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Unformatted text preview: Chapter V: Rigid Body Kinetics Homework Chapter V Rigid Body Kinetics Homework V1 V2 Chapter V: Rigid Body Kinetics Homework Chapter V: Rigid Body Kinetics Homework V3 Rigid Body Kinetics Homework Problems ME 274 Problem V.1 Problem V1 Given: is traveling along a along a horizontal roadway with a of v .of v0 . A some point in time the
A truck A truck is traveling horizontal roadway with a speed speed A some point in time
truck driver starts braking with the truck experiencing a constant0acceleration a. A homogeneous,
the truck driver starts braking with the truck experiencing a constant acceleration a. A
triangular block of mass M rests on the bed of the truck. Assume that the block does not slip on
homogeneous, triangular block of mass M rests on the bed of the truck. Determine the
the truckbed during braking. normal reactions at supports A and B of the block during the braking. Assume that the
block Determine the the truckbed during braking.
Find: does not slip onnormal reactions at supports A and B of the block during the braking.
b h
A B Use the followingparameters in your analysis: M = 400kg , h = = mmb ter3m and a = 6m/sec2 .
Use the following parameters in your analysis: M = 300 kg , h 5 3 , e = s , b = 2 meters and a = 4 m / sec 2 . V4 Chapter V: Rigid Body Kinetics Homework Rigid Body
Problem V.2 Kinetics Homework Problems ME 274 Given: A thin, homogeneous bar of mass M is supported by parallel cables AC and BD, as shown
below Problem V2The bar is released from rest with the cables at an angle of θ measured from
in the ﬁgure.
A thin, homogeneous bar of mass M is supported by parallel cables AC and BD, as
the horizontal. shown below in the figure. The bar is released from rest with the cables at an angle of ! Find: measured from the horizontal. DetermineBD on release. cables AC and BD on release.
Determine the tension in cables AC and the tension in g C ! D L L B
A
3L/4 L/4 Use the following parameters in your analysis: M = 300kg , L = 2.5m and θ = 30◦ . Use the following parameters in your analysis: M = 250 kg , L = 3 meters and
! = 53.13° . Chapter V: Rigid Body Kinetics Homework
mework Problems Problem V.3 V5
ME 274 Given: A uniform slender rod is at rest upon a frictionless horizontal surface when a force F is
applied in the direction shown. Assume that the rod has mass m and length of L. r rod is at rest upon a frictionless horizontal surface when a force F is
ection shown. Assuming that the rod has mass m and length ofrods center of mass G, and b) the rod’s angular
Find: Determine: a) the acceleration of the L,
acceleration ofacceleration.
the rod’s center of mass G and (b) the rod’s angular ◦
Use 5 lb, F = 15 parameters ft.
: ! = 36.87°, mg =the followinglb and L = 2 in your analysis: θ = 30 , mg = 5lb, F = 20lb and L = 3f t. V6 Chapter V: Rigid Body Kinetics Homework Rigid Body Kinetics Homework Problems ME 274 Problem V.4 Problem V4 Given: The system shown below consists of a uniform slender bar of mass m and an attached
The system shown m. Point G represents the midpoint (center of mass) of the bar. The
particle of equal mass below consists of a uniform slender bar of mass m and an attached system
particle from rest in the Point G represents the midpoint (center of mass) of the bar. If
is released of equal mass m.position shown. this system is released from rest in the position shown, determine (a) the value of x for
which the system’s angular acceleration is the system’s (b) the acceleration is maximum, and
Find: Determine: a) the value of x for whichmaximum andangular corresponding angular
b) acceleration.
the corresponding angular acceleration. L/2 L/2
x O G m
m A gid Body Kinetics HChapter V: Rigid
omework Problems ME 274 Body Kinetics Homework V7 roblem V5
Problem V.5 pallet/winch/roller system is used to move crate B of mass M, where the pallet A has a
Given: A
ass of m. Determine the pallet/winch/roller system is used to move crate B of mass M, where the pallet A has
a mass of m. a) maximum winch force that can be applied to the pallet such that the crate does
not tip and
Find: Determine: a) the maximum winch force that can be applied to the pallet such that the
crate does not friction required between the crate and pallet to ensure
b) the coefficient of statictip, and b) the coeﬃcient of static friction required between the crate and pallet to
ensure that tipping occurs
that tipping occurs prior to slip. prior to slip.
d1 d1
d2 G
d2 B
A Use = 4 kg, M parameters 0 your and d = m m.
se the following: mthe1following= 65 kg, d1 =in .3 m, analysis: 0.6= 16kg , M = 80kg , d1 = 0.3m and d2 = 0.6m.
2 V8 Chapter V: Rigid Body Kinetics Homework netics Homework Problems V6 ME 274 Problem V.6
Given: The center of mass G of the disk shown below (mass m) is oﬀset by a distance e from the of mass G of the disk shown below (mass position shown,aG is directly above O as the wheel rolls without slip
geometric center O. At the m) is offset by distance e from the
center O. If Gthrough the bottom of a as the wheel rolls without slip through
is in the position shown circular path shown.
of a circular path shown, find the normal force N that is exerted by the path on
Find: For this position, determine the normal force N that is exerted by the path on the disk. C
! R e
O G
r Use the following e = 0.05 m, ! = 3 analysis: R = 1. m,
llowing: R = 1.25 m, r = 0.25 m, parameters in yourrad/s, and m = 505kg. r = 0.2m, e = 0.1m, ω = 4rad/sec and
m = 75kg . Chapter V: Rigid Body Kinetics Homework V9 netics Homework Problems V7 ME 274 Problem V.7
Given: The system shown below consists of a uniform slender bar of mass m and length L and an shown belowattached of a uniform slender barThemass m and length L and rest in the position shown with the
consists particle of equal mass. of system is released from
particle of equal mass. If the system is released from mass of the roller, and assume its radius is negligible.
roller resting on the incline. Neglect the rest in the position
h the roller resting on the incline, determine the initial acceleration of A.
lect the mass of the roller and assume its radius is negligible.
Find: Determine the initial acceleration of A. A
! L, m m V10 Chapter V: Rigid Body Kinetics Homework Problem V.8
igid Body Kinetics Homework Problems ME 274 roblem V8 Given: System shown with mA and mB being the masses of pallet A and homogeneous disk B,
respectively. ind the maximum force T that can be applied to the pallet A by the winch such that the
Find: not the maximum to roll.
omogeneous disk willFindslip as it beginsforce T that can be applied to the pallet A by the winch such that the
homogeneous disk will not slip as it begins to roll. µ s, µk B
r O
A Use the following parameters r your analysis: mA and kg 0.25.
se the following: mA = 20 kg, mB = 150 kg,in = 0.15 m, µs = 0.3, = 25µk ,=mB = 175kg , r = 0.15m, µs = 0.3 and
µk = 0.25. Problem V9
A thin, homogeneous bar AB (having a length of L and mass m) is pinned to ground at
pointChapter V: Rigid a distance of L/4 from end A). A homogeneous disk of mass M
O (where O is at Body Kinetics Homework
and radius r is welded to the bar AB with end B being at the center of the disk. A constant
forceProblemend A in such a way that the line of action of F is always perpendicular to
F acts at V.9
OA. At instant “1”, the system is released from rest with AB being horizontal. V11 Given: A thin, homogeneous bar system made up of of L and mass m) is pinned to ground at
a) Draw a free body diagram of theAB (having a lengthbar AB and the disk. List
point O (where O is at a distance of L/4 from end A). A homogeneous disk of mass M and radius r
which, if any, nonconservative forces do work on this system. is welded to the bar AB with end B being at the center of the disk. A constant force F acts at end b) Determine thethat the velocity of bar ABis always perpendicular to OA. The system is released
A in such a way angular line of action of F when AB reaches its vertical position
(instant “2”). Note that since the
from rest with AB being horizontal.disk is welded to AB, the disk has the same
angular velocity as bar AB.
Find: For this problem: a) determine the angular disk as particle of mass M reaches
c) Repeat your calculations in b) but here treat the velocityaof bar AB when AB (that its vertical
position, and b) ! 0
is, consider r repeat your calculations in a) but here treat the disk as a particle of mass M.
). F
M r m
A
L/4 O 3L/4 B g
VERTICAL PLANE
Use the following parameters in your analysis: L = rs r 1 1 e m = 40kg , ,
Use the following parameters in your analysis: L = 2.5 mete3m,, r == mmt,er , m = 40 kgM = 50kg and
M = F = g and. F = 100 Newtons .
