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Unformatted text preview: Particle K inetics Homework Problems ME 274 IV6 Chapter IV: Particle Kinetics Homework Problem IV4
Problem IV.4 The mechanism, shown below, consisting of a rotating arm, frictionless slot, and block of mas m, i driven at a shown below, ! by a motor rotating arm, frictionless slot, and
Given:s The smechanism, constant speedconsisting of a attached at point O. Determine: block of
mass m,a) tdriven at a in the cable and ω by a motor exerted on the block at the instant shown
( is he tension constant speed normal force attached at point O.
corresponding to ! = 10 rad/s Find: Determine: (a) the tension in the cable and normal force exerted on the block at the instant
shown (b) the minimum ω = 10rad/sec, andnecessary to keep angular velocity ω necessary to keep
corresponding to angular velocity ! b) the minimum the cable taut at this instant
the cable taut at this instant.
! r g m
O " Use the following parameters in your analysis: r = 0.3m, m = 12kg and θ = 60◦ . Use r = 0.20 m, m = 10 kg and " = 53.13° in your analysis. Kinetics Homework Problems ME 274 IV8 m IV6 Chapter IV: Particle Kinetics Homework Problem IV.6 um of an industrial polisher, shown an industrial polisher, point Obelow,arotates about point O with a constant
Given: The drum of below, rotates about shown with constant
!. A small cube of ω . A small cube of aluminuminner surfacethe inner surface of the drum. It is observed that
speed aluminum is placed on the is placed on of the drum. It is
ed that this piece of metalmetalno motion relative to the the polishers surface as it passes through θ = 0.
this piece of has has no motion relative to polisher’s surface as it
through " = 0. Determine the relationship among r, µS (the static coefficient of
Find: Determine the relationship among r, ( S (the static coeﬃcient
between the block and the drum’s inner surface) and "crit µthe angle of orientation of friction between the block
and the
h the block first slips).drums inner surface) and θcritical (the angle of orientation at which the block ﬁrst slips). ! g µs
O r
" m etics Homework Problems
IV10 IV8 ME 274
Chapter IV: Particle Kinetics Homework Problem IV.8 Given: The collar, from below, of mass A starts from rest is
r, shown below, of mass m, starts shown rest at point A. m, constant force F at point A. A constant force F is
the collar in applied to the shown.in the direction shown.of the that the mechanism lies in the vertical plane.
the direction collar Determine the speed Note collar when it
Assumemechanism liesbe smooth.
oint B. Note that the all surfaces to in the vertical plane. Assume all surfaces
oth.
Find: Determine the speed of the collar when it reaches point B.
A
m
F
r g
B ollowing: mg =Useb, F = 30 lb and r = 1.5 ft.in your analysis: mg = 10lb, 50lb and r = 2f t.
6 l the following parameters Chapter IV: Particle Kinetics Homework inetics Homework Problems m IV9 ME 274 IV11 Problem IV.9 Given: The collar shown below, of mass m is pushed against a spring with stiﬀness k until it is
llar, shown below, of mass mandpushed against a spring ﬃcient of kineticuntil it isbetween the collar and rod is
compressed ∆ is then released. The coe with stiffness k friction
ssed !, and then. released. Assuming that there is a kinetic coefficient of friction
µk een the collar and rod, and that the collar fails to launch clear of the rod,
Find: Determine the maximum height measured the the as measured from the point of release.
ine the maximum height obtained by the collar, as obtained byfromcollarpoint of
Assume that the collar remains on the rod at all times.
. g m µk
!
k following: m Use.5 kg, k = 2000 N/m, ! = 0.3your analysis: and !4= 36.87°. N/m, µk = 0.15 and θ = 30◦ .
= 2 the following parameters in m, µk = 0.20, m = kg , k = 2500 Problem IV11
Pellets A and B (having masses m and 3m, respectively) are placed within a smooth tube
trapping a small compartment of fuel. At a time when the pellets are initially at rest, the
Chapter ignited. The Kinetics Homework over a short time !t, and over this time the
IV13
fuel is IV: Particle combustion occurs
combustion applies equal and opposite forces on the pellets with this force idealized by
the force time h
Problem IV.11 istory F(t) shown below.
a) Draw individual free body diagrams (FBD’s) for pellets A and B. From these
Given: Pellets Adetermine the speed of each pellet at a time t , where t > !within a smooth tube
FBD’s, and B (having masses m and 3m, respectively) are placed t.
