Unformatted text preview: Chapter I: Particle Kinematics Homework Chapter I Particle Kinematics Homework I1 I2 Chapter I: Particle Kinematics Homework Chapter I: Particle
article Kinematics Homework Problems Kinematics Homework I3 ME 274 roblem I/1 Problem I.1
√ 2
article P moves within the xy plane on awithin the xy plane (on = 2 4 xdescribed by y (x) = 2 4x2 + 12 where
Given: Particle P moves path described by y x ) a path + 12 , where
oth x and y areboth x in feet.are given in feet.
given and y Find: Determine the acceleration of as a vector in terms as a xy
etermine the acceleration of P. Write your answer P. Write your answerof itsvector in terms of its xy components.
omponents. y path of P P
x
U parameters in parameters in your analysis: t sec x =
˙
¨
!
se the followingse the followingyour analysis: x = 6 ft , x = !2x f=/2f t, and −10f t/sec and x = 0 ! = !3 ft / sec 2 . I4 Chapter I: Particle Kinematics Homework Problem I.2
Particle Kinematics Homework Problems ME 274 Given: An automobile A is traveling on a circular path centered at O and having a radius of R.
The automobile has a speed of v and is changing this speed at a rate of v .
˙ Problem I/2
Find: For this on a circular path centered at O and having a radius of R.
An automobile A is traveling problem:
!
The automobile has a speed of v , and is changing this speed at a rate of v . • Determine the acceleration of A. Write this as a vector in terms of its xy components. a) Determine the acceleration of A. Write this as a vector in terms of its xy
• Make a sketch of the acceleration vector for A.
components. • Determine the magnitude of for A.
b) Make a sketch of the acceleration vector the acceleration of A in terms of the number of ”g’s” experienced
by a passenger in the automobile. c) Determine the magnitude of the acceleration of A in terms of the number of “g’s”
experienced by a passenger in the automobile.
y
v
A R
!
x
O Use the following parameters in your analysis: 53.13 75 v 20 m / s ◦ nd
˙
Use the following parameters in your analysis: R = 150 m , ! = R = ° , m,= θ = 135a, v = 10m/sec and v = !
v = !2 m / s 2 . −6m/sec2 . Chapter I: Particle Kinematics Homework I5 Problem I.3
Given: A particle P travels in the xy plane with a path whose coordinates are given as a function
of time t as: x (t) = 16 − 12t and y (t) = 2 + 15t − 3t2 .
Find: For this problem:
• Determine the velocity and acceleration of P in terms of their xy components.
• Make a sketch of the velocity and acceleration vectors for P.
• Determine the rate of change of speed of P and the radius of curvature for the path of P.
Use the following parameters in your analysis: t = 10seconds. I6 Chapter I: Particle Kinematics Homework Problem I.4
Given: A particle P moves within a plane with a path given in terms of the polar coordinates of:
R = 2θ2 where R and θ are as deﬁned below in the ﬁgure, and with R and θ given in meters and
in radians, respectively.
mework Problems ME 274 Find: For this problem:
• Determine the velocity and acceleration vectors of P in terms of their polar coordinates. s within a plane with a path given in terms of the polar coordinates of: • Make a sketch of the velocity and acceleration vectors for P.
• Determine the rate of change of speed of P. • Is speed of and with R nd ! given in meters and in
e as defined below in the figure P increasingaor decreasing at this instant? Explain.
ely. the velocity and acceleration vectors of P in
eir polar coordinates. eR etch of the velocity and acceleration vectors
P
the rate of change of speed of P.
f P increasing or decreasing at this instant? e! ! R O
˙
¨
Use the following parameters in your analysis: θ = π /2radians, θ = −3rad/sec and θ = −2rad/sec2 . !
parameters in your analysis: ! = " radians , ! = " 1.5 rad / sec and Chapter I: Particle Kinematics Homework I7 Problem I.5
Given: Automobile A is traveling down a roadway with a speed of vA when it encounters a
hailstorm. Hailstone P is known to be falling with a speed of vP at an angle of θ forward of the
automobile.
Particle Kinematics Homework Problems ME 274 Find: For this problem: • Write down, as a vector, the velocity of P as seen by a passenger in the automobile. Problem I/5
Automobile A is traveling doangle roadway with a speed ofobserved velocity of P? a
• At what wn a with the vertical is this vA when it encounters
hailstorm. Hailstone P is known to be falling with a speed of vP at an angle of ! forward
of the automobile.
a) Write down, as a
vector, the velocity of
P as seen by a
passenger in the
automobile.
b) At what angle with the
vertical is this
observed velocity of P? vP vA ! P A Use the following parameters in your analysis: vA = 40m/sec, vP = 10m/sec and θ = 20◦ . Use the following parameters in your analysis: vA = 20 m / sec , vP = 6 m / sec and
! = 53.13° . be small.
termine the speed of block B when s A = 0 .
