Kinematics
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Particle Kinematics Homework Problems
ME 274 – Spring 2009
Problem 2/I Particle P moves within the x-y plane on a path described by y ( x ) = 2 4x 2 + 12 , where both x and y are given in feet. Determine the acceleration of P. Write your answer as a vector in terms of its x-y components. y
path of P
P x
Use the following parameters in your analysis: x = 1 ft , x! = 0 and !! x = !4 ft / s 2 .
Particle Kinematics Homework Problems
ME 274 – Spring 2009
Problem 2/II 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! . a) Determine the acceleration of A. Write this as a vector in terms of its x-y components. b) Make a sketch of the acceleration vector for A. 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: R = 400 ft , ! = 36.87° , v = 80 ft / s and v! = !12 ft / s 2 (braking).
Particle Kinematics Homework Problems
ME 274 – Spring 2009
Problem 2/III A particle P travels in the x-y plane with a path whose coordinates are given as a function of time t as:
x ( t ) = 16 ! 12t
y ( t ) = 2 + 15t ! 3t 2 where x and y are given in meters and t is given in seconds. a) Determine the velocity and acceleration of P in terms of their x-y components. b) Make a sketch of the velocity and acceleration vectors for P. c) Determine the rate of change of speed of P and the radius of curvature for the path of P. Use t = 2 seconds in your analysis.
Particle Kinematics Homework Problems
ME 274 – Spring 2009
Problem 2/IV A particle P moves within a plane with a path given in terms of the polar coordinates of:
R = 2! 2 where R and ! are as defined below in the figure and with R and ! given in meters and in radians, respectively. a) Determine the velocity and acceleration vectors of P in terms of their polar coordinates.
eR
b) Make a sketch of the velocity and acceleration vectors for P.
P
c) Determine the rate of change of speed of P. d) Is speed of P increasing or decreasing at this instant? Explain.
e!
!
R
O
Use the following parameters in your analysis: ! = 2 radians , !! = "0.5 rad / sec and !!! = 0 .
Particle Kinematics Homework Problems
ME 274 – Spring 2009
Problem 2/V 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 ! 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 = 80 ft / sec , vP = 20 ft / sec and ! = 30° .
Particle Kinematics Homework Problems
ME 274 – Spring 2009
Problem 2/VI Blocks A and B are connected by the pulley-system 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. 1.5 m
a) Determine the speed of block B when s A = 0 . b) Determine the speed of block B when s A = 1.2 m .
O sA
sB
A vA
Use vA = 10 m / sec in your analysis.
B
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-I The rectangular plate ABCD freely rotates about a pivot at point A. If, at the instant shown, the acceleration of point C is given by a C & %3i $ 7.5 j m / s 2 , determine the angular velocity!" and angular acceleration # of the plate.
aC B
C 0.25 m
A
D 0.5 m
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-II The pulley shown below freely rotates about point C and interacts with two rubber belts (one horizontal, one vertical). If the velocity of point A on the drive belt is measured to be v A ! 1.0i m / s , the acceleration of point B on the load belt is measured to be
a B ! "0.2 j m / s 2 , and the belts do not slip, find the velocity and acceleration of point D.
A C D
0.1 m
0.025 m B
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-III The car lift shown below is controlled by a horizontally-mounted hydraulic cylinder attached to point A. If the cylinder extends at a constant speed v, find the vertical component of the platform’s velocity for an arbitrary value of !. Assume the upper platform remains horizontal and all joints are pinned.
L
L
L B
L
!
!
A
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-IV The compound wheel assembly shown below is driven by a cable attached to the outer rim of the assembly at point A. Assuming that the wheel rolls without slip about point B, ! = 45", and the acceleration at point A is given by a A $ 6i # 4 j m / s 2 , determine the acceleration of point C.
A
! C B
0.1 m
0.025 m
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-V The wheel shown below rolls without slip on cart A. Assuming that the cart has velocity v A ! 2i m / s and acceleration a A ! 6i m / s 2 , and the center of the wheel has velocity v C ! "0.5i m / s and acceleration a C ! 8i m / s 2 , determine the velocity and acceleration of point B.
