chapter 5jfrhoads/caldynamics/h5_summer... · chapter 5: rigid body kinetics homework homework 5.b...

Chapter 5: Rigid Body Kinetics Homework

Chapter 5

Rigid Body Kinetics Homework

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Homework 5.A

Given: A uniform slender rod is at rest upon a frictionless horizontal surface when a force F isapplied in the direction shown. Assume that the rod has mass m and length of L.

Find:(a) Determine the acceleration of the rod’s center of mass G; and(b) Determing the rod’s angular acceleration.

Rigid Body Kinetics Homework Problems ME 274

Problem V-3 A uniform slender rod is at rest upon a frictionless horizontal surface when a force F is applied in the direction shown. Assuming that the rod has mass m and length of L, determine the (a) acceleration of the rod’s center of mass G and (b) the rod’s angular acceleration.

Use the following: ! = 36.87°, mg = 5 lb, F = 15 lb and L = 2 ft.

Use the following parameters in your analysis: θ = 30◦, mg = 5 lb, F = 20 lb and L = 3 ft.



Homework 5.B

Given: An inhomogeneous disk (having a mass of m, outer radius R, mass center at G andcentroidal radius of gyration of kG) moves along a SMOOTH horizontal surface. The distance fromthe geometric center O of the disk to the center of mass G is given by e. At the instant shown, Gis at the same vertical height as O as the disk translates to the right with a speed of vO.

Find: Determine the normal force of the ground acting at the contact point C at this instant.


Problem V-59 Given: An inhomogeneous disk (having a mass of m,

outer radius R, mass center at G and centroidal radius of gyration of kG) moves along a SMOOTH horizontal surface. The distance from the geometric center O of the disk to the center of mass G is given by e. At the instant shown, G is at the same vertical height as O as the disk translates to the right with a speed of vO .

Find: Determine the normal force of the ground

acting at the contact point C at this instant. In your analysis, use the following parameter values: m = 10 kg, R = 0.5 meters, e = 0.3 meters, kG = 0.4 meters, and vO = 20 m/sec.

HINT: The contact point C slips on the smooth horizontal surface.

SMOOTH

g

m

R e

O G

C

vO

Use the following parameters in your analysis: m = 10 kg, R = 0.5 m, e = 0.3 m, kG = 0.4 m andvO = 20 m/s.



Homework 5.C

Given: The homogeneous thin bar AB shown below has a mass of M and a length of L. The baris supported by lightweight, inextensible cable BO at end B and by a smooth, inclined wall at endA. The bar is released from rest with the bar in a horizontal orientation and with cable BO beingvertical.

Find: Determine the tension in cable BO immediately after the bar is released.


Problem V-55 Given: The homogeneous thin bar AB shown below has a mass of M = 100 kg and a

length of L = 3 meters. The bar is supported by lightweight, inextensible cable BO at end B and by a smooth, inclined wall at end A, where ! = 53.13°. The bar is released from rest with the bar in a horizontal orientation and with cable BO being vertical.

Find: Determine the tension in cable BO immediately after the bar is released.

A

!

L B

O

smooth wall

Use the following parameters in your analysis: M = 100 kg, L = 3 m and θ = 53.13◦.



Homework 5.D

Given: A gear of mass m and outer radius R meshes with rack A (having a mass of 2m). RackA is constrained to move along a smooth vertical guide. Treat the gear as a homogeneous disk ofradius R. Assume that the rack does not slip on the gear.

Find: Determine the acceleration of rack A when the system is released from rest. Write youranswer as a vector.


Problem V-63 Given: A gear of mass m and outer radius R meshes with rack A (having a mass of

2m). Rack A is constrained to move along a smooth vertical guide. Treat the gear as a homogeneous disk of radius R. Assume that the rack does not slip on the gear.

Find: Determine the acceleration of rack A when the system is released from rest.

Write your answer as a vector.

R

O

A

smooth

no slip

C g

m

2m



Homework 5.E

Given: An inextensible cable connects block B with point A on a rigid, homogeneous disk. At theinstant shown, point A is directly above the disk’s center O. Block B has a mass of m, and the diskhas a mass of M . The disk is able to roll without slipping on a rough horizontal surface with pointC being the contact point. The system is released from rest.

Find: Determine the angular acceleration of the disk immediately after release.


Problem V-49 Given: An inextensible cable connects block B with point A on a rigid, homogeneous

disk. At the instant shown, point A is directly above the disk’s center O. Block B has a mass of m = 40 kg, and the disk has a mass of M = 20 kg. The disk is able to roll without slipping on a rough horizontal surface with C being the contact point.

Find: If the system is released from rest at the position shown, find the angular

acceleration of the disk. Write your answer as a vector.

B

O R/2

no slip R

A

C

m

M

Use the following parameters in your analysis: M = 20 kg, m = 40 kg, and R = 0.1 m.