50 k 100N V12 Chapter V: Rigid Body Kinetics Homework Rigid Body Kinetics Homework Problems ME 274 Problem V.10 Problem V10 A A thin, homogeneous bar mass M is supported by p parallel cables AC and BD, as
Given: thin, homogeneous bar of of mass M is supported byarallel cables AC and BD, as shown
hown ﬁgure. The figure. The bar is r rest with the cables at an angle at θ angle of
below sin the below in thebar is released fromeleased from rest with the cables of an= θ0 measured
from the=horizontal.
! ! 0 measured from the horizontal.
a D problem: a) the speed of the center of of the the bar ! = θ =
Find: For )this etermine the speed of the center of massmass ofbar whenwhen90° . 90◦ , and b) the
tension inbcables AC andthe tension θ = 90◦ . AC and BD when ! = 90° .
) Determine BD when in cables g C ! D L L B
A
3L/4 L/4 Use the following parameters in your analysis: M = 120kg , L = 3.5m and θ0 = 30◦ . Use the following parameters in your analysis: M = 100 kg , L = 4 meters and
! 0 = 36.87° . Chapter V: Rigid Body Kinetics Homework V13 Problem V.11
Given: ConsiderBody Kinetics Homework Problems
Rigid the three systems (each of total mass m) shown below where: (I) 274
ME made up on a
particle of mass m supported within a massless ring by massless spokes, (II) made up of identical,
P total mass
thin spokes of roblem V11 m in a massless ring, and (III) a thin ring of mass m. Each system is
Consider the three systems (each of total mass m) shown below where: (I) made up on a
released from prest, with eachssystem rolling withoutring by massless spokes, (II) made up of
article of mass m upported within a massless slipping on a rough incline.
identical, thin spokes of total mass m in a massless ring, and (III) a thin ring of mass m.
Each the velocity of the mass center O of each system when each system has made
Find: Determine system is released from rest, with each system rolling without slipping on a rough
ncline. Determine the velocity of the inclines. Rank the speeds of O system
two complete irevolutions in rolling down the mass center O of each system when eachfor the three systems.
has made two complete revolutions in rolling down the inclines. Rank the speeds of O for
Explain in words why the speeds endwords why those rankings.with those rankings.
up with the speeds end up
the three systems. Explain in g
O O R
no slip ! (I) massless ring, massless
spokes and particle of
mass m O R R no slip ! (II) massless ring and
identical spokes (with
total mass of spokes m) no slip ! (III) thin ring of mass m V14 Chapter V: Rigid Body Kinetics Homework Problem V.12
Given: A solid drum having a mass of 2m and centroidal radius of gyration kO is pinned to ground
ME 274
at its center of mass O. A cable is wrapped around the outer radius (2R) of the drum supports
block A (having a mass of 3m). A second cable is wrapped around the inner radius (R) of the
drum supports block B (having a mass of m). Assume that the cables do not slip on the drum as
the system moves. The system is released from rest. netics Homework Problems V12
m having a mass of 2m and centroidal radius of gyration kO is pinned to
ts center of mass O. A cable is wrapped around theof the drum after the the has rotated through an angle of θ.
Find: Determine the angular speed outer radius (2R) of drum
orts block A (having a mass of 3m). A second cable is wrapped around the
s (R) of the drum supports block B
ass of m). Assume that the cables
2m
on the drum as the system moves.
is released from rest at instant “1’.
2R
w a free body diagram (FBD) of the
O
tem made up of the drum and
g
ticles A and B. List which, if any,
R
conservative forces do work on this
tem.
termine the angular speed of the
m after the drum has rotated through
angle of !. A m B 3m
lowing parameters theyour analysis: m = 50 kg ,your analysis:ter ,= 60kg , R = 0.5m, k = 0.6m and θ = 180◦ .
Use in following parameters in R = 0.75 me m
0
eters and ! = 180° . Rigid Body Kinetics Homework Problems ME 274 Chapter V:V13 Body Kinetics Homework
Problem Rigid V15 One end of an inextensible cable is attached to a rigid wall at B. This cable is wrapped
around a homogeneous disk of mass m and outer radius R with the free end of the cable
Problem V.13
remaining horizontal at all time. A constant horizontal force F acts at the free end A of
the cable. A spring of stiffness k is attached between the center of the disk O and ground.
Given: One end of an inextensible cable is attached to a rigid wall at B. This cable is wrapped
As the disk moves to the left, it does not slip on the cable. The system is released from
around a homogeneous disk of mass m and outer radius R with the free end of the cable remaining
rest at instant “1’ with the spring unstretched.
horizontal at all time. A constant horizontal force F acts at the free end A of the cable. A spring
of stiﬀness k is attached between the center thethe disk O which, if any, As the disk moves to the
a) Draw a free body diagram (FBD) of of disk. List and ground. nonconservative
left, it does not slip on the this system. system is released from rest with the spring unstretched.
forces do work on cable. The b) Determine the speed of the center of the disk O after O has moved through a Find: Determine the speed of the center of the disk O after O has moved through a distance d to
distance d to the right.
the right. A
R k F no slip O
no slip B C
Use the following parameters in your analysis: m = 50kg , R = 0.6m , e s
Use the following parameters in your analysis: m = 60 kg , R = 0.5 metkr=, 250N/m, F = 2200N
k = 300 N / m
and d = 1.75m., F = 2000 N and d = 2 meters . Rigid Body Kinetics Homework Problems ME 274 V16 Chapter V: Rigid Body Kinetics Homework Problem V14
A thin homogeneous bar OA having a length of L and mass m is pinned to ground at O. A
Problem sV.14 thin homogeneous bar AB (having a length of 2L and mass 2m) is pinned to bar
econd
OA at A, and end B of the bar is constrained to move within a smooth, horizontal track.
Given: A thin homogeneous bar OA having a length of L and mass m is pinned to ground at O.
A constant force F acts horizontally at end B. The bar is released from rest at instant “1”
A second thin homogeneous bar AB (having a length of 2L and mass 2m) is pinned to bar OA at
with link OA being horizontal and with pin B being directly below pin O.
A, and end B of the bar is constrained to move within a smooth, horizontal track. A constant force
a) D at end B. body diagram of the system with up OA being horizontal List
F acts horizontally raw a freeThe bar is released from rest madelink of links OA and AB. and with
which, if pin nonconservative forces do
pin B being directly below any,O, as shown in the ﬁgure. work on this system. b) Determine the speed of end B of bar AB when the system has reached a position Find: Determine the speed of end B of bar AB when the system has reached a position of link OA
of link OA being vertical (instant “2”).
being vertical. m
L O g A 2m
2L F
B VERTICAL PLANE
Use the following parameters in your analysis: F = 600lb, L = 3f t and mg = 200lb. Use the following parameters in your analysis: F = 500 lb , L = 4 ft and mg = 200 lb . Chapter V: Rigid Body Kinetics Homework V17 Rigid Body KineticsProblemProblems
Homework V.15 ME 274 Given: The center of the homogeneous disk of mass m shown below has an initial velocity of v0
Problem V15
up the slope. The center of the homogeneous disk of mass m shown below has an initial velocity of v0
Find: Find the of the of the disk after an elapsed time of ∆t, assuming that
up the slope. Find the velocityvelocitydisk after an elapsed time of !t, assuming that the the disk rolls without
slip.
disk rolls without slip. O r ! ◦
Use m 8 kg, r = 0.3 m, ! = in your analysis: / = and
Use the following: the=following parameters 36.87°, v0 = 36 m msec 10kg ,!rt = 105secs . = 30 , v0 = 30m/sec and
= 0. m , θ ∆t = 12sec. V18
etics Homework Problems Problem V.16 Chapter V: Rigid Body Kinetics Homework
ME 274 16 Given: A homogeneous, slender bar of mass M length L moves is sliding across a frictionless
eous, slender horizontal surface with L moves vG slidingit is suddenly struck at its end by a particle of mass
bar of mass M length a speed is when across a frictionless
m moving with velocity vm in the direction shown. Assume that the particle sticks to the bar
surface with a speed vG when it is suddenly struck at its end by a particle of
immediately on impact. ving with velocity vm in the direction shown. Assuming that the particle
e bar, determine the velocity of the bar’s center of mass G and the angular
Find: Determine the velocity of the bar’s center of mass G and the angular velocity of the bar
the bar postimpact.