2
2
trapping a small compartment of fuel. At a time when the pellets are initially at rest, the fuel is
b) Draw an FBD for pellets A a short time ∆t, and over and B as the combustion
ignited. The combustion occurs over and B together (i.e., treat A this time a single system).applies
F opposite forces on the pellets with this force momentum the force time history F(t)
equal and rom this FBD, provide an argument that linear idealized by for the system of A
and
shown below. B together should be conserved. Use your results from a) above to verify this argument. Find: Determine the speed of each pellet at time t2 , where t2 > ∆t. F(t)
A B m 3m smooth F0 smooth
combusting
fuel 0 !t Use the following parameters in your analysis: F0 = 3000N , ∆t = 0.005sec and m = 0.75kg . Use the following parameters in your analysis: F0 = 2000 N , ! t = 0.006 s and
m = 0.5 kg . t particle impacts the floor with a speed of v1. This impact of the particle with the floor
lasts for a short duration of time !t, and after the impact is complete, the particle
rebounds upward with a speed of v2. The particle continues upward reaching a maximum
IV14 height of h2.
Chapter IV: Particle Kinetics Homework
a) Draw a free body diagram (FBD) of the particle for the time period during its
impact with the floor.
Problem IV.12
b) Based on your FBD in a), determine the average force acting on the particle by Given: A particle of mass m is dropped from rest when at avalue ofha force Fa rigid ﬂoor. The
the floor during impact. Recall that the average height 1 above (t) acting over a
particle impacts the ﬂoor with a speed of v1 . This impact of the particle with the ﬂoor lasts for a
!t
short duration period of time !t after the by: F is= 1
of time ∆t, and is given impact complete,( t ) dtparticle rebounds upward with a
F the .
ave
! tmaximum height of h2 .
speed of v2 . The particle continues upward reaching a 0 ! Find: For c) Rproblem: a)cdetermine the b) aboveforcenow ignorethe particle force during during
this epeat your alculations in average but acting on the weight by the ﬂoor the
time of of gravity, b) determine for the average acting on the particle by the
impact in the presenceimpact. Did your answerthe average forceimpact force change much? ﬂoor
during impact Determine the of gravity, c)/ compare your answers from a) and b), and d) determine
d) in the absence value of h2 h1 .
the value of h2 /h1 . v=0
v=0
h1 h2 v = v2 v = v1 Use the following parameters in your analysis: ∆t = 0.002sec, m = 15kg , v1 = 80m/sec and
Use the
v2 = 50m/sec. following parameters in your analysis: ! t = 0.001 s , m = 10 kg , v1 = 60 meters / sec and v2 = 45 meters / sec . Particle K inetics Homework Problems ME 274 IV16 Chapter IV: Particle Kinetics Homework Problem IV14
A cannonball P of mass m is fired toward a steel barrier on a stationary cart. At some
Problem IV.14
time after rebounding from the barrier, the cannonball is observed to have a speed of vP
a A moving in a of mass m is ﬁred toward the figure. Let M stationary cart. mass of time
Given: nd cannonball P direction shown below in a steel barrier on abe the combined At some the
cannon and cart. Assume that the cart is able to move to have a speed of vP the
after rebounding from the barrier, the cannonball is observed without friction along and moving in
horizontal below and ignore the influence of combined mass
a direction shown surfacein the ﬁgure. Let M be the air resistance. of the cannon/cart. Assume
that the cart is able to move without friction along the horizontal surface and ignore the inﬂuence
a) D
of air resistance.etermine the velocity (both magnitude AND direction) of the cart after the cannonball bounces off the steel barrier at the instant shown below.
