VECTOR MECHANICS FOR ENGINEERS W/CON >B
VECTOR MECHANICS FOR ENGINEERS W/CON >B
12th Edition
ISBN: 9781260804638
Author: BEER
Publisher: MCG
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Chapter 15.6, Problem 15.219P
To determine

The acceleration of the collar A when c=120mm.

Expert Solution & Answer
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Answer to Problem 15.219P

The acceleration of collar A is aA=(8.764mm/s2)j_.

Explanation of Solution

Given information:

The collar B moves towards D at a constant speed of vB=50mm/s.

The length of rod AB is lAB=300mm

The distance c=120mm

Calculation:

Calculate the position vector (r) as shown below.

The position of A.

rA=yj

The position of D.

rD=(90mm)i

The position of C.

rC=(180mm)k

The position of D with respect to C.

rD/C=rDrC

Substitute (90mm)i for rD and (180mm)k for rC.

rD/C=(90mm)i(180mm)k

Calculate the length of CD (lCD) as shown below.

lCD=902+(180)2=40,500=201.246mm

The position of B with respect to C.

rB/C=c(rD/C)180

Substitute 120mm for c and (90mm)i(180mm)k for rD/C.

rB/C=120(90i180k)180=(60mm)i(120mm)k

The position of B.

rB=rC+rB/C

Substitute (60mm)i(120mm)k for rB/C and (180mm)k for rC.

rB=(180mm)k+((60mm)i(120mm)k)=(60mm)i+(60mm)k

The position of A with respect to B.

rA/B=rArB

Substitute yj for rA and (60mm)i+(60mm)k for rB.

rA/B=yj((60mm)i+(60mm)k)=(60mm)i+yj(60mm)k (1)

Calculate the length value of y of AB using the relation as shown below.

lAB2=(60)2+y2+(60)2

Substitute 300mm for lAB.

3002=7,200+y2y2=82,800y=287.75mm

Calculate the position of A with respect to B (rA/B) as shown below.

Substitute 287.75mm for y in Equation (1).

rA/B=(60mm)i+(287.75mm)j(60mm)k

Calculate the velocity at B (vB) as shown below.

vB=vB×rD/ClCD

Substitute 201.246mm for lCD, 50mm/s for vB, and (90mm)i(180mm)k for rD/C.

vB=50(90i180k)201.246=22.3607i44.7214k=(22.3607mm/s)i(44.7214mm/s)k

Calculate the velocity of collar A with respect to B (vA/B) as shown below.

vA/B=ωAB×rA/B

The velocity of collar A with respect to B vA/B is perpendicular to rA/B. Hence, rA/BvA/B=0.

Calculate the velocity of collar A (vA) as shown below.

vA=vB+vA/B

Apply the dot product of rA/B on both sides of the Equation.

rA/BvA=rA/B(vB+vA/B)=rA/BvB+rA/BvA/B

Substitute 0 for rA/BvA/B.

rA/BvA=rA/BvB

Substitute (60mm)i+(287.75mm)j(60mm)k for rA/B, (vA)j for vA, and (22.3607mm/s)i(44.7214mm/s)k for vB.

(60i+287.75j60k)(vAj)=(60i+287.75j60k)(22.3607i44.7214k)287.75vA=1,341.642+2,683.284287.75vA=1,341.642vA=4.66mm/s

Calculate the relative velocity of collar A with respect to B (vA/B) as shown below.

vA/B=vAvB

Substitute (4.66mm/s)j for vA and (22.3607mm/s)i(44.7214mm/s)k for vB.

vA/B=(4.66mm/s)j((22.3607mm/s)i(44.7214mm/s)k)=(22.3607mm/s)i+(4.66mm/s)j(44.7214mm/s)k

(vA/B)2=(22.3607)2+(4.66)2+(44.7214)2=2,521.7(mm/s)2

The acceleration of collar B is aB=0.

The acceleration of collar A is aA=aAj.

Calculate the acceleration of collar A with respect to B (aA/B) as shown below.

aA/B=αA/B×rA/B+ωAB×vA/B

The acceleration αA/B×rA/B is perpendicular to rA/B. Hence, rA/B(αA/B×rA/B)=0.

Calculate the acceleration of collar A (aA) as shown below.

aA=aB+aA/B

Substitute αA/B×rA/B+ωAB×vA/B for aA/B and 0 for aB.

aA=0+αA/B×rA/B+ωAB×vA/B=αA/B×rA/B+ωAB×vA/B

Apply the dot product of rA/B on both sides of the Equation.

rA/BaA=rA/BαA/B×rA/B+rA/BωA/B×vA/B=rA/BαA/B×rA/B+vA/BrA/B×ωA/B

Substitute 0 for rA/BαA/B×rA/B and vA/B for rA/B×ωA/B.

rA/BaA=0+vA/B(vA/B)=(vA/B)2

Substitute (60mm)i+(287.75mm)j(60mm)k for rA/B, aAj for aA, and 2,521.7(mm/s)2 for (vA/B)2.

(60i+287.75j60k)aAj=2,521.7287.75aA=2,521.7aA=8.764mm/s2

Therefore, the acceleration of collar A is aA=(8.76mm/s2)j_.

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Chapter 15 Solutions

VECTOR MECHANICS FOR ENGINEERS W/CON >B

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