postimpact. lowing: M = 3 Ug, L = 0.5 m, vG = 4 m/s, m = 0.3 kg and vm = 20 m/s.kg , L = 0.75m, vG = 5m/sec, m = 0.4kg
k se the following parameters in your analysis: M = 4
and vm = 25m/sec. Chapter V: Rigid Body Kinetics Homework
omework Problems V19
ME 274 Problem V.17
Given: The skater begins to spin in place with his arms extended, and then increases his rate of
rotation his arms extended, and then increases his
ns to spin in place with by bringing his arms closer to his body. rate of ging his arms closer to his body. Estimate the skater’s rotational speed
his arms, if heFind: Estimate the skaters rotational speed afterahe retracts his arms, if he started rotating, with
started rotating, with his arms fully extended, at rate of
his arms fully extended, atas uniformω1 = 1rev/s. tInitially, you can model the extended arms as
itially, you can model the extended arms a rate of slender rods hat are
uniform slender rods that are 28 inches long inches in
nd weigh 24 lb each and the torso as a 124 lb cylinder, 18and weigh 24 lb each and the torso as a 124 lb cylinder,
18 inches in diameter. be modeled as are retracted, the
his arms are retracted, the skater can After his arms a solid cylinder 18skater can be modeled as a solid cylinder
18 inches
ter and weighing 172 lb. in diameter and weighing 172 lb. V20
Body Kinetics Homework Problems Problem V.18 Chapter V: Rigid Body Kinetics Homework
ME 274 blem V18 Given: The slender bar, shown below, of mass m and length L has a downward velocity vG at
the instant shown. Assume that the pivot at the end of bar becomes attached to the support after
slender bar, shown below, of mass m and length L has a downward velocity at the
falling h. nt shown. Assuming that the pivot at the end of bar becomes attached to the support
falling h, determine the angular the angular the bar immediately after impact.
Find: Determine velocity of velocity of the bar immediately after impact. Use the following parameters in your analysis: m = 4kg , L = 0.4m, vG = 3m/sec and h = 4m. the following: m = 3 kg, L = 0.3 m, vG = 4 m/s and h = 3 m. Chapter V: Rigid Body Kinetics Homework V21 Problem V.19
Given: Homogneous bar AB of mass m is supported by a base AC that is translating to the left
with a constant acceleration of a. Cable AC restrains the rotation of the bar as it translates to the
s Homework Problems
ME 274
left. 9 (6/8) Find: Determine: a) the tension in the cable, and b) the reaction force acting on bar AB at A. ar AB of mass m is supported by a
s translating to the left with a constant
a. Cable AC restrains the rotation of
nslates to the left. Determine
sion in the cable.
ction force acting on bar AB at A. Use the following parameters in your analysis: m = 5kg and a = 0.6g . ing parameters in your analysis: m = 5 kg and a = 0.6 g . V22 Chapter V: Rigid Body Kinetics Homework Problem V.20
mework Problems /40) Given: A constant frictional torque from the bearings at O resist the motion of the pulley system
ME 274
shown. The mass and radius of gyration of the pulley are known to be m and kO , respectively.
Find: Determine the angular acceleration of the pulley on release. al torque M f from the bearings at O
f the pulley system shown. The mass
tion of the pulley are known to be m
ely. Determine the angular acceleration
lease. Use the following parameters in your analysis: Mf = 2N m, m = 8kg and kO = 225mm. parameters in your analysis: M f = 2 N ! m , m = 8 kg and Chapter V: Rigid Body Kinetics Homework
Kinetics Homework Problems Problem V.21 V21 (6/94) V23
ME 274 Given: A couple M acts on a slender, uniform bar of mass m. Assume all surfaces to be smooth. M acts on a slender, uniform bar of mass m. Determine the reaction forces
Find: Determine the reaction forces acting on the bar at A and B on release.
the bar at A and B on release. Assume all surfaces to be smooth. V24
etics Homework Problems Problem V.22 22 (6/109) Chapter V: Rigid Body Kinetics Homework
ME 274 Given: A slender, homogeneous bar of mass m is released from rest. omogeneous bar of mass m is released from rest. Determine the contact force
Find: Determine the contact force on the bar due to the smooth horizontal surface on release.
ue to the smooth horizontal surface. Chapter V: Rigid Body Kinetics Homework V25 Problem V.23
Given: The inhomogeneous disk shown has a mass of m, its center of mass at G and a centroidal
radius of gyration of kG . At the position shown, G is on the same horizontal line as the geometric
center O of the disk and the disk is rolling without slipping with a CCW angular speed of ω .
omework Problems (6/110) ME 274 Find: Determine: a) the normal force acting on the disk by the horizontal surface, and b) the
friction force acting on the disk by the horizontal surface. ous disk shown has a mass of m, its
G and a centroidal radius of gyration of
ion shown, G is on the same horizontal
etric center O of the disk and the disk is
lipping with a CCW angular speed of ! .
etermine:
al force acting on the disk by the
l surface.
n force acting on the disk by the
l surface.
Use the following parameters in your analysis: m = 10kg , ω = 2rad/sec and kG = 64mm. g parameters in your analysis: m = 10 kg , ! = 2 rad / sec and V26 Chapter V: Rigid Body Kinetics Homework Problem V.24 mework Problems 6/117) Given: A constant frictional torque Mf from the bearings at O resist the motion of the pulley
ME downward with a speed of v .
system shown. The system is released with A traveling 274
A1
Find: Determine the speed of block A after it drops through a distance of d. nal torque M f from the bearings at O
of the pulley system shown. If the
with A traveling downward with a
ermine the speed of block A after it
istance of d. m, kO A Use the following parameters in your analysis: m = 12kg , Mf = 3N m, vA1 = 0.3m/sec, d = 1.5m
and kO = 210mm. parameters in your analysis: m = 12 kg , M f = 3 N ! m ,
d = 1.5 meters and kO = 210 mm . Chapter V: Rigid Body Kinetics Homework V27 Problem V.25
omework ProblemsGiven: A wheel is made up of a thin ring (mass M ) ME 274 spokes (each of mass m). The wheel
and four
is initially at rest. Assume that the wheel does not slip as it rolls. (6/121)
Find: Determine the speed of the wheel’s center after it has moved a distance d. up of a thin ring (mass M) and four
ass m). If the wheel is initially at
ed of the wheel’s center after it has
d. Assume that the wheel does not Use the following parameters in your analysis: d = 3m, m = 2kg and M = 10kg . g parameters in your analysis: d = 3 meters , m = 2 kg and V28 Chapter V: Rigid Body Kinetics Homework Problem V.26
mework Problems Given: The pulley has a mass of m and a radius ofME 274
gyration about its center O of kO . The spring
is initially stretched by an amount of ∆1 . The system is released from rest. /140)
Find: Determine the speed of O after O has dropped through a distance of d. ass of m and a radius of gyration about its
the spring is initially stretched by an amount
d from rest, find the speed of O after O has
distance of d. Use the following parameters in your analysis: m = 50kg , d = 50mm, k = 1500N/m, ∆1 = 100m
parameters in and kanalysis: m . 50 kg , d = 50 mm ,
your O = 300mm=
1 = 100 mm and kO = 300 mm . Chapter V: Rigid Body Kinetics Homework V29 Problem V.27
work Problems Given: A bullet having a weight of mg strikes a stationary slender, homogeneous bar with a weight
ME 274
of M g . The bullet becomes embedded in the bar after impact. 83)
Find: Determine the angular velocity of the bar immediately after impact. ight of mg strikes a stationary slender,
ith a weight of Mg. The bullet becomes
after impact. Determine the angular
mediately after impact. Use the following parameters in your analysis: M g = 20lb and mg = 2ounces. arameters in your analysis: Mg = 20 lb and mg = 2 ounces . V30 Chapter V: Rigid Body Kinetics Homework Problem V.28
Given: A homogeneous sphere of mass m is projected with an initial angular speed of ω0 and with
an initial speed of its center G of v0 . The coeﬃcient of kinetic friction between the sphere and the
rough incline is µk .
Homework Problems
ME 274 8 (6/196) Find: Determine: a) the time during which sphere slips on the incline, and b) the angular velocity
of the sphere and the speed of G immediately after slipping has ceased. s sphere of mass m is projected with
ar speed of ! 0 and with an initial
ter G of v0 . The coefficient of
between the sphere and the rough
etermine:
e during which sphere slips on the
ular velocity of the sphere and the
f G immediately after slipping has Rigid Body Kinetics Homework Problems ME 274 Chapter V: Rigid Body Kinetics Homework V31 Problem V29
A crate of mass m is being pulled to the right by a cable attached to a winch. The winch
is pulling in
Problem V.29 the cable with the angular acceleration of the winch pulley being ! . The
crate moves along a rough, horizontal surface with the coefficient of kinetic friction
Given: A crate ofsurface andbeing pulled to the .right by athat the crate slides without The winch
between this mass m is the crate being µ k Assume cable attached to a winch. tipping. is pulling in the cable with the angular acceleration of the winch pulley being α. The crate moves
a) Determine the maximum angular acceleration ! of the winch pulley such that the
along a rough, horizontal surface with the coeﬃcient of kinetic friction between this surface and
crate does not tip.
the crate being µk . Assume that the crate slides without tipping. b) Determine the tension in the cable corresponding to the value of ! found above
in a).