Homework 5.F

Given: A pulley of mass M is connected to two blocks as shown. Assume that the axle at C isfrictionless, and that the pulley can be crudely modeled as a homogeneous disk.

Find: Determine the angular acceleration of the pulley and the acceleration of each block.

Am Bm

or

irA BC

Use the following parameters in your analysis: M = 10 kg, mA = 12 kg, mB = 9 kg, ri = 0.06 mand ro = 0.2 m.



Homework 5.G

Given: A homogeneous disk (having a mass of M and outer radius of R) is welded to a thin,homogeneous bar having a mass of m and length L with the end of the bar located at the center Oof the disk. The disk is able to roll without slipping on a horizontal surface. The system is releasedfrom rest at Position 1 with the bar being horizontal at this position. On release, the system movesto the right. At Position 2 shown below the bar is vertical.

Find: Determine the speed of point G when the system is at Position 2.


Problem V-47 Given: A homogeneous disk (having a mass of M = 10 kg and outer radius of R = 1

meter) is welded to a thin, homogeneous bar having a mass of m = 20 kg and length L = 4 meters with the end of the bar located at the center O of the disk. The disk is able to roll without slipping on a horizontal surface. The system is released from rest at position 1 with the bar being horizontal at this position. On release, the system rolls to the right. At position 2 shown below the bar is vertical.

Find:

a) Draw a free body diagram (FBD) of the system of the bar and disk together.

b) Classify all of the forces in your FBD in a) as to whether they do work or will be included in potential energy.

c) Determine the speed of point G when the system is at position 2.

O

C

R

L

G

Position 1

G

O

Position 2

Use the following parameters in your analysis: M = 10 kg, m = 20 kg, R = 1 m and L = 4 m.



Homework 5.H

Given: A thin, homogeneous bar having a mass of M and length L is pinned to ground at its masscenter G. Particle B, having a mass of m, is rigidly attached to the right end of the bar. A spring,having a stiffness of k, is attached between end A of the bar and pin C on a wall. The pin G isa distance of b from the wall. When the bar is horizontal (Position 1 shown below), the spring isunstretched.

Find: If the bar is released from rest in Position 1 above, find the angular velocity of the bar inPosition 2 when the bar is in a vertical position.


Problem V-41 Given: A thin, homogeneous bar having a mass of M and length L is pinned to

ground at its mass center G. Particle B, having a mass of m, is attached to the right end of the bar. A spring, having a stiffness of k, is attached between end A of the bar and pin C on a wall. The pin G is a distance of b from the wall. When the bar is horizontal (Position 1 shown below), the spring is unstretched.

Find: If the bar is released from rest in Position 1 above, find the angular velocity of

the bar in Position 2 when the bar is in a vertical position. Use the following parameters for your final numerical solution:

M = 15 kg m = 25 kg k = 100 N/meter

L = 3 meters b = 2.5 meters

L

m M

k

G

A B C

b

g Position 1

G

A

B

C Position 2

Use the following parameters in your analysis: M = 15 kg, m = 25 kg, k = 100 N/m, L = 3 mand b = 2.5 m.



Homework 5.I

Given: The compound wheel shown below rolls without slipping up the incline on its hubs andis pulled by a constant force F applied to a cord wrapped around its outer rim. The wheel startsfrom rest, has a mass of m, and has a radius of gyration about its center of mass O of kO. Assumethat the cable does not slip on the wheel.

Find: Determine the angular velocity of the wheel after its center O has moved a distance of d upthe incline.


Problem V-60 Given: The compound wheel shown rolls without slipping up the incline on its hubs

and is pulled by a constant force T = 100 N applied to a cord wrapped around its outer rim. The wheel starts from rest, has a mass of m = 40 kg, and has a radius of gyration about its center of mass O of kO = 0.15 meters. Assume that the cable does not slip on the wheel.

Find: The angular velocity of the wheel after its center O has moved 2 meters up the

incline.

R/2

R

F

! O

no slip

Use the following parameters in your analysis: m = 40 kg, R = 0.2 m, d = 2 m, F = 100 N, kO =0.15 m, and θ = 30◦.



Homework 5.J

Given: A drum has a mass of M , outer radius of R, inner radius of r, a centroidal radius ofgyration of kG and a centroid at the geometric center G. A cable is wrapped around the innerradius of the drum and is connected to particle A having a mass of m. The cable is also wrappedover an ideal pulley. A spring of stiffness K is attached between G of the drum and ground. Thesystem is released from rest with the spring unstretched.

Find: Determine the speed of A after it has dropped through a distance of d.


Problem V-54 Given: A drum has a mass of M = 30 kg, outer radius of R = 0.5 m, inner radius of r

= 0.25 m, a centroidal radius of gyration of

kG = 0.6 m and centroid at the geometric center G. A cable is wrapped around the inner radius of the drum and is connected to particle A having a mass of m = 10 kg. The cable is also wrapped over an ideal pulley. A spring of stiffness K = 100 N/m is attached between G of the drum and ground. The system is released from rest with the spring unstretched.