b) Lproblem: a) determine the velocity vector of the cartof the cannonball and bounces
Find: For this et !t represents the elapsed time between the firing after the cannonball the
instant shown below. Determine the average value of the elapsed time between
oﬀ the steel barrier at the instant shown below, and b) if ∆t representsthe horizontal force actingthe
on the combined cannon and cart over the time period of 0 < t < ! of
ﬁring of the cannonball and the instant shown below, determine the average value t. the horizontal
force acting on the combined cannon/cart over the time period of 0 < t < ∆t. vP P !
m
cannon
cart M Use the following parameters in your analysis: mg = 60lb, M g = 200lb, ∆t = 0.2sec, θ = 20◦ and
vP = 120f t/sec. Use the following parameters in your analysis: mg = 50 lb , Mg = 250 lb , ! t = 0.4 s ,
! = 25° and vP = 100 ft / s . Particle K inetics Homework Problems ME 274 Chapter IV: Particle Kinetics Homework IV17 Problem IV15
A particle of mass m is projected horizontally to the right at a height h above a smooth,
Problem IV.15
horizontal floor with a speed of v0. The particle strikes the floor at a horizontal distance d
from where it was is projected horizontally to the right restitution h above a smooth,
Given: A particle of mass m initially projected. The coefficient of at a heightof the impact of the
particle with the floor is The particle strikes the ﬂoor at a horizontal distance d from
horizontal ﬂoor with a speed of v0 . e.
where it was initially projected. The coeﬃcient of restitution of the impact of the particle with the
a) Determine the angle !1 that the velocity of the particle has with the horizontal
ﬂoor is e. before impact. Use conservation of energy prior to impact. Find: For thisbproblem: a) determine thethat the 1 that the velocity of the particle has with the
) Determine the angle !2 angle θvelocity of the particle has with the horizontal
horizontal immediately before impact,impact. determine the angle θ2 that the mentum of the particle
immediately after and b) Use conservation of linear mo velocity in the horizontal
has with the horizontal immediately after impact.
direction along with the coefficient of restitution equation during impact. v0 v2 h !2 d
BEFORE impact !1
v1
immediately AFTER impact Use the following parameters in your analysis: = f t , h v125 ft a t/sec
Use the following parameters in your analysis: e = 0.5, h =e1500.6 and = 0 = 160fnd v0. = 150 ft / sec . A satellite P of mass m is in orbit around a planet whose center is at E. At position 1
shown below, the satellite is at a distance R1 from E and moving with a speed of v1. At
position 2 shown, P is at a distance of R2 from E and moving with a speed of v2 with the
Chapter IV: Particle Kinetics Homework
velocity of P oriented as shown below. Assume that the only force acting on the satellite
is a gravitational force directed toward E, and that the planet is not accelerating. IV19 Problem IV.17 a) Determine the angular momentum of P about E at positions 1 and 2. Given: A satellite P of mass m is in orbit aroundthe planet whose center about E. is position 1
b) Provide an argument to support the claim that a angular momentum is at E At
shown onstant.the satellite is at a distance R1 from E and moving with a speed of v1 . At position
c below,
2 shown, P is at a distance of R2 from E and moving with a speed of v2 with the velocity of P
c) Based on b) below. Assume that speed of P at acting on v2 in terms of gravitational force
oriented as shown above, determine the the only force position 2,the, satellite is aR1, R2,
directedatoward E, and that the planet is stationary.
!2 nd v1. d) Explain why your result in c) does not depend on the gravitational force acting on Find: For this problem: a) determine the speed of P at position 2, v2 , in terms of R1 , R2 , θ2 and v1 ,
the satellite.
and b) provide an argument explaining why your result in a) does not depend on the gravitational
force acting on the satellite. e! eR !2 v2 e!
P
R2 E R1 v1 P eR Problem IV20
Particles A and B each have a mass of m. A is constrained to move on a horizontal arm
OC, and B is constrained to move on the vertical shaft about which arm OC rotates. A
taut cable connects A and B as shown in the figure below. At an insIV: Particle Kinetics Homework
IV22
Chapter tant when the shaft is
rotating with a rate of ! = !1 and when R = R1 , A and B are released from rest relative to
the rotating arm and shaft. Consider all surfaces to be smooth, assume the masses of the
Problem IV.20
arm, shaft and pulley to be negligible, and assume the radius of the pulley to be small.