I8
termine the speed of block B when s A = 4 m . Chapter I: Particle Kinematics Homework Problem I.6
Given: Blocks A and B are connected by the pulleysystem shown below. Block A moves downward
with a constant speed of vA on a vertical guide. Assume the radii of the pulleys to be small.
Find: For this problem:
• Determine the speed of block B when sA = 0.
• Determine the speed of block B when sA = 4m. 8 m / sec in your analysis. 1.5 m O
sA sB A
vA Use the following parameters in your analysis: vA = 25m/sec. B Chapter I: Particle Kinematics Homework I9 Problem I.7
Given: Pin P moves within slots cut into links A and B. The horizontal position of link A is given
by x = 20 + t2 /4 whereas the vertical position of link B is given by y = 15 − t3 /6, where t is given
in seconds and with x and y given in mm.
omework Problems ME 274 Find: Determine the velocity and acceleration of P at t = 2 seconds. roblem 2/66)
thin slots cut into links A and B. The
ion of link A is given by x = 20 + t 2 / 4
tical position of link B is given by
where t is given in seconds and with x
m. Determine the velocity and and
P at t = 2 seconds. I10 Chapter I: Particle Kinematics Homework Problem I.8
Given: A pitcher throws a baseball with an inital speed of v0 at an angle of θ with the horizontal.
Find: Determine the rate of change of speed and radius of curvature of the path when:
• the ball is released by the pitcher. ework Problems blem 2/126) ME 274 • the ball reaches the maximum height of its path. baseball with an inital speed of
! with the horizontal. Determine the
peed and radius of curvature of the v0
! released by the pitcher.
ches the maximum height of its path. ec and ! = 30° in your analysis. Use the following parameters in your analysis: v0 = 100f t/sec and θ = 30◦ . Chapter I: Particle Kinematics Homework I11 Problem I.9
Given: A string is used to pull in particle P in such a way that the radial position of P is given by
r = 0.8 − 0.1t − 0.05t2 while the angular orientation of arm OA is given by θ = 0.4 + 0.12t + 0.06t2 ,
where r, θ and t are given in meters, radians and seconds, respectively.
work Problems ME 274 Find: Determine the velocity and acceleration of P. lem 2/155)
ull in particle P in such a way that the
is given by r = 0.8 ! 0.1t ! 0.05t 2 while
ion of arm OA is given by
06t 3 , where r, ! and t are given in
seconds, respectively. Determine the
ation of P when t = 2 seconds. Use the following parameters in your analysis: t = 2seconds. I12 Chapter I: Particle Kinematics Homework Problem I.10
mework Problems Given: Block A moves to the right with a speed of vA . ME 274 Find: Determine the speed of block B. roblem 2/213) o the right with a speed of 3.6 ft/sec.
eed of block B. Use the following parameters in your analysis: vA = 3.6f t/sec. Chapter I: Particle Kinematics Homework I13 Problem I.11 mework Problems Given: Block B moves downward with a constant speed of vB .
ME 274 Find: Determine the speed of block A as a function of the position y of block A. roblem 2/220)
nward with a speed of vB . Determine
k A as a function of the position y of I14 Chapter I: Particle Kinematics Homework Problem I.12
Given: A skier is moving down a slope. Along the straight slope from A to B, her speed is changing
Particle Kinematics Homework Problems
at a constant rate of vAB . From B to C, the slope is curved with a radius of curvature of ρ. At
˙
point C, her speed is changing at a rate of vC . The skiers speed at points A, B and C are vA , vB
˙
Problem I/12
and vC , respectively. ME 274 A skier is moving down a slope. Along the straight slope from A to B, her speed is
increasing at a constant rate of 2.4 m/s2. From B to C, the slope is curved with a radius of
Find: For this problem:
curvature of ! =150 m. At point C, her speed is decreasing at a rate of 2 m/s2. The skier’s
peed at points A B and vectors m/s, 25 A, and 30 m/s, respectively.
• Sketch the sunit tangent and, normal C are 15at points m/s B and C.
a) Sketch the unit tangent and normal vectors at points A, B and C.
