Chapter 5 - Purdue Engineeringjfrhoads/me274/H5_fall_2016_pwl.pdf · A thin homogeneous bar OA having a length of L and mass m is pinned to ground at O. A A second thin homogeneous

Chapter 5: Rigid Body Kinetics Homework

Chapter 5

Rigid Body Kinetics Homework

Freeform c©2016 5-1


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Homework 5.A

Given: A smooth, massless roller is attached to end A of a thin, homogeneous bar of length L andmass m. The roller is allowed to move on a horizontal guide. The bar is released from rest at theposition shown below.

Find: Determine the acceleration of the center of mass of the bar at the instant of release. Stateyour answer in vector form.

Rigid Body Kinetics Homework Problems ME 274

Problem V-62 Given: A smooth, massless roller is attached to end A of a thin, homogeneous bar of

length L = 2 meters and mass m = 10 kg. The roller is allowed to move on a horizontal guide. The bar is released from rest with the bar being at an angle of ! = 53.13° from the vertical, as shown below.

Find: Determine the acceleration of the center of mass of the bar at the instant of

release. State your answer in vector form.

A smooth,

massless roller

!

m, L g

Use the following parameters in your analysis: m = 10 kg, L = 2 m and θ = 53.13.



Homework 5.B

Given: Homogneous bar AB of mass m is supported by a base AC that is translating to the leftwith a constant acceleration of a. Cable AC restrains the rotation of the bar as it translates to theleft.

Find:(a) Determine the tension in the cable; and(b) Determine the reaction force acting on bar AB at A.

aA

B

0.75 m

1 m

C

Use the following parameters in your analysis: m = 3 kg and a = 7 m/s2.



Homework 5.C

Given: An applied moment M acts on a slender, uniform bar of mass m. Assume all surfaces tobe smooth.

Find: Determine the reaction forces acting on the bar at A and B on release.

M

A

Bx

y

l

Use the following parameters in your analysis: θ = 45.



Homework 5.D

Given: A homogeneous disk of mass m and outer radius r is attached to a horizontal shaft atpoint O, where O is at a radial distance of r/2 from the center C of the disk. The shaft to whichthe disk is attached is rotating at a constant rate of θ = Ω.

Find:(a) Determine the total reaction force acting on the disk at O when θ = 90. Write this answer as

a vector.(b) Determine the total reaction force acting on the disk at O when θ = 180. Write this answer

as a vector.

C

r2

r

θ

O

g

m

Use the following parameters in your analysis: m = 20 kg, r = 0.8 m and Ω = 10 rad/s.



Homework 5.E

Given: A homogeneous plate of mass m is attached to cart B by a pin joint at corner O of theplate. The cart (having a mass of M) is free to move along a smooth horizontal surface. The plateand cart are released from rest with the plate at angle of θ.

Find:(a) Determine the angular acceleration of the plate on release.(b) Determine the acceleration of the cart on release.

smooth O

A

θ

B M

m

h b

C

D

Use the following parameters in your analysis: m = 10 kg, M = 20 kg, h = 0.2 m, b = 0.8 m, andθ = 30.



Homework 5.F

Given: A homogeneous disk having a mass m and outer radius R is supported by the cable-pulleysystem shown, where block A (with a mass of m) is attached to the free end of the cable. The diskdoes not slip on the cable as the system moves.

Find: Determine the acceleration of block A on release.

g

pulley of negligible mass

2R

C B

A

O

m

m

no slip

Use the following parameters in your analysis: m = 20 kg and R = 0.2 m.



Homework 5.G

Given: A thin homogeneous bar of length L and mass m is pinned to ground at point O. Ahomogeneous disk with a mass of M and radius R is PINNED to end A of the bar. The disk rollswithout slipping on the inside of a circular surface. The system is released from rest with θ = 90.

Find: Find the angular velocity of the bar when θ = 0.


Problem V-45 A thin homogeneous bar of length L = 1.5 meters and mass m = 30 kg is pinned to ground at point O. A homogeneous disk with a mass of M = 100 kg and radius R = 0.6 meters is PINNED to end A of the bar. The disk rolls without slipping on the inside of a circular surface. The system is released from rest with ! = 90°. Find the angular velocity of the bar when ! = 0.