Given: Particles A and B each have a mass of m. A is constrained to move on a horizontal arm
OC, and B is constrained to move on the vertical shaft about which arm OC rotates. A taut cable
connects A and the angular speed of the arm ! 2 . At an instant when angular is rotating with a
a) determine B as shown in the ﬁgure below. Consider using the the shaft
rate of omentum equation. R1 , A and B are released from rest relative to the rotating arm and
mω = ω1 when R =
shaft. Consider all surfaces to be smooth, assume the masses of the arm, shaft and pulley to be
b) determine the speed radius of the pulley the workenergy equation along with
negligible, and assume the of B. Consider usingto be small. When A has moved outward to a position of R = R2 , appropriate kinematics relating the motions of A and B. Be careful – the velocity
Find:of A has two components at position 2 (one due = R2 : a) determine the angular speed of the
When A has moved outward to a position of R to the rotation of the arm and one
due to its sliding along particle B.
arm ω2 , and b) the speed of the arm). R C
O
A
g B ! Use the following parameters in your analysis: m = 15kg , R1 = 0.3m, R2 = 0.6m and ω1 = Use rad/sec.
the following parameters in your analysis: m = 10 kg , R1 = 0.2 meters ,
25
R2 = 0.5 meters and !1 = 20 rad / sec . Chapter IV: Particle Kinetics Homework IV23 Problem IV.21
tics Homework Problems
Given: The system shown is released from rest. ME 274 IV21 (3/27) Find: On release, determine: a) the acceleration of each block, and b) the tension in the cable.
shown is released from rest.
the following:
acceleration of each block.
tension in the cable. ! Use the following parameters in your analysis: µs = 0.25, µk = 0.2 and θ = 30◦ . llowing parameters in your analysis: µ s = 0.25 , µ k = 0.20 and ! = 30° . Chapter IV: Particle Kinetics Homework IV31 Problem IV.29
Given: A bullet strikes the lower particle of a stationary pendulum with a speed of v. After
impact, the bullet sticks to the particle.
mework Problems 9 (3/248) ME 274 Find: Determine: a) the angular speed of the pendulum immediately after impact, and b) the
maximum rotation angle through which the pendulum swings after impact. M the lower particle of a stationary
a speed of v. After impact, the bullet
ticle. Determine:
lar speed of the pendulum immediately
pact.
imum rotation angle through which the
m swings after impact. M
m
Use the following parameters in your analysis: v = 300m/sec, m = 50grams and M = 3.2kg . ng parameters in your analysis: v = 300 m / sec , m = 50 grams and Particle K inetics Homework Problems M Chapter IV: Particle Kinetics Homework 31
Problem IV IV33 A basketball is being thrown upward with an initial velocity of vi. Using Newton’s
second law, determine the maximum height of the basketball
Problem IV.31
a) when drag force caused by air resistance is neglected.
2
b) w upward force magnitude is given by
Given: A basketball is being thrown hen dragwith an initial velocity of vi . Fdrag = cv . Find: Determine the maximum height of the basketball: a) in your analysis:caused .62 air resistance
Use the following parameters when drag force m = 0 by kg, vi = 10 m/s, c = 0.013 N
22
is neglected, and b) when drag force magnitude is given by Fdrag = cv 3 .
sec /m h g
vi
Use the following parameters in your analysis: m = 0.7kg , vi = 10m/sec and c = 0.015N − sec2 /m2 . Chapter IV: ParticleParticle KineHomework Problems
Kinetics tics Homework IV35 Problem IV.33 ME 274 Problem IV33
!
A mass is a spring rotating about the about O.
Given: A mass is attached to attached to a spring rotating pin at the pin at O. At the instant when ! = 1 rad/s
2
!
r
= constant, r = 0.8 m, r = 2 m/s and !! = 2.4 m/s . Find the unstretched length of the
spring. (m = 2 length=of 0 N/m)
kg, k 8 the spring.
Find: Determine the unstretched O
! g
r m Use the following parameters in your analysis: m = 3kg , k = 100N/m, r = 0.75m, r = 3m/sec and
˙
r = 3m/sec2 .