• Calculate the accelerationthe acceleration vectors and C. A, B and C.
b) Sketch vectors at points A, B at points
c) Calculate the acceleration vectors at
• Sketch the acceleration vectors at points A, B and C. points A, B and C. A B ! C
Use the following parameters in your analysis: vAB = 5m/sec2 , vC = −1.5/sec2 , ρ = 175m,
˙
˙
vA = 20m/sec, vB = 30m/sec and vC = 40m/sec Particle Kinematics Homework Problems ME 27 Problem I/13
I15
M
PartParticle Kinematics Homework Problems
I
A roller coaster car travels along a short part of the track given by the equation
Problem I.13
Problem m a
r = 50 cos(2! ) I/13nd ! = 0.02t 2 radians. What is the velocity and acceleration of the car
P! =
when art I20°?
PART I A roller coaster car travelscoaster a short part of theshort part of the track given byrthe equation
A roller along car travels along a track given by the equation =
50cos (2θ), where r is in meters and θ is in radians. What 2 the velocity and acceleration of the
r = 50 cos(2! ) m and ! = 0.02t is radians. What is the velocity and acceleration of the
car when θ = 10◦ ?
when ! = 20°?
Chapter I: Particle Kinematics Homework ! r
! r Part II
Along a circular loop at the bottom of the track, the cars will move at a speed of vo with a
!
rate of v . To ensure the safety of the passengers, the magnitude of the acceleration
PART II Along a circular loop at the bottom of the track, the cars will move at a speed of v0 that
Part II
s.hould not exceed 4g. What is the minimum magnituderadius, acceleration of the track?
is changing at a rate of v To ensure the safety of the passengers, the allowable of the R, of this part
˙
Along a circular loop at the bottom of the track, the cars will move at a speed of vo w
should not exceed 4g. What is the minimum allowable radius, R, of this part of the track?
!
Use ratefollowing ensure the safety of analysis: vo = 20 m/s,agnitudem/sthe acceleration
the of v . To parameters in your the passengers, the m v = 15 of 2.
!
should not exceed 4g. What is the minimum allowable radius, R, of this part of the tr !
R
Use the following parameters in your analysis: vo = 20 m/s, v = 15 m/s2. R Use the following parameters in your analysis: v0 = 40m/sec and v = 10m/sec2
˙ I16 Chapter I: Particle Kinematics Homework
Particle Kinematics Homework Problems ME 2 Problem I.14 Problem I/14
Two cars are travelling as shown. Car A is traveling along a circular path of radius, r,
Given: Two cars are travelling as shown. Car A is traveling along a circular path of radius, r,
with a constant speed, v A . Car B is traveling with a speed, vB , in the positive x axis. Th
with a constant speed, vA . Car B is traveling with a speed, vB , in the positive x axis. The speed
!
of B is changing at a ratespeed of B is changing at a rate of vB . At this instant, find:
of vB .
˙
a) The velocity of A as seen by car B.
Find: At this instant:
b) The acceleration of A as seen by car B.
• Determine the velocity of A as seen by car B. Use the following parameters in your analysis: r = 100 m, v A = 15 m/s, vB = 20 m/s,
• Determine the acceleration m/s2. as seen by car B.
!
vB = 2 of A B y A
r x Use the following parameters in your analysis: r = 150m, vA = 20m/sec, vB = 30m/sec and
vB = 4m/sec2
˙ Chapter I: Particle Kinematics Homework I17 Particle Kinematics Homework Problems Problem I.15 ME 274 Problem I/15
A worker is lifting a large crate, P, using a pulley shown in the diagram. He is holding
Given: A worker tis lifting at point crate, P, using a pulley shownat athe diagram. He v .holdingis the speed
he rope a large A and is walking to the right in constant speed is What the
A
rope at point A and is walking to the right at a constant speed vA .
of the crate when the worker has walked a distance x?
Find: What is the speed of the crate when the worker has walked a distance x? Use the following parameters in your analysis: v A = 1.5 m/s, x = 2 m, h = 5 m. O h
P’ P A A’ x
Use the following parameters in your analysis: vA = 2m/sec, x = 3m and h = 8m I18 Chapter I: Particle Kinematics Homework Particle Kinematics Homework Problems ME 274 Problem I/16
Problem I.16
Particle P travels on a circular path having a radius of 2.5 feet with its center at point A, Given: Particle P travels on a circularseen by an observer at point with its center at point A, as
as shown in Figure (a). This particle is path having a radius of L O that is 2 feet
shown in Figure (a). This particle is seen by an observer at point O that is a distance of d directly
directly to the left of point A. At the position shown in Figure (b) (! = 0), P is directly
to the left of point A. At the position shown in Figure (b) (θ = 0), P is directly below point A, has
below point A, has a speed of 10 ft/sec and has an acceleration that points directly toward
a speed of vP and has an acceleration that points directly toward point A. point A. Find: In Figure (b) (b) shown below:
i) For position below, make a sketch of the path unit vectors for P ( et and en ) and the polar unit of the path unit ! .