!

R

O

A

L no slip

g

Use the following parameters in your analysis: L = 1.5 m, R = 0.6 m, m = 30 kg and M = 100 kg.



Homework 5.H

Given: Consider the three systems (each of total mass m) shown below where: (I) is made upof a particle of mass m supported within a massless ring by massless spokes, (II) is made up ofidentical, thin spokes of total mass m in a massless ring, and (III) is a thin ring of mass m. Eachsystem is released from rest, and rolls without slipping on a rough incline.

Find: Determine the velocity of the mass center O of each system when each system has madetwo complete revolutions in rolling down the inclines. Rank the speeds of O for the three systems.Explain in words why the speeds end up with those rankings.


Problem V-11 Consider the three systems (each of total mass m) shown below where: (I) made up on a particle of mass m supported within a massless ring by massless spokes, (II) made up of identical, thin spokes of total mass m in a massless ring, and (III) a thin ring of mass m. Each system is released from rest, with each system rolling without slipping on a rough incline. Determine the velocity of the mass center O of each system when each system has made two complete revolutions in rolling down the inclines. Rank the speeds of O for the three systems. Explain in words why the speeds end up with those rankings.

no slip

O

R

!

g

O O

R R

no slip no slip

! !

(II) massless ring and identical spokes (with total mass of spokes m)

(I) massless ring, massless spokes and particle of mass m

(III) thin ring of mass m



Homework 5.I

Given: One end of an inextensible cable is attached to a rigid wall at B. This cable is wrappedaround a homogeneous disk of mass m and outer radius R with the free end of the cable remaininghorizontal at all times. A constant horizontal force F acts at the free end A of the cable. A springof stiffness k is attached between the center of the disk O and ground. As the disk moves to theleft, it does not slip on the cable. The system is released from rest with the spring unstretched.

Find: Determine the speed of the center of the disk O after O has moved through a distance d tothe right.


Problem V-13 One end of an inextensible cable is attached to a rigid wall at B. This cable is wrapped around a homogeneous disk of mass m and outer radius R with the free end of the cable remaining horizontal at all time. A constant horizontal force F acts at the free end A of the cable. A spring of stiffness k is attached between the center of the disk O and ground. As the disk moves to the left, it does not slip on the cable. The system is released from rest at instant “1’ with the spring unstretched.

a) Draw a free body diagram (FBD) of the disk. List which, if any, nonconservative forces do work on this system.

b) Determine the speed of the center of the disk O after O has moved through a distance d to the right.

Use the following parameters in your analysis:m = 60 kg , R = 0.5 meters , k = 300 N / m , F = 2000 N and d = 2 meters .

A

k

O

C

F

R

no slip

no slip

B

Use the following parameters in your analysis: m = 50 kg, R = 0.6 m, k = 250 N/m, F = 2200 Nand d = 1.75 m.



Homework 5.J

Given: A thin homogeneous bar OA having a length of L and mass m is pinned to ground at O.A second thin homogeneous bar AB (having a length of 2L and mass 2m) is pinned to bar OA atA, and end B of the bar is constrained to move within a smooth, horizontal track. A constant forceF acts horizontally at end B. The bar is released from rest with link OA being horizontal and withpin B being directly below pin O, as shown in the figure.

Find: Determine the speed of end B of bar AB when the system has reached a position of link OAbeing vertical.


Problem V-14 A thin homogeneous bar OA having a length of L and mass m is pinned to ground at O. A second thin homogeneous bar AB (having a length of 2L and mass 2m) is pinned to bar OA at A, and end B of the bar is constrained to move within a smooth, horizontal track. A constant force F acts horizontally at end B. The bar is released from rest at instant “1” with link OA being horizontal and with pin B being directly below pin O.

a) Draw a free body diagram of the system made up of links OA and AB. List which, if any, nonconservative forces do work on this system.

b) Determine the speed of end B of bar AB when the system has reached a position of link OA being vertical (instant “2”).

Use the following parameters in your analysis: F = 500 lb , L = 4 ft and mg = 200 lb .

VERTICAL PLANE

g

A

B

O L

2L

m

2m

F

Use the following parameters in your analysis: F = 600 lb, L = 3 ft and mg = 200 lb.



