Vector Mechanics for Engineers: Statics and Dynamics
Vector Mechanics for Engineers: Statics and Dynamics
12th Edition
ISBN: 9781259638091
Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self
Publisher: McGraw-Hill Education
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Chapter 3.1, Problem 3.19P

(a)

To determine

The moment about origin O.

(a)

Expert Solution
Check Mark

Answer to Problem 3.19P

The moment about origin O is (8)i+(7)j+(1)k.

Explanation of Solution

Write an expression to calculate the moment of the force at the origin O due to the force acting at point A.

MO=|r×F| (I)

Here MO is the magnitude of the moment of the force at the origin O due to the force acting at point A, r is the point vector of point A directing from the origin and F is the force vector acting at point A.

Write an expression to represent force vector.

F=Fxi+Fyj+Fzk (II)

Here, Fx is the x component of the force vector, Fy is the y component of the force vector and Fz is the z component of the force vector.

Write an expression to represent position vector of point A.

r=rxi+ryj+rzk (III)

Here, rx is the x component of the position vector, ry is the y component of the position vector and rz is the z component of the position vector.

Write an expression to calculate the moment of force.

r×F=|ijkrxryrzFxFyFz| (IV)

Conclusion:

Substitute (2)i+(3)j+(5)k for F and (1)i+(1)j+(1)k for r in equation (IV) to find MO.

MO=((1)i+(1)j+(1)k)×((2)i+(3)j+(5)k)=|ijk111235|=(5+3)i+(25)j+(3+2)k=(8)i+(7)j+(1)k

Thus, the moment about origin O is (8)i+(7)j+(1)k.

(b)

To determine

The moment about origin O.

(b)

Expert Solution
Check Mark

Answer to Problem 3.19P

The moment about origin O is 0.

Explanation of Solution

Write an expression to calculate the moment of the force at the origin O due to the force acting at point A.

MO=|r×F| (V)

Write an expression to represent force vector.

F=Fxi+Fyj+Fzk (VI)

Write an expression to represent position vector of point A.

r=rxi+ryj+rzk (VII)

Write an expression to calculate the moment of force.

r×F=|ijkrxryrzFxFyFz| (VIII)

Conclusion:

Substitute (2)i+(3)j+(5)k for F and (4)i+(6)j+(10)k for r in equation (VIII) to find MO.

MO=((4)i+(6)j+(10)k)×((2)i+(3)j+(5)k)=|ijk4610235|=(3030)i+(2020)j+(12+12)k=0

Thus, the moment about origin O is 0.

(c)

To determine

The moment about origin O.

(c)

Expert Solution
Check Mark

Answer to Problem 3.19P

The moment about origin O is (10)j+(6)k.

Explanation of Solution

Write an expression to calculate the moment of the force at the origin O due to the force acting at point A.

MO=|r×F| (IX)

Write an expression to represent force vector.

F=Fxi+Fyj+Fzk (X)

Write an expression to represent position vector of point A.

r=rxi+ryj+rzk (XI)

Write an expression to calculate the moment of force.

r×F=|ijkrxryrzFxFyFz| (XII)

Conclusion:

Substitute (2)i+(3)j+(5)k for F and (4)i+(3)j+(5)k for r in equation (IV) to find MO.

MO=((4)i+(3)j+(5)k)×((2)i+(3)j+(5)k)=|ijk435235|=(1515)i+(1020)j+(12+6)k=(10)j+(6)k

Thus, the moment about origin is (10)j+(6)k.

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Chapter 3 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

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