Chapter 3.1, Problem 3.19P

(a)

To determine

The moment about origin $O$.

Answer to Problem 3.19P

The moment about origin $O$ is $\left(8\right)i+\left(-7\right)j+\left(-1\right)k$.

Explanation of Solution

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

$M_O=\left|r\times F\right|$ (I)

Here $M_O$ 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.

$F=F_{x}i+F_{y}j+F_{z}k$ (II)

Here, $F_x$ is the x component of the force vector, $F_y$ is the y component of the force vector and $F_z$ is the z component of the force vector.

Write an expression to represent position vector of point $A$.

$r=r_{x}i+r_{y}j+r_{z}k$ (III)

Here, $r_x$ is the x component of the position vector, $r_y$ is the y component of the position vector and $r_z$ is the z component of the position vector.

Write an expression to calculate the moment of force.

$r\times F=\left|\begin{array}{ccc}i& j& k\\ r_x& r_y& r_z\\ F_x& F_y& F_z\end{array}\right|$ (IV)

Conclusion:

Substitute $\left(-2\right)i+\left(-3\right)j+\left(5\right)k$ for $F$ and $\left(1\right)i+\left(1\right)j+\left(1\right)k$ for $r$ in equation (IV) to find $M_O$.

$\begin{array}{c}{M}_O=\left(\left(1\right)i+\left(1\right)j+\left(1\right)k\right)\times \left(\left(-2\right)i+\left(-3\right)j+\left(5\right)k\right)\\ =\left|\begin{array}{ccc}i& j& k\\ 1& 1& 1\\ -2& -3& 5\end{array}\right|\\ =\left(5+3\right)i+\left(-2-5\right)j+\left(-3+2\right)k\\ =\left(8\right)i+\left(-7\right)j+\left(-1\right)k\end{array}$

Thus, the moment about origin $O$ is $\left(8\right)i+\left(-7\right)j+\left(-1\right)k$.

(b)

To determine

The moment about origin $O$.

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 $A$.

$M_O=\left|r\times F\right|$ (V)

Write an expression to represent force vector.

$F=F_{x}i+F_{y}j+F_{z}k$ (VI)

Write an expression to represent position vector of point $A$.

$r=r_{x}i+r_{y}j+r_{z}k$ (VII)

Write an expression to calculate the moment of force.

$r\times F=\left|\begin{array}{ccc}i& j& k\\ r_x& r_y& r_z\\ F_x& F_y& F_z\end{array}\right|$ (VIII)

Conclusion:

Substitute $\left(-2\right)i+\left(-3\right)j+\left(5\right)k$ for $F$ and $\left(4\right)i+\left(6\right)j+\left(-10\right)k$ for $r$ in equation (VIII) to find $M_O$.

$\begin{array}{c}{M}_O=\left(\left(4\right)i+\left(6\right)j+\left(-10\right)k\right)\times \left(\left(-2\right)i+\left(-3\right)j+\left(5\right)k\right)\\ =\left|\begin{array}{ccc}i& j& k\\ 4& 6& -10\\ -2& -3& 5\end{array}\right|\\ =\left(30-30\right)i+\left(20-20\right)j+\left(-12+12\right)k\\ =0\end{array}$

Thus, the moment about origin $O$ is $0$.

(c)

To determine

The moment about origin $O$.

Answer to Problem 3.19P

The moment about origin $O$ is $\left(-10\right)j+\left(-6\right)k$.

Explanation of Solution

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

$M_O=\left|r\times F\right|$ (IX)

Write an expression to represent force vector.

$F=F_{x}i+F_{y}j+F_{z}k$ (X)

Write an expression to represent position vector of point $A$.

$r=r_{x}i+r_{y}j+r_{z}k$ (XI)

Write an expression to calculate the moment of force.

$r\times F=\left|\begin{array}{ccc}i& j& k\\ r_x& r_y& r_z\\ F_x& F_y& F_z\end{array}\right|$ (XII)

Conclusion:

Substitute $\left(-2\right)i+\left(-3\right)j+\left(5\right)k$ for $F$ and $\left(4\right)i+\left(3\right)j+\left(-5\right)k$ for $r$ in equation (IV) to find $M_O$.

$\begin{array}{c}{M}_O=\left(\left(4\right)i+\left(3\right)j+\left(-5\right)k\right)\times \left(\left(-2\right)i+\left(-3\right)j+\left(5\right)k\right)\\ =\left|\begin{array}{ccc}i& j& k\\ 4& 3& -5\\ -2& -3& 5\end{array}\right|\\ =\left(15-15\right)i+\left(10-20\right)j+\left(-12+6\right)k\\ =\left(-10\right)j+\left(-6\right)k\end{array}$

Thus, the moment about origin is $\left(-10\right)j+\left(-6\right)k$.