0.1 m B
C A
vA aA
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-VI If, at the instant shown, ! = 30" and the piston has velocity v A $ #2.0i m / s , determine the resulting angular velocity%&CB of the crankshaft.
B 0.06 m C
v
! A
0.03 m
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5/VII Consider the four mechanisms shown below. Respond to the questions posed. Feel free to draw directly on these sheets and submit for your homework solution. Use a straight edge when making your drawings. Mechanism #1 Link OA is rotating in the clockwise sense. Determine the location of the instant center (IC) of link AB. From the location of this IC, determine the sense of rotation for links AB and BC. Justify your answers in words.
Mechanism #2 Determine the location of the instant center (IC) of link AD. From the location of this IC, determine the sense of rotation for links AD and DB. Justify your answers in words.
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Mechanism #3 Link OA is rotating in the clockwise sense. Determine the location of the instant center (IC) of link AB. From the location of this IC, determine the sense of rotation for link AB and the sense of translation for the piston. Justify your answers in words.
Mechanism #4 Racks A and B are moving in the directions shown with the speeds provided. Assume that the gear does not slip on the racks. Determine the location of the instant center (IC) of the gear. From the location of this IC, determine the sense of rotation for the gear. Justify your answers with numbers and words.
vA = 0.8 m/s
vB = 0.6 m/s
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-VIII The mechanism below is driven by a motor at point A which produces an angular velocity of " AB ! 6 rad / s in the direction shown. Using the method of instantaneous centers, find the velocity of point C. Note that at this instant # = 60$, % = 45$, and BC is horizontal.
" AB
B
C
0.1 m A
0.2 m
#
%
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-IX The sliding collar moves horizontally along the shaft producing an oscillation in AB. If, at the instant shown, AB and BC are horizontal and vertical, respectively, and the velocity of C is not changing, find the angular acceleration of AB. Note that v C ! 2i m / s .
A 0.1 m B
0.3 m
C
vC
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-X The mechanism below is driven by a motor at point A, which produces an angular velocity of " AB ! 6 rad / s and angular acceleration of & AB ! %2 rad / s 2 in the direction shown. Provided BCE and CD are horizontal and vertical, respectively, and # = 45$ at the instant shown, find the velocity and acceleration of point E.
0.3 m
E
C
B
" AB & AB D
A 0.1 m
0.5 m '#
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5/XI A particle P moves within a straight slot cut into a rotating disk with the x-position of P increasing at a constant rate of x! . The disk is rotating with counter-clockwise sense at a ! . The xyz coordinate system is attached to speed of ! , with ! changing at a rate of ! the disk with its origin at the center O of the disk. y a) Determine the velocity and acceleration of particle P. Express your answers as vectors in terms of their x-y components.
! x
b) Make a sketch of the velocity and acceleration vectors found in a) above.
P
h
x O
Use the following parameters in your analysis: x = 0.5 ft , h = 0.75 ft , x! = 10 ft / s , ! = 4 rad / s 2 . ! = 20 rad / s and !
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5/XII A telescoping arm OP is pinned to a cart at end O. The cart moves along a horizontal surface with a constant speed of vO . The angle of the arm, !, is increasing at a constant rate of !! and is extending at a rate of L! . The xyz coordinate system is attached to the telescoping arm with its origin at end O of the arm. a) Determine the velocity and acceleration of particle P. Express your answers as vectors in terms of their x-y components. b) Make a sketch of the velocity and acceleration vectors found in a) above.
P L y O
! vO
Use the following parameters in your analysis: ! = 90° , vO = 20 ft / s , !! = 2 rad / s , L = 4 ft , L! = 2 ft / s and L!! = 3 ft / s 2 .
x
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5/XIII An L-shaped arm OBE is pinned to ground at O (section OB is perpendicular to section BE of OBE). Arm OBE is known to be rotating in the counterclockwise sense with a constant rotational speed of !. Rigid link AP is pinned to ground at A, and with P able to slide within a slot that is cut into section BE of arm OBE, as shown. The xyz coordinate system is attached to arm OBE with its origin at pin O. For the position shown, section OB is aligned with the direction of OA, and P is at a position of yP within the slot. For this position: E a) Determine the angular velocity of link AP. Write P y your answer as a vector. b) Determine the angular acceleration of link AP. Write your answer as a vector.
