Vector Mechanics for Engineers: Dynamics
Vector Mechanics for Engineers: Dynamics
11th Edition
ISBN: 9780077687342
Author: Ferdinand P. Beer, E. Russell Johnston Jr., Phillip J. Cornwell, Brian Self
Publisher: McGraw-Hill Education
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Chapter 17.1, Problem 17.17P
To determine

(a)

The distance b for which the angular velocity of the rod is maximum when it passes through a vertical position.

Expert Solution
Check Mark

Answer to Problem 17.17P

The distance b: b=l12

Explanation of Solution

Given information:

Length of the slender rod is l. The distance between the pivot and the center of rod is b.

The rod is released from rest in a horizontal position and swings freely.

Calculation:

Vector Mechanics for Engineers: Dynamics, Chapter 17.1, Problem 17.17P , additional homework tip  1

At Position 1._v¯=0,ω=0Kinetic energy:T1=0Elevation:h=0Potential energy:V1=mgh=0At Position 2._v¯2=bω2I=112ml2Kinetic energy:T2=112mv¯22+112I¯ω22T2=112m(b2+112l2)ω22Elevation:h=bPotential energy:V2=mgbApplying the principle of conservation of energy:T1+V1=T2+V2:0+0=12m(b2+112l2)ω22mgbω22=2gbb2+112l2 ................(1)

For maximum value of ω2;ddb(ω2)=0ddb(bb2+112l2)=(b2+112l2)b(2b)(b2+112l2)=0b2=112l2b=l12

Conclusion:

For b=l12, the angular velocity of the rod is maximum when it passes through a vertical position.

To determine

(b)

The corresponding value of the angular velocity and the reaction at C.

Expert Solution
Check Mark

Answer to Problem 17.17P

The corresponding value of the angular velocity: ω2=1.861gl

The reaction at C: C=2mg

Explanation of Solution

Given information:

Length of the slender rod is l. The distance between the pivot and the center of rod is b.

The rod is released from rest in a horizontal position and swings freely.

From the result of part (a), b=l12.

Calculations:

Using eq.(1), from part (a), The angular velocity of the rod when it passes through a vertical position is:ω2=(2gbb2+112l2)12substituting, b=l12ω2=(2gl12l212+l212)12ω22=12glω2=121/4glω2=1.861gl

Vector Mechanics for Engineers: Dynamics, Chapter 17.1, Problem 17.17P , additional homework tip  2

From the free body and kinetic diagram of the rod when it passes through the vertical position (as shown above):an=bω22an=l1212glan=gApplying force balance in y-direction:+Fy=man:Cymg=mgCy=2mgMC=0:mbat+I¯α=0with, (at=bα)(mb2+I¯)α=0α=0,hence, at=0Applying force balance in x-direction:Fx=mat:Cx=mat=0Thus, the reaction at C:C=2mg

Conclusion:

The corresponding value of the angular velocity is ω2=1.861gl and the reaction at C is C=2mg.

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Chapter 17 Solutions

Vector Mechanics for Engineers: Dynamics

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