¨ Particle K inetics Homework Problems ME 274 Problem IV44 IV46 Chapter IV: Particle Kinetics Homework A thin homogeneous bar OA (having a mass of M = 10 kg and length L = 3 meters) is
pinned to ground at point O in such a way that the bar moves in a HORIZONTAL plane.
Problem IV.44
Particle B (with a mass of 10 kg) is able to slide without friction on OA, and is attached
to A with a spring having a stiffness of k = 500 N/m and an unstretched length of 1.5
Given: A thinthe instant shown, bar OA is rotating CW M and speed of ) is pinned to ground at
meters. At homogeneous bar OA (having a mass of with a length L
,
point O in such a way that the bar moves in a HORIZONTAL plane. Particle B (with a mass of
particle B is not moving relative to bar OA and the spring is unstretched.
m) is able to slide without friction on OA, and is attached to A with a spring having a stiﬀness of
k andind the velocity of particle d. after it has moved 0.5 meters outward on the bar. Write
F an unstretched length of B At the instant shown, bar OA is rotating CW with a speed of
ω1 , particle B is notamoving relative to bar OA and the spring is unstretched.
your answer as vector. H Determine thebar OA and particle B as a single sytstem. a distance of b meters outward on
Treat velocity of particle B after it has moved
Find:INT:
the bar. Write your answer as a vector.
O
M (mass of bar)
!1
B L m (mass of B) (B can slide on rod)
k
A
HORIZONTAL PLANE Use the following parameters in your analysis: ω1 = 9rad/sec, m = 10kg , M = 10kg , L = 3m,
k = 500N/m, d = 1.5m and b = 0.5m. Particle K inetics Homework Problems ME 274 Problem IV45
Chapter IV: Particle Kinetics Homework masses of mA = mB = 10 kg) are interconnected by
IV47
Given: Particles A and B (having the cablepulley system shown in the figure. Both particles are constrained to
vertical motion with particle A able to slide on a smooth vertical rod. The
Problem IV.45
system is released at sA = 0 with A traveling downward with a speed of 5
m A and B (having masses of m small, massless and frictionless.
Given: Particles /sec. Assume the pulleys to beA and mB ) are interconnected by the cablepulley
system shown in the ﬁgure. Both particles are constrained to vertical motion with particle A able
to slide on a smoothare asked rod. The system peed of particleA = 0 with A travelingthe
Find:
You vertical here to find the s is released at s A when A has reached downward
with a speed of position of s the pulleys to be small, massless and frictionless.
vA1 . Assume = 2 m.
A Find: Find the speed of particle A when A has reached the position of sA . 1.5 m O
sA g O
sB A B A B smooth
rod Use the following parameters in your analysis: mA = 10kg , mB = 10kg , vA1 = 5m/sec and
sA = 2m. s A and B (having masses of m and M, respectively) are initially
IV48
Chapter IV: Particle Kinetics Homework
ing in directions perpendicular to each other with speeds of vA1 and vB1 ,
tively, as shown below in the figure. After impacting each other, A is
ing to the RIGHT with a speed of vA 2 , and B travels with a speed of vB 2 .
Problem IV.46
der all surfaces to be smooth.
Given: Blocks A and B (having masses of m and M , respectively) are initially traveling in directions perpendicular to each other with speeds of vA1 and vB 1 , respectively, as shown below in the
is problem,
ﬁgure. After impacting each other, A is traveling to the RIGHT with a speed of vA2 , and B travels
) draw FBD’swith A speed of vB 2 (direction of Imotion a set of coordinate is not known). Consider all surfaces to
for a alone, B alone and A+B. nclude for B after impact
axes with thesesmooth.
be FBD’s. ) determine the mass M of block B. Find: For this problem: a) determine the mass M of block B, and b) determine the coeﬃcient of ) determine the coefficient, of restitution, e,of A the impact of A and B.