L
yP
O
B
!
x
b Use the following parameters in your analysis: ! = 6 rad / s , yP = 3 ft , L = 5 ft and b = 4 ft .
A
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5/XIV Arm OB is pinned to ground at end O. Rigid link AP is pinned to ground at A, with end P able to slide within a straight slot that is cut into arm OB. Arm AP is rotating in the clockwise sense with a constant rate of !. The xyz coordinate system is attached to arm OB with its origin at pin O. For the position shown, link OB is perpendicular to the direction of OA, with arm AP at an angle of " measured from OB. For this position: a) Determine the angular velocity of arm OB. Write your answer as a vector.
A
!
b) Determine the angular acceleration of arm OB. Write your answer as a vector. c) Determine the values of x! P and !! xP .
L y
"
P
O xP
Use the following parameters in your analysis: ! = 3 rad / s , L = 0.5 m and ! = 53.13° .
B
x
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 7/I A disk and shaft OC are mounted in a clevis that rotates about a fixed vertical axis at a rate of !. The shaft and disk rotate with respect to the clevis with a rate of "0 in the direction shown below, with the angle # held constant. The XYZ coordinate system is fixed with the Z-axis aligned with the fixed rotation axis of the clevis. The xyz coordinate system is attached to the disk with the y-axis aligned with OC for all time. For the position shown below, the x- and X-axes are aligned. For the position shown:
z
a) Determine the angular velocity of the disk. Write your answer as a vector in terms of its xyz components.
Z y
b) Determine the angular acceleration of the disk. Write your answer as a vector in terms of its xyz components.
O
C x Y
X
Use the following parameters in your analysis: ! = 36.87° , ! = 6 rad / s = constant and ! 0 = 2 rad / s = constant .
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 7/II A disk and shaft OC are mounted in a clevis that is attached to a fixed vertical axis ( ! = 0 ). The shaft and disk rotate with respect to the clevis with a rate of !0 in the direction shown below, with the angle " increasing at a constant rate of !! . The XYZ coordinate system is fixed with the Z-axis aligned with the fixed vertical direction. The xyz coordinate system is attached to the disk with the y-axis aligned with OC for all time. For the position shown below, the x- and X-axes are aligned. For the position shown: z
a) Determine the angular velocity of the disk. Write your answer as a vector in terms of its xyz components.
Z y
b) Determine the angular acceleration of the disk. Write your answer as a vector in terms of its xyz components.
O
C x Y
X
Use the following parameters in your analysis: ! = 36.87° , !! = 5 rad / s and ! 0 = 2 rad / s = constant .
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 7/III Arm OB rotates about a fixed axis with a constant rate of !. A disk rotates about its central axis BC with a constant rate of "0 relative to arm OB. The XYZ coordinate system is fixed with the Y-axis aligned with the fixed rotation axis of OB. The xyz coordinate system is attached to the disk with the z-axis aligned with BC for all time. For the position shown below, the xyz axes are aligned with the XYZ axes. For the position shown: a) Determine the angular velocity of the disk. Write your answer as a vector in terms of its xyz components. b) Determine the angular acceleration of the disk. Write your answer as a vector in terms of its xyz components.
"0
d R
h
!
Use the following parameters in your analysis: ! = 2 rad / s and ! 0 = 6 rad / s .
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 7/IV Yoke A rotates about a fixed axis with a constant rate of !1. A disk rotates about its center C with a constant rate of p relative to yoke A. The XYZ coordinate system is fixed with the X-axis aligned with the fixed rotation axis of yoke A. The xyz coordinate system is attached to the disk with the z-axis aligned with the rotation axis of the disk for all time. For the position shown below, the xyz axes are aligned with the XYZ axes. For the position shown: a) Determine the angular velocity and angular acceleration of the disk. Write your answers as vectors in terms of their xyz components. b) Determine the acceleration of point B of the disk. Write your answer as a vector in terms of its xyz components.
d
C R B
b O
Use the following parameters in your analysis: !1 = 4.5 rad / s , p = 2.5 rad / s , d = 6 in , b = 8 in and R = 3 in .
