# Planar Rigid Body Kinematics Homework

### Transcript of Planar Rigid Body Kinematics Homework

Chapter II: Planar Rigid Body Kinematics Homework II-1

Chapter II

Planar Rigid Body KinematicsHomework

II-2 Chapter II: Planar Rigid Body Kinematics Homework

Chapter II: Planar Rigid Body Kinematics Homework II-3

Homework 2.A

Given: Point A of plate ABCD moves to the right with a speed of vA as the plate rotates with anangular speed of ω.

Find: For the instant shown, determine the velocity of point C on the plate. Write your answeras a vector.

A

B

C

D

Av

y

x

0.5 m

0.5 m

0.5 m

0.5 m

Use the following parameters in your analysis: vA = 1 m/s and ω = 3 rad/s (CCW).

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Homework 2.B

Given: The orientation angle θ for a circular disk of radius R is given by: θ (t) = π3 sinπt (with θ

in radians and t in seconds).

Find: Determine the velocity and acceleration vectors for point A on the outer edge of the diskfor:

(a) t = 0 s

(b) t = 1/3 s

Make sketches of these vectors for the two instants in time noted above.

Planar Kinematics Homework Problems ME 274

Problem II-16 The orientation angle ! for the circular disk of radius R is

given by: ! t( ) = "3sin"t (with ! in radians and t in

seconds). Determine the velocity and acceleration vectors for point A on the outer edge of the disk for:

a) t = 0 b) t = 1 / 3 seconds

Make sketches of these vectors for the above two instants in time. Use R = 3 ft in your analysis.

!

A

OR

Use the following parameter in your analysis: R = 0.5 m.

Chapter II: Planar Rigid Body Kinematics Homework II-5

Homework 2.C

Given: Cart A moves to the right with a speed of vA and an acceleration aA. A disk of radius R isattached to a shaft on the cart at the center O of the disk. The disk is in contact with a horizontalsurface at its top surface; as the cart moves, the disk rolls without slipping on this horizontalsurface.

Find: For this problem:

(a) Determine the angular velocity and angular acceleration of the disk. Write your answers asvectors.

(b) Determine the acceleration of point B on the circumference of the disk and with B beingimmediately to right of O at the instant shown.

no slip

A vA,aA

R

O B

Use the following parameters in your analysis: R = 2 ft, vA = 4 ft/s and aA = 5 ft/s2.

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Homework 2.D

Given: Thin bar AB (having a length of L) moves in a way that ends A and B slide along straight,fixed guides as shown in the figure. End A has a constant speed of vA to the right as the bar moves.

Find: Determine the velocity and acceleration of end B.

Planar Kinematics Homework Problems ME 274

Problem II-17 Thin bar AB (having a length of L) moves in a way that ends A and B slide along straight, fixed guides as shown in the figure. End A has a constant speed of vA to the right as the bar moves. Determine the velocity and acceleration of end B. Use L = 4 meters , vA = 3m / sec , ! = 90° and ! = 36.87° in your analysis.

! A

B

L

vA!

Use the following parameters in your analysis: L = 2 ft, vA = 4 ft/s, θ = 36.87◦ and φ = 45◦.

Chapter II: Planar Rigid Body Kinematics Homework II-7

Homework 2.E

Given: Link OA rotates at a constant rate of ω. At the instant shown, link OA is aligned withthe horizontal line of OC and link AB is vertical.

Find: For this position, determine the angular accelerations of links AB and BC. Write youranswers as vectors.

O

A C

B

d d

d

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Homework 2.F

Given: A mechanism is made up of links AE, BD, and DE. Links AE and BD are pinned toground at pins A and B, respectively, and link DE is pinned to links AE and BD at pins D andE, respectively. At the instant shown, link AE is vertical and link BD is horizontal. Also at thisinstant, link BD is rotating CW at a constant rate of ωBD.

Find: For this position:

(a) Determine the angular velocity of links AE and DE.

(b) Determine the angular acceleration of links AE and DE.

Planar Kinematics Homework Problems ME 274

Problem II-28 Given: A mechanism is made up of links AE, BD, and DE. Links AE and BD are

pinned to ground at pins A and B, respectively, and link DE is pinned to links AE and BD at pins D and E, respectively. At the instant shown, link AE is vertical and link BD is horizontal. Also at this instant, link BD is rotating clockwise at a constant rate of !BD = 6 rad/sec.

Find: For the position shown:

a) determine the angular velocity of links AE and DE. b) determine the angular acceleration of links AE and DE.

B

A

D

E !BD b

c d

a

Use the following parameters in your analysis: a = 5 ft, b = 3 ft, c = 4 ft, d = 5 ft and ωBD = 6rad/s.