Find: Determine the speed of A after it has dropped 2 meters.

no slip

G r

R K

A

M

m

pulley has negligible mass

Use the following parameters in your analysis: M = 30 kg, m = 10 kg, R = 0.5 m, r = 0.25 m, kG= 0.3 m, K = 10 N/m and d = 2 m.



Homework 5.K

Given: The center of the homogeneous disk of mass m shown below has an initial velocity of v0up the slope.

Find: Find the velocity of the disk after an elapsed time of ∆t, assuming that the disk rolls withoutslip.

rO

Use the following parameters in your analysis: m = 10 kg, r = 0.5 m, θ = 30◦, v0 = 30 m/s and∆t = 12 s.



Homework 5.L

Given: A uniform disk (of mass m and radius r) is pinned at its center of mass O. Force F isapplied to a rope wrapped around the disk. The disk is released from rest. Assume that the ropedoes not slip on the disk.

Find: At time t, determine:(a) the angular velocity of the disk; and(b) the reaction force components on the disk at pin O.


Problem V-36 A uniform disk (m, r) is pinned at its center of mass O. Force F is applied to a rope wrapped around the disk. The disk is released from rest. After 2 seconds,

a) determine the angular velocity of the disk. b) determine the reaction force components at the pin.

Use the following parameters for your analysis: m = 10 kg, r = 0.3 m, F = 250 N.

F O

r

Use the following parameters in your analysis: m = 15 kg, r = 0.4 m, F = 200 N and t = 2 s.



Homework 5.M

Given: A homogeneous disk (with mass M and outer radius of R) is placed on a rough surface.When placed on this surface, the center of the disk A is moving to the right with a speed of vA1

and has a counterclockwise rotation rate of ω1, as shown in Figure 1 below. In Figure 2 below isshown the instant at which the disk ceases to slip as it continues to move on the horizontal surface.

Find: For this problem:(a) Determine the speed of A, vA2, at the instant in Figure 2 when the disk ceases to slip on the

horizontal surface; and(b) Determine the elapsed time during the motion as the disk moves from the position in Figure 1

to the position in Figure 2.


Problem V-57 Given: A homogeneous disk (with mass M = 50 kg and outer radius of R = 0.5

meters) is placed on a rough surface (µk = 0.3). When placed on this surface, the center of the disk A is moving to the right with a speed of vA1 = 5 m/sec and has a counterclockwise rotation rate of !1 = 8 rad/sec, as shown in Figure 1 below. In Figure 2 below is shown the instant at which the disk ceases to slip as it continues to move on the horizontal surface.

Find: For this problem:

a) Determine the speed of A, vA2, at the instant in Figure 2 when the disk ceases to slip on the horizontal surface.

b) Determine the elapsed time during the motion as the disk moves from the position in Figure 1 to the position in Figure 2.

You must provide an accurate free body diagram of the disk in order to receive full credit for your work. HINT: Consider using both the linear impulse-momentum equation and the angular

impulse-momentum equation on the disk in solving this problem.

vA2

A

!2

FIGURE 2

no slip

vA1

A

!1

R

FIGURE 1

µk

slipping

M

Use the following parameters in your analysis: M = 50 kg, R = 0.5 m, µk = 0.3, vA1 = 5 m/s andω1 = 8 rad/s.



Homework 5.N

Given: Particle B (having a mass of m) initially travels on a smooth HORIZONTAL surface witha speed of vB1. This particle strikes point E on a stationary thin homogeneous bar (having a massof M), with end A of the bar being pinned to ground. The coefficient of restitution between theparticle and the bar at point E is known to be e. Furthermore, the contact surface between B andpoint E is smooth.

Find: Determine the angular velocity of the bar immediately after being impacted by the particle.


Problem V-50 Given: Particle B (having a mass of m = 10 kg) initially travels on a smooth

HORIZONTAL surface with a speed of vB1 = 20 m/sec. This particle strikes point E on a stationary thin homogeneous bar (having a mass of M = 15 kg), with end A of the bar being pinned to ground. The coefficient of restitution between the particle and the bar at point E is known to be e = 0.6. Furthermore, the contact surface between B and point E is smooth.

Find: Determine the angular velocity of the bar immediately after being impacted

by the particle. Write your answer as a vector.

B

A

! vB1

E

d

L

m M

Use the following parameters in your analysis: L = 3 m, d = 2 m, m = 10 kg, M = 15 kg, e = 0.6,θ = 60◦ and vB1 = 20 m/s.


chapter 5jfrhoads/caldynamics/h5_summer... · chapter 5: rigid body kinetics homework homework 5.b...

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