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 7/V Shaft AB rotates about a fixed axis with a constant rotational speed of !. A tube is hinged on shaft AB with the angle " between the tube and shaft increasing at a constant rate of !! . Particle P moves outward within the tube at a constant rate of x! P . The XYZ coordinate system is fixed with the X-axis aligned with the fixed rotation axis of the shaft AB. The xyz coordinate system is attached to the tube with the x-axis aligned with the tube for all time. For the position shown below, the z- and Z-axes are aligned. For the position shown: a) Determine the angular velocity and angular acceleration of the tube. Write your answers as vectors in terms of their xyz components. b) Determine the acceleration of particle P. Write your answer as a vector in terms of its xyz components. x Y y
xP
"
P X
O A
! B
Use the following parameters in your analysis: ! = 10 rad / s , ! = 90° , !! = 3 rad / s , xP = 0.4 m and x! P = 15 m / sec .
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 7/VI The collar and clevis A are given a constant downward velocity of z! < 0 . This motion causes end B of bar AB to slide outward within a radial slot of the rotating disk. The disk has a constant rotational speed of ! about a fixed axis passing through the center of the disk. The XYZ coordinate system is fixed with the Z-axis aligned with the fixed rotation axis of the disk. The xyz coordinate system is attached to bar AB. For the position shown below, the xyz axes are aligned with the XYZ axes. z, Z
For the position shown: a) Determine the angular velocity of bar AB. Write your answer as a vector in terms of its xyz components. b) Determine the angular acceleration of bar AB. Write your answer as a vector in terms of its xyz components.
L
y, Y x, X
!
Use the following parameters in your analysis: ! = 7 rad / s , L = 10 in , z = 6 in and z! = ! 24 in / s .
ME 274 – Spring 2009
Problem 2/I Particle P moves within the x-y plane on a path described by y ( x ) = 2 4x 2 + 12 , where both x and y are given in feet. Determine the acceleration of P. Write your answer as a vector in terms of its x-y components. y
path of P
P x
Use the following parameters in your analysis: x = 1 ft , x! = 0 and !! x = !4 ft / s 2 .
Particle Kinematics Homework Problems
ME 274 – Spring 2009
Problem 2/II 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! . a) Determine the acceleration of A. Write this as a vector in terms of its x-y components. b) Make a sketch of the acceleration vector for A. 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: R = 400 ft , ! = 36.87° , v = 80 ft / s and v! = !12 ft / s 2 (braking).
Particle Kinematics Homework Problems
ME 274 – Spring 2009
Problem 2/III A particle P travels in the x-y plane with a path whose coordinates are given as a function of time t as:
x ( t ) = 16 ! 12t
y ( t ) = 2 + 15t ! 3t 2 where x and y are given in meters and t is given in seconds. a) Determine the velocity and acceleration of P in terms of their x-y components. b) Make a sketch of the velocity and acceleration vectors for P. c) Determine the rate of change of speed of P and the radius of curvature for the path of P. Use t = 2 seconds in your analysis.
Particle Kinematics Homework Problems
ME 274 – Spring 2009
Problem 2/IV A particle P moves within a plane with a path given in terms of the polar coordinates of:
R = 2! 2 where R and ! are as defined below in the figure and with R and ! given in meters and in radians, respectively. a) Determine the velocity and acceleration vectors of P in terms of their polar coordinates.
eR
b) Make a sketch of the velocity and acceleration vectors for P.
P
c) Determine the rate of change of speed of P. d) Is speed of P increasing or decreasing at this instant? Explain.
e!
!
R
O
Use the following parameters in your analysis: ! = 2 radians , !! = "0.5 rad / sec and !!! = 0 .
Particle Kinematics Homework Problems
ME 274 – Spring 2009
Problem 2/V 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 ! 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 = 80 ft / sec , vP = 20 ft / sec and ! = 30° .