Find: Determine: a) the maximum angular acceleration α of the winch pulley such that the crate
does not tip, and b) the tension in the cable corresponding to the value of α found above in a). b/2 b/2 !
r
h d µk
A B Use the following parameters in your analysis: r = 0.6f t, mg = 120lb, d = 2.5f t, h = 3f t, b = 4f t
and µk = 0.2. Use the following parameters in your analysis: r = 0.5 ft , mg = 100 lb , d = 2.5 ft ,
h = 3 ft , b = 4 ft and µ k = 0.2 . V32 Chapter V: Rigid Body Kinetics Homework
Rigid Body Kinetics Homework Problems ME 274 Problem V.30 Problem V30
Given: A homogeneous bar of mass M and length L L is pinned to ground O. TheThe other of
A homogeneous
of mass M and length is pinned to ground at at O. other end end
of the bar, pointpoint O, is supportedaby a cable thatpulled over a smooth pulley and attached to
the bar, O, is supported by cable that is is pulled over a smooth pulley and attached
a block Boof block m.of mass B isBlockto slide on a slide on ahorizontal surface. At the instant of
t a mass B Block m. able B is able to smooth, smooth, horizontal surface. At the
release with barof release withat an angle of θ with the horizontal, the bar is stationary and the
instant OB oriented bar OB oriented at an angle of ! with the horizontal, the bar is
cable is taut with the two free sections with the two oriented horizontally and vertically, as shown
stationary and the cable is taut of the cable free sections of the cable oriented
in the ﬁgure. Assumeand vertically, aspulley to be negligible compared to the of theof the bar be
horizontally the mass of the shown in the figure. Assume the mass mass pulley to and
block. negligible compared to the mass of the bar and block. Determine the acceleration block B on release.
Find: Determine the acceleration ofof block B on release. B
O m
smooth ! M, L g A
Use the followingfollowing parameters in your analysis: M m = 20g , ,m = 25m g , θ = 60◦ .
Use the parameters in your analysis: M = 50kg , = 40 k kg L = 2 k and L = 1.5 meters and ! = 53.13° . Chapter V: Rigid Body Kinetics Homework V33 Rigid Body Kinetics Homework Problems ME 274 Problem V.31 Problem V31 A thin homogeneous bar OA (of negligible mass mass len of L) is L) is to ground at
Given: A thin homogeneous bar OA (of negligible and ofand gthlengthpinnedpinned to ground at
O.O. The other end of the bar pinned to tosecond thin homogeneous barbar AB (of massand length
The other end of the bar is is pinned a a second thin homogeneous AB (of mass m m
2Land length 2L)of bar AB is ablebar slideis able to slidehorizontal surface. At thesurface. of release
) at A. End B at A. End B of to AB on a smooth, on a smooth, horizontal instant At
the instant of horizontal and end B of bar AB is located B of bar AB is located directly
from rest, OA isrelease from rest, OA is horizontal and end directly below O.
below O.
Find: Determine the angular acceleration of bar AB on release. Write your answer as a vector. Determine the angular acceleration of bar AB on release. Write your answer as a vector. O L A 2L
g
smooth B
Use se the following parameters your analysis: m = 50kg and L = 3m. meters .
U the following parameters in in your analysis: m = 40 kg and L = 2.5 Rigid Body Kinetics Homework Problems ME 274 V34 Chapter V: Rigid Body Kinetics Homework Problem V32
Particle P (having a mass of m) is able to slide within a smooth slot cut in bar OA where
Problem rotates about a vertical axis passing through end O. A torque T acts on OA. At a
OA V.32
particular instant, P is at a distance of R from O and is moving outward from O in the slot
Given:t a rate of P! (having ato the arm asis able to slide within a smooth of !cut in OA has a
a Particle R relative mass of m) arm OA is rotating with a rate slot . Bar bar OA where
OA rotates of M andvertical axis gyration through bout O. A torque T acts on OA. At a particular
mass about a a radius of passing of kO a end point O. ˙
instant, P is at a distance of R from O and is moving outward from O in the slot at a rate of R
relative to a) Determine the angular accelerationratebar ω . Bar this instant.
the arm as arm OA is rotating with a of of OA at OA has a mass of M and a radius
of gyration of kO about point O. b) Determine the acceleration of particle P. Find:Write your answer as a vector.
Determine: a) the angular acceleration of bar OA at this instant, and b) the acceleration of
particle P. ! T R
smooth O P A HORIZONTAL PLANE
˙
Use the following parameters in your analysis: m = 30kg , M = 40kg , R = 0.5m, R = 4m/sec,
Use the, following parameters and kO = 1.5m. m = 40 kg , M = 50 kg ,
in your analysis:
ω = 5rad/sec T = 175N m (CCW) !
R = 0.6 meters , R = !3m / sec , ! = 4 rad / sec , T = 150 N ! m ( CCW ) and
kO = 2 meters . Chapter V: Rigid Body Kinetics Homework V35 Problem V.33
Rigid Body Kinetics Homework Problems ME 274 Given: A Tshaped bar consisting of a bar AB (of mass m and length L) is welded to the end of
an Problem V33 bar that is pinned at to ground at O. Tip A is pressed against a spring with a
identical second
spring constant bar consisting of a bar (m, L) welded to the end 1). a second bar (m, L) is
A Tshaped k and the spring is compressed by ∆ (Position of pinned at O. Tip A is pressed against a spring with a spring constant k and the spring is Find: Determine the(Position velocity of the Tbar at position 2. of the Tbar at position 2.
compressed by ! angular 1). Determine the angular velocity Position 1 Position 2
B O O G1 G1 G2 A g
A G2 B Use following parameters in your analysis: m = 0 = 0.2 = L = 0 k = k = 80 N/m, ∆ =
Use thethe following parameters for your analysis: m.4kg , Lkg, 0.8m, .6 m,100N/m and ! = 0.6m.
0.5 m. V36 Chapter V: Rigid Body Kinetics Homework Problem V.34
Rigid Body Kinetics Homework Problems ME 274 Given: A homogeneous disk (mass M and radius R) is attached to a homogeneous, thin rod OA
(mass m) atVits center O. A constant force F is applied at O and point A is conﬁned so that it
Problem 34
moves along a smooth, verticalis attached disk rolls without m) at its center O. A constant
A disk (mass M) of radius R slot. The to rod OA (mass slipping. If the system is released from
rest with is = θ1 . at O and point A is confined so that it moves along a smooth vertical
force F θ applied slot. The disk rolls without slipping. If the system is released from rest when !1 = 60°,
find the velocity of point A when !2 = 0. Find: Determine the velocity of point A when θ = θ2 . smooth A
M, R g m, L
! F O no slip C UUse the following parameters in your analysis: M = 10 ,kg, m 6kg ,kg, = 1m, m,= 2=52 m, F 80N ,
se the following parameters in your analysis: M = 12kg m = = 5 R R = 1 L L . m, F =
== 0 N. ◦ and θ2 = 0.
θ1 5 53.13 Chapter V: Rigid Body Kinetics Homework V37 Rigid Body Kinetics Homework Problems ME 274 Problem V.35 Problem V35 Given: Bars OA and AB each have a mass of m and a length of L. B is constrained to move in
Bthe horizontal direction only. mass of mMO is appliedof L. B OA at O. The system isin
ars OA and AB each have a A couple and a length to link is constrained to move released from
rest when thedangle between A couple AB is applied. at O. Theall surfacesreleased from
the horizontal irection only. OA and M is θ = θ1 Assume system is are smooth. rest when the angle between OA and AB is !1. Assume all surfaces are smooth.
Find: Determine the of point B when ! when
Determine the velocity velocity of point B = !2. θ = θ2 .
B O
! MO L L g A
Use the following parameters for your analysis: mm 10 12kgL = 2 m2.!m= θ 80°, 36.87◦ ,0° = 0 and
Use the following parameters in your analysis: = = kg, , L = , 5 1 , 11 = !2 = 3 θ2
and MO 6040m.
M = = N Nm.
O V38 Chapter V: Rigid Body Kinetics Homework Rigid Body Kinetics Homework Problems ME 274 Problem V.36 Problem V36 Given: A uniform, disk (of mass m and radius r) is pinned at is applied to amass O. Force F is
A uniform disk (m r) is pinned at its center of mass O. Force F its center of rope
applied toaround the disk. The diskthe released from rest.released from rest. Assume that the rope
wrapped a rope wrapped around is disk. The disk is
does not seconds,
After 2 slip on the disk. a) determine the angular velocity of the disk.