Chapter II: Planar Rigid Body Kinematics Homework II-9

Homework 2.G

Consider the four mechanisms shown below. Respond to the questions posed. Feel free to drawdirectly on these sheets and submit for your homework solution. Use a straight edge when makingyour drawings.

MECHANISM NO. 1Link OA is rotating in the clockwise sense. Determine the location of the instant center (IC) oflink AB. From the location of this IC, determine the sense of rotation for links AB and BC. Justifyyour answers in words.

AB

C

O

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MECHANISM NO. 2Determine the location of the instant center (IC) of link AD. From the location of this IC, determinethe sense of rotation for links AD and DB. Justify your answers in words.

A

B

D

CO

v

Chapter II: Planar Rigid Body Kinematics Homework II-11

MECHANISM NO. 3Link OA is rotating in the clockwise sense. Determine the location of the instant center (IC) oflink AB. From the location of this IC, determine the sense of rotation for link AB and the sense oftranslation for the piston. Justify your answers in words.

O

A

B

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MECHANISM NO. 4Racks A and B are moving in the directions shown with the speeds provided. Assume that the geardoes not slip on the racks. Determine the location of the instant center (IC) of the gear. From thelocation of this IC, determine the sense of rotation for the gear. Justify your answers with numbersand words.

O

A

B

0.1 m

0.2 m

vA = 0.3 m/s

vB = 0.1 m/s

Chapter II: Planar Rigid Body Kinematics Homework II-13

Homework 2.H

Given: The mechanism shown below is made up of links AB, BD, DF, EG and GH. Link AB isknown to be rotating clockwise with an angular speed of ωAB.

Find: For this problem:

(a) Locate the instant centers for all links.

(b) Determine the sense of rotation (clockwise or counterclockwise) for all links.

(c) Use the locations of the instant centers found above to approximate the ratio of ωGH/ωAB.Assume that the drawing of the mechanism has been done to scale.

A

B D

E

F

G

H

! AB

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Homework 2.I

Given: A mechanism is made up of links AB and BC and a wheel pinned to BC at the wheel’scenter C. The wheel rolls without slipping on a horizontal surface. Link AB rotates counterclockwisewith a constant rate of ωAB. At the instant shown, link AB is vertical.

Find: For this position:

(a) Determine the angular velocity of link BC and the wheel. Write your answers as vectors.

(b) Determine the angular acceleration of link BC and the wheel. Write your answers as vectors.

Planar Kinematics Homework Problems ME 274

Problem II-26 Given: A mechanism is made up of links AB and BC and a wheel pinned to

BC at the wheel’s center C. The wheel rolls without slipping on a horizontal surface. Link AB rotates counterclockwise with a constant rate of 3 rad/sec. At the instant shown, link AB is vertical.

Find: At the instant shown,

a) determine the angular velocities of link BC and the wheel. b) determine the angular accelerations of link BC and the wheel. c) determine the velocity and acceleration of point E on the perimeter

of the wheel (at the instant shown, E is on the same horizontal line as C).

Write your answers as vectors.

L d

A

B

C

r

no slip

E !AB

Use the following parameters in your analysis: r = 0.5 m, L = 2 m, d = 1 m and ωAB = 3 rad/s.You may solve the problem using the Method of Instant Centers and/or by adopting a vectorapproach.

Chapter II: Planar Rigid Body Kinematics Homework II-15

Homework 2.J

Given: A mechanism is made up of links AB and BD, with point A being pinned to ground andpoint B being a pin joint connecting links AB and BD. A slider is connected to BD at pin D withthe slider being constrained to move on a circular path with a radius of R. At the position shownbelow, link AB is vertical, link BD is horizontal and D is directly above point O. At this position,D is traveling to the right with a constant speed of vD.

Find: For this position:

(a) Determine the angular velocities of links AB and BD. Write your answers as vectors.

(b) Determine the angular accelerations of links AB and BD. Write your answers as vectors.

Planar Kinematics Homework Problems ME 274

Problem II-23

A mechanism is made up of links AB and BD, with point A being pinned to ground and point B being a pin joint connecting links AB and BD. A slider is connected to BD at pin D with the slider being constrained to move on a circular path with a radius of 2.5 meters. At the position shown below, link AB is vertical, link BD is horizontal and D is directly above point O. At this position, D is traveling to the right with a constant speed of 5 m/sec. For the position shown, find:

i) the angular velocity of links AB and BD, and

ii) the angular acceleration of links AB and BD.

d

L v D

A

B

D

x

y

O

R

Use the following parameters in your analysis: R = 0.5 m, L = 2 m, d = 1.5 m and vD = 5 m/s. Youmay solve the problem using the Method of Instant Centers and/or by adopting a vector approach.