Kinematics

Particle Kinematics Homework Problems ME 274 – Spring 2009

x

y path of P

P

Problem 2/I

Particle P moves within the x-y plane on a path described by y x( ) = 2 4x2+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.

Use the following parameters in your analysis: x = 1 ft , !x = 0 and !!x = !4 ft / s

2 .


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.

Use the following parameters in your analysis: R = 400 ft , ! = 36.87° , v = 80 ft / s and

!v = !12 ft / s

2 (braking).

x

y

A

v

O

R

!


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 ! 3t2

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.


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.

b) Make a sketch of the velocity and acceleration vectors

for P.

c) Determine the rate of change of speed of P.

d) Is speed of P increasing or decreasing at this instant?

Explain.

Use the following parameters in your analysis: ! = 2 radians , !! = "0.5 rad / sec and

!!! = 0 .

R

P

O

!

eR

e!


P

A

vA

vP !

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?

Use the following parameters in your analysis: vA = 80 ft / sec , vP = 20 ft / sec and

! = 30° .


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.

a) Determine the speed of block B

when sA = 0 .

b) Determine the speed of block B

when sA = 1.2 m .

Use vA = 10 m / sec in your analysis.

B

A

sA sB

1.5 m

O

vA

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 , determine the angular velocity!" and angular acceleration # of the plate.

2/5.73 sm jiaC $%&

A

B C

D

Ca

0.5 m

0.25 m

A

B C

D

Ca

0.5 m

0.25 m


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 , the acceleration of point B on the load belt is measured to be

, and the belts do not slip, find the velocity and acceleration of point D. sm /0.1 ivA !

2/2.0 sm jaB "!

0.025 m

0.1 m

A

B

CD

0.025 m

0.1 m

A

B

CD


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.

LB

AL! !

LL

LB

AL! !

LL


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 , determine the acceleration of point C.

2/46 sm jiaA #$

0.025 m

0.1 m

A

C

B

!

0.025 m

0.1 m

A

C

B

!


Problem 5-V The wheel shown below rolls without slip on cart A. Assuming that the cart has velocity

and acceleration , and the center of the wheel has velocity and acceleration , determine the velocity and acceleration

of point B.

sm /2 ivA ! 2/6 sm iaA !sm /5.0 ivC "! 2/8 sm iaC !

0.1 m

A

CB

Aa

Av

0.1 m

A

CB

Aa

Av


Problem 5-VI If, at the instant shown, ! = 30" and the piston has velocity sm /0.2 ivA #$ , determine the resulting angular velocity%&CB of the crankshaft.

0.03 m

A

B

C !

0.06 m

v

0.03 m

A

B

C !

0.06 m

v


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.


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

vB = 0.6 m/s

vA = 0.8 m/s

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.


Problem 5-VIII The mechanism below is driven by a motor at point A which produces an angular velocity of s/rad6 AB !" 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.

0.2 m

C

BAB"

A

0.1 m

#%

0.2 m

C

BAB"

A

0.1 m

#%


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 . sm /2 ivC !

0.3 m

C

BA 0.1 m

Cv

0.3 m

C

BA 0.1 m

Cv


Problem 5-X The mechanism below is driven by a motor at point A, which produces an angular velocity of s/rad6 AB !" and angular acceleration of 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.

2/rad2 s AB %!&

0.3 mC

B

A

0.5 m

'#

E

D

0.1 m

AB"AB&

0.3 mC

B

A

0.5 m

'#

E

D

0.1 m

AB"AB&

0.3 mC

B

A

0.5 m

'#

E

D

0.1 m

AB"AB&

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

speed of ! , with ! changing at a rate of !! . The xyz coordinate system is attached to

the disk with its origin at the center O of the disk.

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.

Use the following parameters in your analysis: x = 0.5 ft , h = 0.75 ft , !x = 10 ft / s ,

! = 20 rad / s and !! = 4 rad / s

2 .

O

P

x

y

h

x

!


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.

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 .

P

L

O !

x

y

vO


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:

a) Determine the angular

velocity of link AP. Write

your answer as a vector.

b) Determine the angular

acceleration of link AP. Write

your answer as a vector.

Use the following parameters in your analysis: ! = 6 rad / s , yP = 3 ft , L = 5 ft and

b = 4 ft .

P

O A

L

x

y

b

!

B

E

yP


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:


velocity of arm OB.

Write your answer as a

vector.


acceleration of arm OB.


vector.

c) Determine the values of

!xP

and !!xP

.

Use the following parameters in your analysis: ! = 3 rad / s , L = 0.5 m and ! = 53.13° .

O

L

!

P

A

B x

y

"

xP


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:


velocity of the disk. Write

your answer as a vector in

terms of its xyz

components.


acceleration of the disk.


vector in terms of its xyz

components.

Use the following parameters in your analysis: ! = 36.87° , ! = 6 rad / s = constant and

!0 = 2 rad / s = constant .

Y

Z

X

y

z

x

O C


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.





terms of its xyz

components.





components.

Use the following parameters in your analysis: ! = 36.87° , !! = 5 rad / s and

!0 = 2 rad / s = constant .

Y

Z

X

y

z

x

O C


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.





terms of its xyz

components.





components.

Use the following parameters in your analysis: ! = 2 rad / s and !0 = 6 rad / s .

h

!

d "0

R


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.


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.

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 .

d

b

R

O

C

B


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

!xP

. 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.


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.

Use the following parameters in your analysis: ! = 10 rad / s , ! = 90° , !! = 3 rad / s ,

xP= 0.4 m and

!xP= 15 m / sec .

xP

O

P

X

y

Y

x

"

!

A B


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.


a) Determine the angular velocity

of bar AB. Write your answer

as a vector in terms of its xyz

components.


acceleration of bar AB. Write


terms of its xyz components.

Use the following parameters in your analysis: ! = 7 rad / s , L = 10 in , z = 6 in and

!z = !24 in / s .

!

L

y, Y

x, X

z, Z

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