Find: At time t, determine: a) the angular velocity of the disk, and b) the reaction force components on)theeterminepin O.
b d disk at the reaction force components at the pin. r O F Use the following parameters in your analysis: m = 15kg , r = 0.4m, F = 200N and t = 2sec. Use the following parameters for your analysis: m = 10 kg, r = 0.3 m, F = 250 N. Chapter V: Rigid Body Kinetics Homework V39 Problem V.37 Homework Problems
Rigid Body Kinetics ME 274 Given: A disk37 mass m and radius r rolls without slipping.
Problem V with A disk with mass m and radius r rolls without slip. Find the minimum velocity of G for
Find: disk to the minimum velocity of G for the d. Assume that the disk does not bounce
the Find roll over an obstruction with height disk to roll over an obstruction with height d.
Assume that the disk obstruction and remains in contact with the obstruction as it rolls over with
when it strikes the does not bounce when it strikes the obstruction and remains in contact
thehe obstruction. it rolls over the obstruction.
t obstruction as r
G vG
d U the following parameters in for your analysis: m = 2 , kg, 0 4 0 and d = 0 03 m.
Use se the following parameters your analysis: m = 18kg0 r = r .=m.3 m, d = 0..02m. V40 Chapter V: Rigid Body Kinetics Homework dy Kinetics Homework Problems ME 274 Problem V.38 em V38
Given: After traveling at a constant forward speed of vG1 for some distance, the driver of the car
shown forward speed applied for some distance, all four of the
raveling at a constant below suddenlyof 12 m/sthe brakes, causing the driver of thecars wheels to instantaneously
lock up. The car skidded to rest in a distance all with a constant
wn below (m = 2500 kg) suddenly applied the brakes, causingof Lfour of the car’s acceleration. The combined
mass of the car and driver is m.
to instantaneously lock up. The car skidded to rest in 35 m. Determine the tude of the normal reactions and friction forces at each wheel as the car skidded to
Find: Determine the assumed, 1 the normal = 2.25 m, and h =
ote that constant deceleration can bemagnitudedof= 1.75 m, d2 reactions and friction forces at each wheel as the
car skids to rest.
. G
h
A B
d1 d2 Use the following parameters in your analysis: m = 2600kg , L = 35m, d1 = 1.7m, d2 = 2.1m and
h = 1.15m. Chapter V: Rigid Body Kinetics Homework V41 ody Kinetics Homework Problems ME 274 Problem V.39 lem V39
Given: A pulley of mass M is connected to two blocks as shown. Assume that the axle at C is
ley of mass 10frictionless.1 = 0.08 m and d2 = 0.2 m is connected to two blocks as
kg with d . Assuming that the axle at C is frictionless, mA = 12 kg, and mB = 8 kg, determine
Find: Determine the angular acceleration of block.
gular acceleration of the pulley and the acceleration of each the pulley and the acceleration of each block.
d1 C
B A mA d2 mB Use the following parameters in your analysis: M = 10kg , mA = 12kg , mB = 8kg , d1 = 0.08m and
d2 = 0.2m. Problem V40
Given: A homogeneous wheel having a mass of M and outer radius of r is able to roll
without slipping on a horizontal surface. A cable that is attached to the center
V42
O of the wheel is hung over an idealChapterand Rigid Body Kinetics particle B
pulley V: is also attached to a Homework
having a mass of m. A second cable, attached to point E on the wheel, is also
Problem V.40 hung over an ideal pulley and is attached to particle A having a mass of m.
The system is released from rest with point E directly below point O.
Given: A homogeneous wheel having a mass of M and outer radius of r is able to roll without
slipping on a horizontal surface. A cable that is attached to the center O of the wheel is hung over
an ideal pulley and is also attached to a particle B having a mass of m. A second cable, attached
to point E on the wheel, is also hung over an ideal pulley and is attached to particle A having a
mass of m. The system is released from rest withangular acceleration of the wheel. Express your
Find:
At the time of release, find the point E directly below point O. final answer as a vector. For your numerical answer, use:
Find: At the time of release, ﬁnd the angular acceleration of the wheel. Express your ﬁnal answer
as a vector.
m = 10 kg
M = 20 kg
r = 2 meters y
x
ideal
pulley r A ideal
pulley r/2
E m M O no slip B Use the following parameters in your analysis: m = 10kg , M = 20kg and r = 2m. m Problem V41
Given:
A thin, homogeneous bar having a mass of M and length L is pinned to
ground at its mass center G.
Chapter V: Rigid Body Kinetics Homework Particle B, having a mass of m, is attached to the
V43
right end of the bar. A spring, having a stiffness of k, is attached between end
A of the bar and pin C on a wall. The pin G is a distance of b from the wall.
Problem V.41 hen the bar is horizontal (Position 1 shown below), the spring is
W
unstretched. Given: A thin, homogeneous bar having a mass of M and length L is pinned to ground at its mass
center G. Particle B, having a mass of m, is rigidly attached to the right end of the bar. A spring,
having a stiﬀness of k , is attached between end A of the bar and pin C on a wall. The pin G is
Find:
If the bar is released from rest in Position 1 above, find the angular velocity of
a distance of b from the wall. When the bar is horizontal (Position 1 shown below), the spring is
the bar in Position 2 when the bar is in a vertical position. Use the following
unstretched. parameters for your final numerical solution: Find: If the bar is 15 kg from m = 25 kg
rest in Position 1 above,00 N/meter
M = released
k = 1 ﬁnd the angular velocity of the bar in
Position 2 when the bar is in a vertical position. L = 3 meters b = 2.5 meters L
M
Position 1 C G
m A k B g b A G
Position 2 C B Use the following parameters in your analysis: M = 15kg , m = 25kg , k = 100N/m, L = 3m and
b = 2.5m. Rigid Body Kinetics Homework Problems ME 274 Problem V42
A thin homogeneous bar having a mass of M = 100 kg and length of 5 meters is pinned to
V44
Chapter V: Rigid Body Kinetics Homework
ground at point O in such a way that the bar can rotate about O in a horizontal plane.
Puck A, with a mass of m = 50 kg, strikes the bar at point B (located at a distance of 3
Problem V.42
meters from the pin at O) with a speed of vA1 = 30 m / sec , with A initially moving in
Given: A the direction shown below. The mass of M and length of Lstruck by the groundAssume
thin homogeneous bar having a bar is at rest before being is pinned to puck. at point
O in such that the puck sticks to the bar after impact. a horizontal plane. Puck A, with a mass of
a way that the bar can rotate about O in
m, strikes the bar at point B (located at a distance of d from the pin at O) with a speed of vA1 ,
Determine the angular velocity of the bar after the puck strikes the bar. Assume all
with A initially moving in the direction shown below. The bar is at rest before being struck by the
surfaces to be smooth. Treat the puck as a particle.
puck. Assume that the puck sticks to the bar after impact. HINT: Treat the puck and the bar as a single system in your analysis. You will need to
s angular velocity of the diagram of puck strikes order to receive full credit
Find: Determine thehow a correct free bodybar after the the system inthe bar. Assume all surfacesfor
to be smooth. Treatthe problem. a particle.
the puck as O O
! d vA1
G
B ! vA1 A M m
Top View Use the following parameters in your analysis: M = 100kg , m = 50kg , L = 5m, d = 3m, vA1 =
30m/sec and θ = 30◦ . L Rigid Body Kinetics Homework Problems ME 274 Problem V43
Chapter traight, rigid bar is welded to a wheel with a radius of R = 1.5 meters and center at O.
V45
A s V: Rigid Body Kinetics Homework The combined barwheel system has a total mass of m = 60 kg, its mass center at point G
(at a distance of d = 0.5 meters from O), and a centroidal radius of gyration of
Problem V.43
kG = 0.8 meters . The barwheel system is released from rest with points O and G lying
Given: a horizontalrigid bar is welded to a horizontal surface that R and center at O. The combined
on A straight, line. Assume that the wheel with a radius of supports the system is
barwheel system has a total mass of m, its mass center at point G (at a distance of d from O),
SMOOTH.
and a centroidal radius of gyration of kG . The barwheel system is released from rest with points
F G lying on the same system is line. Assume that the
O andor the instant after thehorizontal released, determine: horizontal surface that supports the
system isa) the angular acceleration of the wheel.
SMOOTH. b) the acceleration of the system center of mass G. Find: For the instant when the system is released, determine a) the angular acceleration of the
wheel, xpress your acceleration VECTORS. center of mass G.