Particle Kinematics Homework Problems
ME 274 – Spring 2009
Problem 2/VI Blocks A and B are connected by the pulley-system 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. 1.5 m
a) Determine the speed of block B when s A = 0 . b) Determine the speed of block B when s A = 1.2 m .
O sA
sB
A vA
Use vA = 10 m / sec in your analysis.
B
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-I The rectangular plate ABCD freely rotates about a pivot at point A. If, at the instant shown, the acceleration of point C is given by a C & %3i $ 7.5 j m / s 2 , determine the angular velocity!" and angular acceleration # of the plate.
aC B
C 0.25 m
A
D 0.5 m
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-II The pulley shown below freely rotates about point C and interacts with two rubber belts (one horizontal, one vertical). If the velocity of point A on the drive belt is measured to be v A ! 1.0i m / s , the acceleration of point B on the load belt is measured to be
a B ! "0.2 j m / s 2 , and the belts do not slip, find the velocity and acceleration of point D.
A C D
0.1 m
0.025 m B
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-III The car lift shown below is controlled by a horizontally-mounted hydraulic cylinder attached to point A. If the cylinder extends at a constant speed v, find the vertical component of the platform’s velocity for an arbitrary value of !. Assume the upper platform remains horizontal and all joints are pinned.
L
L
L B
L
!
!
A
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-IV The compound wheel assembly shown below is driven by a cable attached to the outer rim of the assembly at point A. Assuming that the wheel rolls without slip about point B, ! = 45", and the acceleration at point A is given by a A $ 6i # 4 j m / s 2 , determine the acceleration of point C.
A
! C B
0.1 m
0.025 m
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-V The wheel shown below rolls without slip on cart A. Assuming that the cart has velocity v A ! 2i m / s and acceleration a A ! 6i m / s 2 , and the center of the wheel has velocity v C ! "0.5i m / s and acceleration a C ! 8i m / s 2 , determine the velocity and acceleration of point B.
0.1 m B
C A
vA aA
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-VI If, at the instant shown, ! = 30" and the piston has velocity v A $ #2.0i m / s , determine the resulting angular velocity%&CB of the crankshaft.
B 0.06 m C
v
! A
0.03 m
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5/VII Consider the four mechanisms shown below. Respond to the questions posed. Feel free to draw directly on these sheets and submit for your homework solution. Use a straight edge when making your drawings. Mechanism #1 Link OA is rotating in the clockwise sense. Determine the location of the instant center (IC) of link AB. From the location of this IC, determine the sense of rotation for links AB and BC. Justify your answers in words.
Mechanism #2 Determine the location of the instant center (IC) of link AD. From the location of this IC, determine the sense of rotation for links AD and DB. Justify your answers in words.
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Mechanism #3 Link OA is rotating in the clockwise sense. Determine the location of the instant center (IC) of link AB. From the location of this IC, determine the sense of rotation for link AB and the sense of translation for the piston. Justify your answers in words.
Mechanism #4 Racks A and B are moving in the directions shown with the speeds provided. Assume that the gear does not slip on the racks. Determine the location of the instant center (IC) of the gear. From the location of this IC, determine the sense of rotation for the gear. Justify your answers with numbers and words.
vA = 0.8 m/s
vB = 0.6 m/s
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-VIII The mechanism below is driven by a motor at point A which produces an angular velocity of " AB ! 6 rad / s in the direction shown. Using the method of instantaneous centers, find the velocity of point C. Note that at this instant # = 60$, % = 45$, and BC is horizontal.
" AB
B
C
0.1 m A
0.2 m
#
%
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-IX The sliding collar moves horizontally along the shaft producing an oscillation in AB. If, at the instant shown, AB and BC are horizontal and vertical, respectively, and the velocity of C is not changing, find the angular acceleration of AB. Note that v C ! 2i m / s .
A 0.1 m B
0.3 m
C
vC
Planar Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5-X The mechanism below is driven by a motor at point A, which produces an angular velocity of " AB ! 6 rad / s and angular acceleration of & AB ! %2 rad / s 2 in the direction shown. Provided BCE and CD are horizontal and vertical, respectively, and # = 45$ at the instant shown, find the velocity and acceleration of point E.