• Make a sketch vectors e R and evectors for P (t and n ) and the polar unit vectors R and
e
e
e
φ .
e ii) For the position in Figure (b), find numerical values for the rate of change of speed
• Determinefor R, !, R values .for the rate of change of speed of P.
!!!
!
of P and numerical! and !!
¨
˙¨˙
• Determine numerical values for R, R, φ and φ.
d O O
A "
R A
R L aP !
P P
(a) vP
(b) Use the following parameters in your analysis: L = 2.5f t, d = 2f t and vP = 10f t/sec Particle Kinematics Homework Problems ME 274 Chapter I: Particle Kinematics Homework I19 Problem I/17
Given: An inextensible cable is attached to block B, is wrapped around a pulleys C
Problem I.17
and D and is attached to a fixed support at E. Block B is constrained to move
on a vertical guide, and block A is constrained to slide along a horizontal
Given: An inextensible cable is attached to block B, is wrapped around a pulleys C and D and
surface. When sB = 4 ft , block B is moving downward with a speed of 15
is attached to a ﬁxed support at E. Block B is constrained to move on a vertical guide, and block
ft/sec
A is constrained .to slide along a horizontal surface. Block B is moving downward with a constant
speed of vB . Find: Determine the speed of block A at this instant. Find: Determine the speed of block A. sA
D d
C A sB
E
B vB Use the following parameters in your analysis: d = 3f t, sB = 4f t and vB = 15f t/sec I20 Chapter I: Particle Kinematics Homework Problem I.18
Given: Particle P travels within the xy plane along a path given by y (x) = x2 /2 − 10x, where x
and y are given in feet. The ycomponent of the position for P is changing at a constant rate of y .
˙
Find: For this problem:
• determine the velocity vector of P.
• determine the acceleration vector of P.
• determine the rate of change of speed of P.
Use the following parameters in your analysis: y = 10f t/sec and x = 9f t.
˙ Chapter I: Particle Kinematics Homework I21 Problem I.19
Given: A projectile is launched from ground with a speed of v0 at a angle of θ with respect to the
horizontal. After ﬁring, it can be assumed that only the gravitational force acts on the projectile;
i.e., the acceleration of the particle is straight down.
Particle Kinematics Homework Problems ME 274 Find: Considering the motion of the projectile at a time t: a) determine the radius of curvature
Problem I/19
of its path, and b) determine the rate of change of speed of the projectile. g v0
! Use the following parameters in your analysis: v0 = 450m/sec, θ = 53.13◦ and t = 25sec. I22 Chapter I: Particle Kinematics Homework Problem I.20
Given: At the instant shown, the Cartesian components for the position of particle P are known
as (xP , yP ). In addition, the Cartesian form for the velocity and acceleration of P are known as P
v
and P , respectively.
a
Particle Kinematics Homework Problems ME 274 Find: For this instant: a) write down the polar unit vectors eR and eφ in terms of their Cartesian
ˆ
ˆ
Problem I/20
¨
˙˙¨
components, and b) determine numerical values for R, R, φ, R and φ . ˆ
eR
P yP
! ˆ
e! R xP
Use the following parameters in your analysis: xP = 15m, yP = 20m, P = −10ˆ + 20ˆ m/sec
v
i
j
and P = −8ˆ m/sec2 .
a
j Chapter I: Particle Kinematics Homework I23 Problem I.21
Given: At the instant shown, the polar components for the velocity and acceleration of particle P
are known as P and P , respectively.
v
a
Particle Kinematics Homework Problems ME 274 Find: For this instant, determine the speed, rate of change of speed and radius of curvature for
the path of P. I/21
Problem ˆ
eR ˆ
e!
P R
! Use the following parameters in your analysis: P = (30ˆR − 40ˆθ ) f t/sec and P = (−4ˆθ ) f t/sec2 .
v
e
e
a
e ...
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 Fall '09
 Acceleration, Velocity, Particle Kinematics

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