Homework 5.K

Given: A cylinder with an outer radius of R rolls without slipping on a horizontal surface. Thecenter of the disk O has an acceleration that is known in terms of the position x of: aO = 3x2/2(x in feet and aO in ft/s2). At position 1 (x = 0 ft), point O has a velocity of vO1 to the right.

Find: For a Position 2 of point O:(a) Determine the velocity of point O;(b) Determine the angular velocity and angular acceleration of the cylinder;(c) Determine the velocity of point A (at this position A is on the same horizontal line as O); and(d) Determine the acceleration of point A (at this position A is on the same horizontal line as O).


Problem V-46 Given: A cylinder with an outer radius of R = 3 ft rolls without slipping on a

horizontal surface. The center of the disk O has an acceleration that is known in terms of the position x of:

aO = 32x2 (x in feet and aO in ft/sec2)

At position 1 (x = 0), point O has a velocity of 1 ft/sec to the right. Find: At position 2, where x = 2 ft,

a) determine the velocity of point O. b) determine the angular velocity and angular acceleration of the

cylinder. Write your answers as vectors. c) determine the velocity and acceleration of point A (at this position A is

on the same horizontal line as O). Write your answers as vectors. d) sketch the velocity vector for A (from c) above) in the Position 2 figure

below.

no slip

O O

R

aO aO A

x

Position 1 Position 2

Use the following parameters in your analysis: R = 3 ft, vO1 = 1 ft/s and x = 2 ft.



Homework 5.L

Given: A homogeneous, slender bar of mass M and length L is sliding across a frictionless hori-zontal surface with a speed vG when it is suddenly struck at its end by a particle of mass m movingwith velocity vm in the direction shown. Assume that the particle sticks to the bar upon impact.

Find: Determine the velocity of the bar’s center of mass G and the angular velocity of the barpost-impact.


Problem V-16 A homogeneous, slender bar of mass M length L moves is sliding across a frictionless horizontal surface with a speed vG when it is suddenly struck at its end by a particle of mass m moving with velocity vm in the direction shown. Assuming that the particle sticks to the bar, determine the velocity of the bar’s center of mass G and the angular velocity of the bar post-impact.

Use the following: M = 3 kg, L = 0.5 m, vG = 4 m/s, m = 0.3 kg and vm = 20 m/s.

Use the following parameters in your analysis: M = 4 kg, L = 0.75 m, vG = 5 m/s, m = 0.4 kgand vm = 25 m/s.



Homework 5.M

Given: Particle A, of mass m, is able to slide on a smooth homogeneous bar of length L and massm. The bar is pinned to ground at end O and particle B (of mass m) is rigidly attached to theother end. The bar is given an initial rotation rate of ω1 when the A is at the midpoint of thebar, after which A slides outward on the bar. Eventually particle A impacts particle B, an impacthaving a coefficient of restitution of e.

Find: Determine the velocity of A immediately after impact. Express your answer in terms of itsR-φ components.

HORIZONTAL'plane'

eR

B

O

L / 2

L / 2

m

ω

eφ

m

smoothA

m

Use the following parameters in your analysis: e = 0.8, m = 10 kg and L = 0.5 m.



Homework 5.N

Given: A homogeneous rectangular plate of mass m is pinned to cart B at corner O, where cartB is constrained to move along a smooth horizontal surface. The system is released from rest withcorner D displaced slightly to the right of a vertical line passing through the pin at O. As a result,the plate eventually impacts bumper C on the cart, with the coefficient of restitution between theplate and the bumper being e.

Find: For this problem:(a) Determine the velocity of the center of mass of the plate immediately before the plate contacts

the bumper C. Write your answer as a vector.(b) Determine the velocity of the center of mass of the plate immediately after the plate contacts

the bumper C. Write your answer as a vector.

smooth O

A

B M

m

h

b

C

D

Use the following parameters in your analysis: m = 10 kg, M = 25 kg, b = 2 m, h = 1 m and e =0.


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Chapter 5 - Purdue Engineeringjfrhoads/me274/H5_fall_2016_pwl.pdf · A thin homogeneous bar OA having a length of L and mass m is pinned to ground at O. A A second thin homogeneous