E and b) the answers as of the system R g d O G smooth Use the following parameters in your analysis: R = 1.5m, m = 60kg , d = 0.5m and kG = 0.8m. V46 Chapter V: Rigid Body Kinetics Homework
Rigid Body Kinetics Homework Problems ME 274 Problem V.44 Problem V44
Given: A homogeneous bar ABbar AB mass of M andof M andis supported by cables AD and
A thin, homogeneous with a with a mass length L length L is supported by cables
G, where G is the cecenter mass of the bar. At At the position shown below, the bar is
AD andEEG, where G is the nter of of mass of the bar. the position shown below, the bar is
horizontal, point A is directly D, and D, and end B is directly below E.bar is released from
horizontal, point A is directly below below end B is directly below E. The
rest in thethe bar isshown. from rest in the position shown, determine the angular acceleration
If position released of the bar immediately after release. In your calculations, use M = 20 kg, L = 4 meters Find: Determine the angular acceleration of the bar immediately after release.
and h = 1.5 meters. Ignore the mass of cables AD and EG. D E h
g A G B L
Use the following parameters in your analysis: M = 20kg , L = 4m and h = 1.5m. Chapter V: Rigid Body Kinetics Homework Rigid Body Kinetics Homework Problems ME 274 V47 Problem VV.45
Problem 45
A thin homogeneous bar of length L = 1.5 meters and mass m = 30 kg is pinned to
ground at point O. homogeneous bardof lengtha L andof M = m is kg and radius R = 0.6 point O. A
Given: A thin A homogeneous isk with mass mass 100 pinned to ground at
mhomogeneous diskto enda mass of bar.and radius R iswithout slipping on thethe bar.of a disk rolls
eters is PINNED with A of the M The disk rolls PINNED to end A of inside The
circular surface. The system is rof a circular surface. The= 90°. is released from rest with θ = 90◦ .
without slipping on the inside eleased from rest with ! system
Find: angular angular velocity of the ! = 0.
Find the Find thevelocity of the bar whenbar when θ = 0. O
L no slip !
g
R
A Use the following parameters in your analysis: L = 1.5m, R = 0.6m, m = 30kg and M = 100kg . V48 in terms of the position x of:
3
2
aO = x 2 (x in feet and aO in ft/sec )
2
At position 1 (x = 0), point O has a velocity Chapter V:to the right. Kinetics Homework
of 1 ft/sec Rigid Body FProblem At position 2, where x = 2 ft,
ind:
V.46
a) determine the velocity of point O. Given: A cylinder with an outer radius of R rolls without slipping on a horizontal surface. The
) d O has an acceleration that is known in terms of the position
center of thebdisk etermine the angular velocity and angular acceleration of the x of: aO = 3x2 /2
c in f t/sec2 ). At position 1 ( = 0), point
(x in feet and aO ylinder. Write your answersxas vectors. O has a velocity of vO1 to the right. c) determine the velocity and acceleration of point A (at this position A is
Find: For a Position 2 same horizontal line as O). Write your answers asb) determine the angular
on the of point O: a) determine the velocity of point O, vectors.
velocity and d) sketch the velocity vector for A (from c) velocityin the acceleration of point A (at
angular acceleration of the cylinder, c) the above) and Position 2 figure
this position A is elow. same horizontal line as O), and d) sketch the velocity vector for A (from
b on the
c) above. x
O aO
no slip Position 1 A O
R Position 2 Use the following parameters in your analysis: R = 3f t, vO1 = 1f t/sec and x = 2f t. aO length L = 4 meters with the end of the bar located at the center O of the disk.
The disk is able to roll without slipping on a horizontal surface. The system is
released from rest at position 1 with the bar being horizontal at this position.
On Body Kinetics Homework
Chapter V: Rigid release, the system rolls to the right. At position 2 shown below the bar is V49
vertical.
Problem V.47
Find: a) Draw a free body diagram (FBD) of the system of the bar and disk Given: A homogeneous disk (having a mass of M and outer radius of R) is welded to a thin,
together.
homogeneous bar having a mass of m and length L with the end of the bar located at the center O
b) is able all of the forces in your FBD in a) as surface. The system is released
of the disk. The diskClassifyto roll without slipping on a horizontal to whether they do work
from rest at Positionor with be included inhorizontal energy.position. On release, the system moves
1 will the bar being potential at this
to the right. At Position 2 shown below the point G when the system is at position 2.
c) Determine the speed of bar is vertical.
Find: Determine the speed of point G when the system is at Position 2. Position 1 Position 2 L
R O O
G C G Use the following parameters in your analysis: M = 10kg , m = 20kg , R = 1m and L = 4m. Problem V48
Given: A system is made up of two identical homogeneous disks (with each disk
Chapter V: Rigid Body Kinetics Homework
having a mass of M = 20 kg and outer radius of R = 0.5 m) and an
inextensible cable. One end of the cable is attached to point O of disk 1 and
Problem V.48other end of the cable is wrapped around the outer radius of disk 2. Disk 1
the
sits on a SMOOTH horizontal surface, whereas disk 2 is on a rough incline
Given: A system is made up ofon the incline.homogeneousis released from rest. having a mass of
and does NOT slip two identical The system disks (with each disk
V50 M and outer radius of R) and an inextensible cable. One end of the cable is attached to point O
of ind: 1 and the other end of the cable is wrapped around the outer radius of disk 2. Disk 1 sits
F disk
on a SMOOTH horizontal surface, whereas disk 2 is on a rough incline and does NOT slip on the
a) Draw a free body diagram of each disk individually.
incline. The system is released from rest. b) Determine the acceleration of the center A of disk 2 immediately after
release. Find: Determine the acceleration of the center A of disk 2 immediately after release.
1
O
smooth R ideal pulley B
R 2 A
C ! no slip Use the following parameters in your analysis: M = 20kg and R = 0.5m. Rigid Body Kinetics Homework Problems ME 274 Problem V49
Chapter V: Rigid Body Kinetics Homework V51 Given: An inextensible cable connects block B with point A on a rigid, homogeneous
disk. At the instant shown, point A is directly above the disk’s center O. Block
Problem V.49
B has a mass of m = 40 kg, and the disk has a mass of M = 20 kg. The disk is
able to roll without slipping on a rough horizontal surface with C being the
Given: An inextensible cable connects block B with point A on a rigid, homogeneous disk. At the
contact oint
instant shown, pointpA is .directly above the disks center O. Block B has a mass of m, and the disk
has a mass of M . The disk is able to roll without slipping on a rough horizontal surface with C
Find:
If the system is released is released the rest.
being the contact point. The system from rest atfromposition shown, find the angular acceleration of the disk. Write your answer as a vector. Find: Determine the angular acceleration of the disk immediately after release. M A
R/2 O
no slip R C
B m Use the following parameters in your analysis: M = 20kg and m = 40kg . inetics Homework Problems ME 274 V50
V52 Chapter V: Rigid Body Kinetics Homework Particle B (having a mass of m = 10 kg) initially travels on a smooth
HORIZONTAL surfaceV.50 a speed of vB1 = 20 m/sec. This particle strikes
Problem with
point E on a stationary thin homogeneous bar (having a mass of M = 15 kg),
Given: being pinned to ground. The coefficient of restitution
with end A of the bar Particle B (having a mass of m) initially travels on a smooth HORIZONTAL surface with
a speed of v the This point E is known to be e a stationary thin homogeneous bar (having a mass
between the particle and B 1 . bar atparticle strikes point E on = 0.6.
