the circular disk of radius R is
"
sin" t (with ! in radians and t in
3
ine the velocity and acceleration vectors
e outer edge of the disk for: A !
R O conds
these vectors for the above two instants
Use the following parameter in your analysis: R = 0.5m our analysis. II22 Chapter II: Planar Rigid Body Kinematics Homework Problem II.17
Given: Thin bar AB (having a length of L) moves in a way that ends A and B slide along straight,
ﬁxed guides as shown in the ﬁgure. End A has a constant speed of vA to the right as the bar moves.
omework Problems ME 274 Find: Determine the velocity and acceleration of end B. ving a length of L) moves in a way
B slide along straight, fixed guides as
ure. End A has a constant speed of
s the bar moves. B elocity and acceleration of end B. L ! ! A vA Use the following parameters in your analysis: L = 2f t, vA = 4f t/sec, θ = 36.87◦ and φ = 120◦ . rs , vA = 3 m / sec , ! = 90° and ! = 36.87° in your analysis. Chapter II: Planar Rigid Body Kinematics Homework II23 Problem II.18
Given: Wedge A moves to left with a speed of vA . A circular disk is able to roll without slipping
on the inclined surface of the wedge. At the instant shown, the velocity of the center of the disk O
is moving straight down with an unknown speed of vO .
Find: For this problem:
• Determine the angular velocity of the disk and the velocity of O. Write your answers as
vectors.
• Determine the velocity of point B on the disk that is directly above O on the outer surface
of the disk. Write your answer as a vector. nematics Homework Problems ME 274 m II18 B
A moves to left with a speed of
ircular disk is able to roll
slipping on the inclined surface
edge. At the instant shown, the
of the center of the disk O is
straight down with an unknown
f vO . O R
no slip vA A vO ! Determine the angular velocity
of the disk and the velocity of O. Write your answers as vectors. Use the following parameters in your analysis: R = 2f t, vA = 40f t/sec and θ = 36.87◦ . Determine the velocity of point B on the disk that is directly above O on the outer
surface of the disk. Write your answer as a vector. = 0.4 meters , vA = 20 m / sec and ! = 53.13° in your analysis. II24 Chapter II: Planar Rigid Body Kinematics Homework ar Kinematics Homework Problems blem II19 ME 274 Problem II.19 echanism is made up of links AB, BD and DE. At the instant shown, links AB the instant shown, links AB and
Given: A mechanism is made up of links AB, BD and DE. At and
are aligned horizontally, and link DE is vertical. ForDE isposition:
BD are aligned horizontally, and link this vertical.
a) locate the instant center for link BD.
Find: For this position: b) determine the angular velocities of links BD and DE. Write your answers as
• locate the instant center for link BD.
vectors. L = 1 ft and ! AB• =determinesthe iangularanalysis. of links BD and DE. Write your answers as vectors.
"4 rad / ec n your velocities E 2L ! AB
L 2L A
B D Use the following parameters in your analysis: L = 1f t and ωAB = 8rad/sec. atics Homework Problems II20 ME 274 Chapter II: Planar Rigid Body Kinematics Homework II25 Problem II.20
ism is made up of links AB, BD and DE. At the instant shown, links AB and
rtical, and link BD is horizontal. For this position:
Given: A mechanism is made up of links AB, BD and DE. At the instant shown, links AB and termine the angular velocities of links AB and horizontal. your answers as
BD. Write
DE are vertical, and link BD is
ctors. Find: For this position:
termine the angular accelerations of links AB and BD. Write your answers as
ctors. • determine the angular velocities of links AB and BD. Write your answers as vectors. ft and ! DE = 2 r• determineonstant in your analysis. of links AB and BD. Write your answers as vectors.
ad / sec = c the angular accelerations E ! DE
2L B 2L
D L
A Use the following parameters in your analysis: L = 3f t and ωDE = 3rad/sec. II26 Chapter II: Planar Rigid Body Kinematics Homework Problem II.21
Given: Link OB rotates with a constant rate of ωOB . At the instant shown, link OB is vertical.
omework Problems (5/143) ME 274 Find: Determine the angular acceleration of link AC at this instant. Write your answer as a vector. with a constant rate of ! OB . At the
ink OB is vertical. Determine the
tion of link AC at this instant. Write
a vector. Use the following parameters in your analysis: ωOB = 0.5rad/sec (CW) and tanθ = 4/3. rad / sec ( CW ) and tan! = 4 / 3 in your analysis. Chapter II: Planar Rigid Body Kinematics Homework II27 Problem II.22
Given: Link OA of the mechanism shown rotates with a constant angular speed of ω0 . At the
position shown, link BC is vertical, and the coordinates of pin A are given by (xA , yA ).
Find:
s Homework Problems 22 (5/145) For this position, determine the angular acceleration of link AB. Write your answer as a
ME 274 vector. he mechanism shown rotates with an
lar speed of ! 0 . At the position
C is vertical, and the coordinates of
en by ( x A , yB ) . For this position,
angular acceleration of link AB.
swer as a vector. Use the following parameters in your analysis: ω0 = 10rad/sec and (xA , yA ) = (−60, 80) mm. rad / sec and ( x A , yB ) = ( !60, 80 ) mm in your analysis. II28 Chapter II: Planar Rigid Body Kinematics Homework Problem II.23 Planar Kinematics Homework Problems ME 274 Given: 23
Problem IIA mechanism is made up of links AB and BD, with point A being pinned to ground and
point B being a pin joint connecting links AB and BD. A slider is connected to BD at pin D with A mechanismbeing constrained to AB and BD, with point A being radius of R. At the position shown
the slider is made up of links move on a circular path with a pinned to ground and
point B being aAB is vertical, link BD is horizontal andA slider is connected to BD at pin position,
below, link pin joint connecting links AB and BD. D is directly above point O. At this
D with the slider being constraineda constant speed of vD path with a radius of 2.5 meters.
D is traveling to the right with to move on a circular .
At the position shown below, link AB is vertical, link BD is horizontal and D is directly
above point O.this this position, D is traveling to the right with a constant speed of 5
Find: For At position:
m/sec. For the position shown, find:
• determine the...
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 Fall '09
 Angular velocity, Velocity, planar rigid body, Rigid Body Kinematics

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