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 12.2, Problem 12.91P

A 1-Ib ball A and a 2-Ib ball B are mounted on a horizontal rod that rotates freely about a vertical shaft. The balls are held in the positions shown by pins. The pin holding B is suddenly removed and the ball moves to position C as the rod rotates. Neglecting friction and the mass of the rod and knowing that the initial speed of A is v A = 8 ft/s, determine (a) the radial and transverse components of the acceleration of ball B immediately after the pin is removed, (b) the acceleration of ball B relative to the rod at that instant, (c) the speed of ball A after ball B has reached the stop at C.

  Chapter 12.2, Problem 12.91P, A 1-Ib ball A and a 2-Ib ball B are mounted on a horizontal rod that rotates freely about a vertical

Expert Solution
Check Mark
To determine

(a)

Radial and transverse component of acceleration of ball B immediately after the pin is removed.

Answer to Problem 12.91P

Radial component:

(aB)r=0

Transverse component:

(aB)θ=0

Explanation of Solution

Given:

Vector Mechanics for Engineers: Dynamics, Chapter 12.2, Problem 12.91P , additional homework tip  1

Let r and θ be polar coordinates with the origin lying at the shaft.

Constraint of rod:

θB=θA+πradians

θ˙B=θ˙A=θ˙

θ¨B=θ¨A=θ¨

Components of acceleration:

Sketch the free body diagrams of the balls showing the radial and transverse components of the forces acting on them.

Owing to frictionless sliding of B along the rod,

(FB)r=0

Vector Mechanics for Engineers: Dynamics, Chapter 12.2, Problem 12.91P , additional homework tip  2

Radial component of acceleration of B.

Fr=mB(aB)r

(aB)r=0

Transverse components of acceleration,

(aA)θ=rAθ¨+2r˙Aθ˙=raθ˙

(aB)θ=rBθ¨+2r˙Bθ˙=raθ˙   ... (1)

Since the rod has no mass, it must be in equilibrium.

Draw its free body diagram, applying Newton’s third law:

Vector Mechanics for Engineers: Dynamics, Chapter 12.2, Problem 12.91P , additional homework tip  3

Vector Mechanics for Engineers: Dynamics, Chapter 12.2, Problem 12.91P , additional homework tip  4m:math display='block'>

M0=0 :

rA(FA)θ+rB(FB)θ=rAmA(aA)θ+rBmB(aB)θ=0

rAmArAθ¨+rBmB(rBθ¨+2r˙Bθ˙)=0

θ¨=2mBr˙Bθ˙mArA2+mBrB2

At t=0 ,

r˙B=0

So that

θ¨=0

From Eq. (1),

(aB)θ=rBθ¨+2r˙Bθ˙=raθ˙

(aB)θ=0

Expert Solution
Check Mark
To determine

(b)

Acceleration of ball B relative to rod.

Answer to Problem 12.91P

Acceleration of ball B relative to rod:

r¨B=61.4ft/s2

Explanation of Solution

Given:

Vector Mechanics for Engineers: Dynamics, Chapter 12.2, Problem 12.91P , additional homework tip  5

Acceleration of B relative to the rod,

At t=0 ,

(vA)θ=8ft/s=96in/s

θ˙=(vA)θrA=9610=9.6rad/s

r¨BrBθ˙2=(aB)r=0

r¨B=rBθ˙2=(8)(9.6)2=737.28in/s2

r¨B=61.4ft/s2

Expert Solution
Check Mark
To determine

(c)

Acceleration of ball B relative to rod.

Answer to Problem 12.91P

Acceleration of ball B relative to rod:

(vA)f=2.98ft/s

Explanation of Solution

Given:

Vector Mechanics for Engineers: Dynamics, Chapter 12.2, Problem 12.91P , additional homework tip  6

Speed of A:

Substituting ddt(mr2θ˙) for rFθ in each term of moment equation gives,

ddt(mArA2θ˙)+ddt(mBrB2θ˙)=0

Integrating with respect to time,

mArA2θ˙+mBrB2θ˙=(mArA2θ˙)0+(mBrB2θ˙)0

Applying to the final state with ball B moved to the stop at C,

(WAgrA2+WBgrC2)θ˙f=(WAgrA2+WBg(rB)02)θ˙0

θ˙f=WArA2+WB(rB)02WArA2+WBrC2θ˙0

θ˙f=(1)(10)2+(2)(8)2(1)(10)2+(2)(16)2(9.6)

θ˙f=3.5765rad/s

(vA)f=rAθ˙f=(10)(3.5765)=35.765in/s

θ˙=(vA)θrA=9610=9.6rad/sddt(mArA2θ˙)+ddt(mBrB2θ˙)=0

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Students have asked these similar questions
A 1-lb ball A and a 2-lb ball B are mounted on a horizontal rod that rotates freely about a vertical shaft. The balls are held in the positions shown by pins. The pin holding B is suddenly removed and the ball moves to position C as the rod rotates. Neglecting friction and the mass of the rod and knowing that the initial speed of A is vA = 8 ft/s, determine (a) the radial and transverse components of the acceleration of ball B immediately after the pin is removed, (b) the acceleration of ball B relative to the rod at that instant, (c) the speed of ball A after ball B has reached the stop at C.
(a) Determine the radial and transverse components of the acceleration of ball B immediately after the pin is removed, (b) the acceleration of ball B relative to the rod at that instant, (c) the speed of ball A after ball B has reached the stop at C.
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Chapter 12 Solutions

Vector Mechanics for Engineers: Dynamics

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