Furthermore,of Mcontact surface of the bar beingpoint E is smooth. The coeﬃcient of restitution between the
the ), with end A between B and pinned to ground.
particle and the bar at point E is known to be e. Furthermore, the contact surface between B and
point E is smooth. Determine the angular velocity of the bar immediately after being impacted
by the particle. Write your answer as a vector. Find: Determine the angular velocity of the bar immediately after being impacted by the particle. A B
d m M
L vB1
E ! Use the following parameters in your analysis: L = 3m, d = 2m, m = 10kg , M = 15kg , e = 0.6,
θ = 60◦ and vB 1 = 20m/sec. Problem V51
Given: A homogeneous, thin bar AB (of length L = 3 meters and mass m = 50 kg) is
Chapter V: Rigid Body Kinetics Homework
pinned to a block at A. The block has a mass of M = 100 kg and is constrained
to move along a smooth horizontal surface. A horizontal force F = 200 N is
Problem V.51 the bar’s center of mass G at an instant when the bar is vertical and
applied to
at rest. V53 Given: A homogeneous, thin bar AB (of length L and mass m) is pinned to a block at A. The
block has a this problem, is constrained to move along a smooth horizontal surface. A horizontal
Find:
For mass of M and
force F is) applied to the bar’s centerdiagramsG atthe bar and the block below.vertical and at rest.
a Complete the free body of mass for an instant when the bar is b) Determine the angular acceleration of the bar for the instant when the
Find: Determine F is angular acceleration of the bar for the instant when the force F is applied to
force the applied to the bar.
the bar. M g A F G L m B
Use the following parameters in your analysis: L = 3m, m = 50kg , M = 100kg and F = 200N . Rigid Body Kinetics Homework Problems ME 274 Problem V52
V54
Chapter V: Rigid Body Kinetics Homework
Given: A homogeneous disk having a mass of m = 100 kg and radius r = 0.5 meters
rolls without slipping on a rough, inclined plane. Pin C at the center of the
Problem isk is constrained to move within a smooth slot in an arm that is pinned to
d V.52
ground at O. The arm has a mass of M = 300 kg with its center of mass at
Given: point G (with d = 0.6 having ) and with mradius of gyration about point O of on a rough,
A homogeneous disk meters a mass of a and radius r rolls without slipping
inclined plane.0.4 meters .the center of is released from rest when move .within a smooth slot in an
kO = Pin C at The system the disk is constrained to ! = 0 arm that is pinned to ground at O. The arm has a mass of M with its center of mass at point G
and with a radius of gyration about point O of kO . The system is released from rest when θ = 0. Find: Determine the angular speed of the arm when ! = 90°. At this position, C is at
a distance of h = 1.5 meters directly below O.
◦ Find: Determine the angular speed of the arm when θ = 90 . O smooth !
r g
h d G
C
no slip " Use the following parameters in your analysis: m = 100kg , M = 300kg , r = 0.5m, d = 0.6m,
h = 1.5m, kO = 0.4 and φ = 36.87◦ . Problem V53
CGiven:V: Block Body Kinetics Homework
hapter
Rigid B with mass M is being transported up the slope by using two cylindrical V55 rollers A and D, each having a mass m and radius r, as shown. A constant force
F is
Problem V.53 applied on the block and the system starts at rest. Assume that there is no
slip at all surfaces.!
Given: Block B with mass M is being transported up the slope by using two cylindrical rollers A
and D,: eachThe speed mass m andwhen the,block has moved a distance d up the hill. on the block
having a of block B radius r as shown. A constant force F is applied
Find
and the system starts at rest. Assume that there is no slip at all surfaces. Use the following parameters: d = 2 m, m = 5 kg, M = 50 kg, r = 0.5m, ! = 30o, F = 400
Find: Determine the speed of block B when the block has moved a distance d up the hill.
N. M
B F A m, r D no slip m, r no slip ! Use the following parameters in your analysis: d = 2m, m = 5kg , M = 50kg , r = 0.5m, θ = 30◦
and F = 400N . Rigid Body Kinetics Homework Problems ME 274 Problem V54
Chapter V: Rigid Body Kinetics Homework
Given: A drum has a mass of M = 30 kg, outer radius of R = 0.5 m, inner radius of r
= 0.25 m, a centroidal radius of gyration of kG = 0.6 m and centroid at the
Problem V.54
geometric center G. A cable is wrapped around the inner radius of the drum
and is connected to particle A having a mass of m = 10 kg. The cable is also
Given: A drum has aover an ideal pulley.radius of R, stiffness K = 100 ,N/m is attached
wrapped mass of M , outer A spring of inner radius of r a centroidal radius of
gyration of kGbetween G of at the geometric center G. A cable is released from rest with theradius
and centroid the drum and ground. The system is wrapped around the inner
of the drum and is connected to particle A having a mass of m. The cable is also wrapped over an
spring unstretched.
V56 ideal pulley. A spring of stiﬀness K is attached between G of the drum and ground. The system is
released from rest with the spring unstretched. Find: Determine the speed of A after it has dropped 2 meters. Find: Determine the speed of A after it has dropped through a distance of d. pulley has
negligible mass M R K
r G
no slip m A Use the following parameters in your analysis: M = 30kg , m = 10kg , R = 0.5m, r = 0.25m,
kG = 0.3m, K = 100N/m and d = 2m. Rigid Body Kinetics Homework Problems ME 274 Chapter V: Rigid Body Kinetics Homework V57 Problem V55
Given: The homogeneous thin bar AB shown below has a mass of M = 100 kg and a
Problem V.55 length of L = 3 meters. The bar is supported by lightweight, inextensible cable
BO at end B and by a smooth, inclined wall at end A, where ! = 53.13°. The
Given: The homogeneous thin bar AB shown below has a mass of M and a length of L. The bar
bar is released from rest with the bar in a horizontal orientation and with cable
is supported by lightweight, inextensible cable BO at end B and by a smooth, inclined wall at end
BO being vertical. A. The bar is released from rest with the bar in a horizontal orientation and with cable BO being
vertical. Find: Determine the tension in cable BO immediately after the bar is released. Find: Determine the tension in cable BO immediately after the bar is released. O smooth
wall A L B ! Use the following parameters in your analysis: M = 100kg , L = 3m and θ = 53.13◦ . Rigid Body Kinetics Homework Problems ME 274 V58 Chapter V: Rigid Body Kinetics Homework Problem V56
Problem V.56
Given: A thin, homogeneous bar OA (having a length of 4 meters and a mass of M =
100 kg), is pinned to ground at O. An inextensible cable is attached to end A
Given: Aothin, homogeneous bar OA (having a to block B, with a mass of M ), is mass ofto ground
f the bar. This cable is also attached length of L and B also having a pinned
at O. An inextensible cable is attachedsmall pulley the bar. Thispulley is also attached to block B,
M. The cable is hung over a to end A of C where the cable has negligible
with B having a The system Theeleased from rest wa small= 90°. Assume that the cable negligible
mass. mass of m. is r cable is hung over hen ! pulley C where the pulley has
mass. The emains taut for all time. rest when θ = 90◦ . Assume that the cable remains taut for all
r system is released from
time. FFind: Determine thethe velocity of block B when bar OA has reached theposition of θ!= 0. Write
ind:
Determine velocity of block B when bar OA has reached the position of
= as a vector.
your answer 0. Write your answer as a vector.
d
A L/2
M g C
small, lightweight
pulley ! L/2 O m
B Use the following parameters in your analysis: m = 20kg , M = 100kg , L = 3m and d = 4m. the center of the disk A is moving to the right with a speed of vA1 = 5 m/sec
and has a counterclockwise rotation rate of !1 = 8 rad/sec, as shown in Figure
1 below. In Figure 2 below is shown the instant at which the disk ceases to
slip as it continues to move on the horizontal surface.
Chapter V: Rigid Body Kinetics Homework
V59
Find: For this problem:
a) Determine the speed of A, vA2, at the instant in Figure 2 when the disk
ceases to slip on the h outer radius of R
Given: A homogeneous disk (with mass M andorizontal surface. ) is placed on a rough surface.
b) Determine the e the disk A is moving motion as the disk speed from
When placed on this surface, the center oflapsed time during the to the right with a moves of vA1
t rotation rate of ω1 , 1 shown in Figure Figure 2.
and has a counterclockwise he position in Figureas to the position in 1 below. In Figure 2 below is
Problem V.57 shown the instant at which the disk ceases to slip as it continues to move on the horizontal surface. You must provide an accurate free body diagram of the disk in order to receive full credit
f this problem:
Find: Foror your work. a) determine the speed of A, vA2 , at the instant in Figure 2 when the
disk ceases to slip on the horizontal surface, and b) determine the elapsed time during the motion
HINT: from the using both the linear impulsemomentum equation and the angular
as the disk moves Consider position in Figure 1 to the position in Figure 2. impulsemomentum equation on the disk in solving this problem. !1 M
FIGURE 1 A
R vA1
µk slipping !2
FIGURE 2 A
vA2 no slip Use the following parameters in your analysis: M = 50kg , R = 0.5m, µk = 0.3, vA1 = 5m/sec and
ω1 = 8rad/sec. Rigid Body Kinetics Homework Problems ME 274 V60 Chapter V: Rigid Body Kinetics Homework Problem V58
Given: Link AB (having a mass of m with uniform mass distribution and length L) is
Problem V.58
supported by a vertical cable OA (of length 0.6L and negligible mass) at A,
and end B is constrained to move within a smooth slot, as shown below. The
Given: Link AB (having a mass of m with uniform mass distribution and length L) is supported
bar is released from rest when it is horizontal. by a vertical cable OA (of length 0.6L and negligible mass) at A, and end B is constrained to move
within a smooth slot, as shown below. The bar is released from rest when it is horizontal. Find: Determine the angular acceleration of AB on release. Use the following
parameters: m = acceleration of meters.