0.3 m
E
C
B
" AB & AB D
A 0.1 m
0.5 m '#
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5/XI A particle P moves within a straight slot cut into a rotating disk with the x-position of P increasing at a constant rate of x! . The disk is rotating with counter-clockwise sense at a ! . The xyz coordinate system is attached to speed of ! , with ! changing at a rate of ! the disk with its origin at the center O of the disk. y a) Determine the velocity and acceleration of particle P. Express your answers as vectors in terms of their x-y components.
! x
b) Make a sketch of the velocity and acceleration vectors found in a) above.
P
h
x O
Use the following parameters in your analysis: x = 0.5 ft , h = 0.75 ft , x! = 10 ft / s , ! = 4 rad / s 2 . ! = 20 rad / s and !
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5/XII A telescoping arm OP is pinned to a cart at end O. The cart moves along a horizontal surface with a constant speed of vO . The angle of the arm, !, is increasing at a constant rate of !! and is extending at a rate of L! . The xyz coordinate system is attached to the telescoping arm with its origin at end O of the arm. a) Determine the velocity and acceleration of particle P. Express your answers as vectors in terms of their x-y components. b) Make a sketch of the velocity and acceleration vectors found in a) above.
P L y O
! vO
Use the following parameters in your analysis: ! = 90° , vO = 20 ft / s , !! = 2 rad / s , L = 4 ft , L! = 2 ft / s and L!! = 3 ft / s 2 .
x
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5/XIII An L-shaped arm OBE is pinned to ground at O (section OB is perpendicular to section BE of OBE). Arm OBE is known to be rotating in the counterclockwise sense with a constant rotational speed of !. Rigid link AP is pinned to ground at A, and with P able to slide within a slot that is cut into section BE of arm OBE, as shown. The xyz coordinate system is attached to arm OBE with its origin at pin O. For the position shown, section OB is aligned with the direction of OA, and P is at a position of yP within the slot. For this position: E a) Determine the angular velocity of link AP. Write P y your answer as a vector. b) Determine the angular acceleration of link AP. Write your answer as a vector.
L
yP
O
B
!
x
b Use the following parameters in your analysis: ! = 6 rad / s , yP = 3 ft , L = 5 ft and b = 4 ft .
A
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 5/XIV Arm OB is pinned to ground at end O. Rigid link AP is pinned to ground at A, with end P able to slide within a straight slot that is cut into arm OB. Arm AP is rotating in the clockwise sense with a constant rate of !. The xyz coordinate system is attached to arm OB with its origin at pin O. For the position shown, link OB is perpendicular to the direction of OA, with arm AP at an angle of " measured from OB. For this position: a) Determine the angular velocity of arm OB. Write your answer as a vector.
A
!
b) Determine the angular acceleration of arm OB. Write your answer as a vector. c) Determine the values of x! P and !! xP .
L y
"
P
O xP
Use the following parameters in your analysis: ! = 3 rad / s , L = 0.5 m and ! = 53.13° .
B
x
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 7/I A disk and shaft OC are mounted in a clevis that rotates about a fixed vertical axis at a rate of !. The shaft and disk rotate with respect to the clevis with a rate of "0 in the direction shown below, with the angle # held constant. The XYZ coordinate system is fixed with the Z-axis aligned with the fixed rotation axis of the clevis. The xyz coordinate system is attached to the disk with the y-axis aligned with OC for all time. For the position shown below, the x- and X-axes are aligned. For the position shown:
z
a) Determine the angular velocity of the disk. Write your answer as a vector in terms of its xyz components.
Z y
b) Determine the angular acceleration of the disk. Write your answer as a vector in terms of its xyz components.
O
C x Y
X
Use the following parameters in your analysis: ! = 36.87° , ! = 6 rad / s = constant and ! 0 = 2 rad / s = constant .