restitution, e for the impact for and B
write down the linear momentum vector
ns for A, B and A+B in terms of their
and tangential components, as well as
ficient of restitution equation. Use these
ns to find M and e. B A vA1
m g parameters in your analysis: m = 3kg ,
eters / sec , vA 2 = 2 meters / sec
rs / sec . M vB1
HORIZONTAL PLANE Use the following parameters in your analysis: m = 3kg , vA1 = 4m/sec, vB 1 = 4m/sec, vA2 =
2m/sec and vB 2 = 5m/sec. Problem IV50
Given: A smooth, circular slot is cut into block A with block A being constrained to
move along a smooth, horizontal surface. The slot is vertical at the top surface
of the block with the slot being horizontal at IV: Particle Kinetics Homework
IV52
Chapter the right edge of the block, as
shown in the figures below. In Position 1, block A is stationary, and a particle
B is released from rest into the upper opening of the slot. At Position 2 shown
Problem IV.50
below, particle B is exiting the slot at the right edge of the block. The masses
of A and B are M = 40 kg and m = 20 kg, respectively. The radius of the
Given: A smooth, circular slot is is given by r = 2 meters.
circular slot cut into block A with block A being constrained to move along
a smooth, horizontal surface. The slot is vertical at the top surface of the block with the slot being
horizontal at the right edge of the block, as shown in the ﬁgures below. In Position 1, block A is
Find
Determine the velocity of block the upper elocity of the slot. at Position 2
stationary, and :a particle B is released from rest into A and the vopeningof particle B At position 2.
shown below, particle B is exiting the slot at the right edge of the block. The masses of A and B
You must provide an radius of the circular slot is r
are M and m, respectively. Theaccurate free body diagram of .the system used in your solution in order to receive full credit for your work. Consider using both the linear impulsemomentum and work block equations in solving particle B at
Find: Determine the velocity ofenergyA and the velocity of this problem. position 2. m VA2 B
r g
B A M POSITION 1 A POSITION 2 Use the following parameters in your analysis: m = 20kg , M = 40kg and r = 2m. VB2 oblem IV52
ven: Particle P (of mass m = 8 kg) moves within a vertical plane inside a rough,
circular slot. The coefficient of kinetic friction between particleChapter IV: Particle Kinetics Homework
P and the slot
IV54
is µk = 0.2, and the radius of the slot is r = 2 meters. At the position shown
below, the speed of P is known to be v = 3 m/sec.
Problem IV.52 d: For this position:
Given: Particle P (of mass m) moves within a vertical plane inside a rough, circular slot. The
a) coeetermine the numerical value of the normalPcontact force of the slot the radius of the slot is r.
D ﬃcient of kinetic friction between particle and the slot is µk , and
Atn P. position shown below, the speed of P is known to be v .
o the
b) Find:in contact position: ia) determine the numerical value of the normal contact force of the slot
Is P For this with the nner or outer surface of the slot?
c) on etermine the rate of change of speed of of speed of P.
D P, and b) determine the rate of change P.
v
rough
surfaces, m
P µk r
53.13° O
g Use the following parameters in your analysis: m = 8kg , µk = 0.2, r = 2m and v = 3m/sec. Chapter IV: Particle Kinetics Homework IV55 Problem IV.53
Given: Blocks A and B (having masses of 2m and m, respectively) are connected by rigid, massless
rod AB of length L. Block A is constrained to move along a smooth vertical guide, and B moves
along a smooth, horizontal surface. The system is released from rest when A is at a height of h1
Homework Problems
ME 274
ABOVE the path of B. 53
Find:
cks A and B (having Determine 2m and m, of block A when A has dropped to a position that is at a distance of
masses of the speed respectively) are connected by
h BELOW the path of B.
id, massless rod 2AB of length L. Block A is constrained to move along a
ooth vertical guide, and B moves
ng a smooth, horizontal surface.
e system is released from rest when
2m
s at a height of h1 ABOVE the path
A
B.
r this problem, g ) Draw a free body diagram of the
system made up of block A,
block B and rod AB. m L h1 B ) Determine the speed of block A
when A has dropped to a
position that is at a distance of
h2 BELOW the path of B.
ysis, use the following parameter
10 kg, k = 1000 N/m, L = 0.5
0.4 meters and h2 = 0.3 meters. Use the following parameters in your analysis: m = 10kg , k = 1000N/m, L = 0.5m, h1 = 0.4m
and h2 = 0.3m. ...
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