Find: Determine the angular40 kg and L = 2AB on release. g O
0.6 L
massless
cable m, L B A
! smooth Use the following parameters in your analysis: m = 40kg and L = 2m. Chapter V: Rigid Body Kinetics Homework V61 Problem V.59
Given: An inhomogeneous disk (having a mass of m, outer radius R, mass center at G and
centroidal radius of gyration of kG ) moves along a SMOOTH horizontal surface. The distance from
omework Problems the geometric center O of the disk to the center of mass G is given by e. At the instant shown, G
ME 274
is at the same vertical height as O as the disk translates to the right with a speed of vO .
Find: Determine the normal force of the ground acting at the contact point C at this instant. mogeneous disk (having a mass of m,
dius R, mass center at G and centroidal
f gyration of kG) moves along a
H horizontal surface. The distance
e geometric center O of the disk to the
f mass G is given by e. At the instant
G is at the same vertical height as O as
translates to the right with a speed of m
g R
e
G vO O SMOOTH
ine the normal force of the ground
t the contact point C at this instant. In
C
alysis, use the following parameter
m = 10 kg, R = 0.5 meters, e = 0.3
kG = 0.4 meters,se nd vO = 20 m/sec.
U a the following parameters in your analysis: m = 10kg , R = 0.5m, e = 0.3m, kG = 0.4m and vO = 20m/sec. ct point C slips on the smooth horizontal surface. Rigid Body Kinetics Homework Problems V62 Problem V60 M Chapter V: Rigid Body Kinetics Homework Given: The compound wheel shown rolls without slipping up the incline on its hub
and is pulled by a constant force T = 100 N applied to a cord wrapped arou
Given: The compound wheel shown rollsouter rim. The wheel starts from rest, hubs and is of m = 40 kg, and has
its without slipping up the incline on its has a mass pulled
by a constant force T applied to a cord wrappedgaround its outer rim. The mass O of kOfrom.15 meters. Assume
radius of yration about its center of wheel starts = 0 rest,
has a mass of m, and has a radius of gyration about not center of mass O of kO . Assume that the
the cable does its slip on the wheel.
Problem V.60 cable does not slip on the wheel. F velocity The angular after its center wheel after itsacenter O hasd up 2 meters up
Find: Determine the angularind:
of the wheel velocity of the O has moved distance of moved
incline.
the incline.
F no slip
R/2 O
R ! Use the following parameters in your analysis: m = 40kg , R = 0.2m, d = 2m, T = 100N and
kO = 0.15m. Problem V61
Given: A uniform rectangular plate of mass m is able to move on a smooth
Chapter V: HORIZONTAL plane with corner A of the plate being constrained to move
Rigid Body Kinetics Homework V63 within a smooth guide in the plane of motion. The plate is initially at rest with
edge AB aligned with the guide for A. A force P is applied at corner D.
Problem V.61
Find
Determine the initial angular acceleration to move on smooth HORIZONTAL
Given:: A uniform rectangular plate of mass m is able of the plate.aWrite your answer as a plane
with corner vector. plate being constrained to move within a smooth guide in the plane of motion.
A of the
The plate is initially at rest with edge AB aligned with the guide for A. A force P is applied at
corner the following parameters in your analysis: m = 200 kg , b = 4 meters , h = 3 meters ,
Use D. ! = 36.87° and P = 800 N . Find: Determine the initial angular acceleration of the plate. A E b P h ! B
D
smooth guide for pin A HORIZONTAL PLANE Use the following parameters in your analysis: m = 200kg , b = 4m, h = 3m, θ = 36.87◦ and
P = 800N . Rigid Body Kinetics Homework Problems ME 274 V64
Problem V62 Given: Chapter V: Rigid Body Kinetics Homework A smooth, massless roller is attached to end A of a thin, homogeneous bar of Problem V.62
length L = 2 meters and mass m = 10 kg. The roller is allowed to move on a horizontal guide. The bar is released from rest with the bar being at an angle Given: A smooth, masslessrom the vertical, asto end A of a thin, homogeneous bar of length L
of ! = 53.13° f roller is attached shown below.
meters and mass m. The roller is allowed to move on a horizontal guide. The bar is released from
rest at the position shown below. Find: Determine the acceleration of the center of mass of the bar at the instant of
release. acceleration of the center of mass
Find: Determine the State your answer in vector form. of the bar at the instant of release. State
your answer in vector form. A smooth,
massless roller m, L g ! Use the following parameters in your analysis: m = 10kg , L = 2m and θ = 53.13◦ . cs Homework Problems
Chapter ME 274 V: Rigid Body Kinetics Homework V65 63
Problem V.63
gear of mass m and outer radius R meshes with rack A (having a mass of
). Rack A is constrained to move along a smooth vertical guide. Treat the
Given: A gear of mass m and outer radius R meshes with rack A (having a mass of 2m). Rack
ar as a homogeneous disk of radius R. Assume that the rack does not slip on
A is constrained to move along a smooth vertical guide. Treat the gear as a homogeneous disk of
e gear.
radius R. Assume that the rack does not slip on the gear. etermine the acceleration of rack A when the system is released from rest.
Find: Determine the acceleration of rack A when the system is released from rest. Write your
rite your answer as a vector.
answer as a vector.
A
no slip m
C O g R
2m
smooth Rigid Body Kinetics Homework Problems ME 274 V66 Chapter V: Rigid Body Kinetics Homework Problem V64
Given: A crate of mass 2m rests on block C of mass m that, in turn, rests on a smooth
!
Problem V.64
horizontal surface. A horizontal force F acts to the right on block C. Given: A crate of mass 2m rests on block C of mass m that, in turn, rests on a smooth horizontal
!
Find:
Determine the acts to the right f block C.
surface. A horizontal force F maximum force Fonor which the crate does not tip on C. Assume that sufficient friction exists between the crate and C such that the crate does
Find: Determine the maximum C. Use the which the crate does not tip on analysis: that suﬃcient
not slip on block force F for following parameters in your C. Assume
friction exists between the ,crate 3 meC rs and b = the meters . not slip on block C.
m = 100 kg h = and te such that 1.5 crate does
b b
2m G
no slip C A B h m !
F smooth Use the following parameters in your analysis: m = 100kg , h = 3m and b = 1.5m. Rigid Body Kinetics Homework Problems ME 274 Chapter V Rigid
V67
Problem V:65 Body Kinetics Homework
Given: A cart is made up of a body (of mass M) and FOUR wheels (each wheel of
mV.65
Problem ass m and outer radius R). The rear wheels are solid and homogeneous,
whereas the front wheels are spoked rims (with the spokes being massless).
T cart is made up moving to the right with a FOUR v. At (each wheel
Given: Ahe cart is initiallyof a body (of mass M ) andspeed ofwheelssome point, aof mass m and
spring of stiffness k attached to the front of the cart whereas fixed wall.
outer radius R). The rear wheels are solid and homogeneous,contacts athe front wheels are spoked
rims (with the spokes being massless). The cart is initially moving to the right with a speed of v .
At :
point, a spring of stiﬀness k attached of the spring the contacting the ﬁxed
Findsome Determine the maximum compressionto the front ofafter cart contacts a wall. wall. Use the following parameters in your analysis: M = 14 kg , m = 3 kg ,
R = 0.15 meters , k = 800 N / m and v = 0.5 m / sec . Find: Determine the maximum compression of the spring after contacting the wall.
v k M R A no slip two wheels
(each of mass m ) B
no slip R two wheels
(each of mass m ) Use the following parameters in your analysis: M = 14kg , m = 3kg , R = 0.15m, k = 800N/m and
v = 0.5m/sec. Rigid Body Kinetics Homework Problems ME 274 Problem V66
V68 Chapter V: Rigid Body Kinetics Homework Given: Particle P (of mass m) strikes end A of a stationary homogeneous Lshaped bar
( V.66
Problemof mass M) with a speed of v. The coefficient of restitution for the impact of P
with end A of the bar is known to be e. Given: Particle P (of mass m) strikes end A of a stationary homogeneous Lshaped bar (of mass
M)
a speed of he angular speed of restitution for the impact of impact. Use the
Find:with Determine tv . The coeﬃcient of bar OA immediately after the P with end A of the bar is
known to ollowing parameters in your analysis: m = 10kg , M = 30 kg , L = 2 meters ,
f be e. v = 40 m / sec and e = 0.5 . Find: Determine the angular speed of bar OA immediately after the impact. O L P
m v
A L HORIZONTAL PLANE
Use the following parameters in your analysis: M = 30kg , m = 10kg , L = 2m, v = 40m/sec and
e = 0.5. ...
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