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 7/II A disk and shaft OC are mounted in a clevis that is attached to a fixed vertical axis ( ! = 0 ). The shaft and disk rotate with respect to the clevis with a rate of !0 in the direction shown below, with the angle " increasing at a constant rate of !! . The XYZ coordinate system is fixed with the Z-axis aligned with the fixed vertical direction. The xyz coordinate system is attached to the disk with the y-axis aligned with OC for all time. For the position shown below, the x- and X-axes are aligned. For the position shown: z
a) Determine the angular velocity of the disk. Write your answer as a vector in terms of its xyz components.
Z y
b) Determine the angular acceleration of the disk. Write your answer as a vector in terms of its xyz components.
O
C x Y
X
Use the following parameters in your analysis: ! = 36.87° , !! = 5 rad / s and ! 0 = 2 rad / s = constant .
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 7/III Arm OB rotates about a fixed axis with a constant rate of !. A disk rotates about its central axis BC with a constant rate of "0 relative to arm OB. The XYZ coordinate system is fixed with the Y-axis aligned with the fixed rotation axis of OB. The xyz coordinate system is attached to the disk with the z-axis aligned with BC for all time. For the position shown below, the xyz axes are aligned with the XYZ axes. For the position shown: a) Determine the angular velocity of the disk. Write your answer as a vector in terms of its xyz components. b) Determine the angular acceleration of the disk. Write your answer as a vector in terms of its xyz components.
"0
d R
h
!
Use the following parameters in your analysis: ! = 2 rad / s and ! 0 = 6 rad / s .
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 7/IV Yoke A rotates about a fixed axis with a constant rate of !1. A disk rotates about its center C with a constant rate of p relative to yoke A. The XYZ coordinate system is fixed with the X-axis aligned with the fixed rotation axis of yoke A. The xyz coordinate system is attached to the disk with the z-axis aligned with the rotation axis of the disk for all time. For the position shown below, the xyz axes are aligned with the XYZ axes. For the position shown: a) Determine the angular velocity and angular acceleration of the disk. Write your answers as vectors in terms of their xyz components. b) Determine the acceleration of point B of the disk. Write your answer as a vector in terms of its xyz components.
d
C R B
b O
Use the following parameters in your analysis: !1 = 4.5 rad / s , p = 2.5 rad / s , d = 6 in , b = 8 in and R = 3 in .
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 7/V Shaft AB rotates about a fixed axis with a constant rotational speed of !. A tube is hinged on shaft AB with the angle " between the tube and shaft increasing at a constant rate of !! . Particle P moves outward within the tube at a constant rate of x! P . The XYZ coordinate system is fixed with the X-axis aligned with the fixed rotation axis of the shaft AB. The xyz coordinate system is attached to the tube with the x-axis aligned with the tube for all time. For the position shown below, the z- and Z-axes are aligned. For the position shown: a) Determine the angular velocity and angular acceleration of the tube. Write your answers as vectors in terms of their xyz components. b) Determine the acceleration of particle P. Write your answer as a vector in terms of its xyz components. x Y y
xP
"
P X
O A
! B
Use the following parameters in your analysis: ! = 10 rad / s , ! = 90° , !! = 3 rad / s , xP = 0.4 m and x! P = 15 m / sec .
Moving Reference Frame Kinematics Homework Problems
ME 274 – Spring 2009
Problem 7/VI The collar and clevis A are given a constant downward velocity of z! < 0 . This motion causes end B of bar AB to slide outward within a radial slot of the rotating disk. The disk has a constant rotational speed of ! about a fixed axis passing through the center of the disk. The XYZ coordinate system is fixed with the Z-axis aligned with the fixed rotation axis of the disk. The xyz coordinate system is attached to bar AB. For the position shown below, the xyz axes are aligned with the XYZ axes. z, Z
For the position shown: a) Determine the angular velocity of bar AB. Write your answer as a vector in terms of its xyz components. b) Determine the angular acceleration of bar AB. Write your answer as a vector in terms of its xyz components.
L
y, Y x, X
!
Use the following parameters in your analysis: ! = 7 rad / s , L = 10 in , z = 6 in and z! = ! 24 in / s .
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