Chapter 3.1, Problem 3.20P

Determine the moment about the origin O of the force F = 4i + 10j + 6k that acts at a point A. Assume that the position vector of A is (a) r = 2i – 3j + 4k, (b) r = 2i + 6j + 3k, (c) r = 2i + 5j + 6k.

To determine

The moment about origin $O$.

Answer to Problem 3.20P

The moment about origin $O$ is $\left(-58\right)i+\left(4\right)j+\left(32\right)k$.

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

$\begin{array}{c}{M}_O=\left(\left(2\right)i+\left(-3\right)j+\left(4\right)k\right)\times \left(\left(4\right)i+\left(10\right)j+\left(6\right)k\right)\\ =\left|\begin{array}{ccc}i& j& k\\ 2& -3& 4\\ 4& 10& 6\end{array}\right|\\ =\left(-18-40\right)i+\left(16-12\right)j+\left(20+12\right)k\\ =\left(-58\right)i+\left(4\right)j+\left(32\right)k\end{array}$

Thus, the moment about origin $O$ is $\left(-58\right)i+\left(4\right)j+\left(32\right)k$.

To determine

The moment about origin $O$.

Answer to Problem 3.20P

The moment about origin $O$ is $\left(6\right)i+\left(-4\right)k$.

Explanation of Solution

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

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

$\begin{array}{c}{M}_O=\left(\left(2\right)i+\left(6\right)j+\left(3\right)k\right)\times \left(\left(4\right)i+\left(10\right)j+\left(6\right)k\right)\\ =\left|\begin{array}{ccc}i& j& k\\ 2& 6& 3\\ 4& 10& 6\end{array}\right|\\ =\left(36-30\right)i+\left(12-12\right)j+\left(20-24\right)k\\ =\left(6\right)i+\left(-4\right)k\end{array}$

Thus, the moment about origin $O$ is $\left(6\right)i+\left(-4\right)k$.

To determine

The moment about origin $O$.

Answer to Problem 3.20P

The moment about origin $O$ is $\left(-30\right)i+\left(12\right)j$.

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

$\begin{array}{c}{M}_O=\left(\left(2\right)i+\left(5\right)j+\left(6\right)k\right)\times \left(\left(4\right)i+\left(10\right)j+\left(6\right)k\right)\\ =\left|\begin{array}{ccc}i& j& k\\ 2& 5& 6\\ 4& 10& 6\end{array}\right|\\ =\left(30-60\right)i+\left(24-12\right)j+\left(20-20\right)k\\ =\left(-30\right)i+\left(12\right)j\end{array}$

Thus, the moment about origin is $\left(-30\right)i+\left(12\right)j$.