The Moment of a Force as a Vector Product

Returning to our consideration of the moment of a force, and referring to Fig. 1 (same as Fig. 1 of Section: The Definition of a Moment), we now see that we can write the moment vector as: $𝐌 = 𝐫 \times 𝐅$

Illustration for The Moment of a Force as a Vector Product — Fig. 1

By the definition of the vector product, this statement is equivalent to the definition of a moment given previously. The vector $𝐫$ joins the point about which the moment is to be taken and any point on the line of action of the force.

The components of the vector $𝐌$ can be written as follows: $𝐫 = x 𝐢 + y 𝐣 + z 𝐤$ $𝐅 = F_{x} 𝐢 + F_{y} 𝐣 + F_{z} 𝐤$ $𝐌 = 𝐫 \times 𝐅 = (y F_{z} - z F_{y}) 𝐢 + (z F_{x} - x F_{z}) 𝐣 + (x F_{y} - y F_{x}) 𝐤$ so that the components of $𝐌$ are: $M_{x} = y F_{z} - z F_{y}$ $M_{y} = z F_{x} - x F_{z}$ $M_{z} = x F_{y} - y F_{x}$

These are, of course, the same expressions obtained from Fig. 1. The introduction of the vector product makes possible a concise notation for moment vectors. Note that the sign convention for moments previously mentioned is consistent with the sign convention for vector products.

1.10.1 PROBLEMS

1. Show that two vectors $𝐚$ and $𝐛$ are perpendicular if $𝐚 \cdot 𝐛 = 0$ . What is the significance of $𝐚 \times 𝐛 = 0$ ?

2. Show that both the scalar product and the vector product obey the distributive law of ordinary multiplication, i.e., $(𝐚 + 𝐛) \cdot 𝐜 = 𝐚 \cdot 𝐜 + 𝐛 \cdot 𝐜$ $(𝐚 + 𝐛) \times 𝐜 = 𝐚 \times 𝐜 + 𝐛 \times 𝐜$

3. Given two vectors $𝐚 = 3 𝐢 + 2 𝐣 + 5 𝐤$ $𝐛 = 2 𝐢 + 𝐣$

Find the following vectors: $𝐚 + 𝐛; 𝐚 - 𝐛; 𝐚 \cdot 𝐛; 𝐚 \times 𝐛$

answer

$𝐚 + 𝐛 = 5 𝐢 + 3 𝐣 + 5 𝐤$

$𝐚 - 𝐛 = 𝐢 + 𝐣 + 5 𝐤$

$𝐚 \cdot 𝐛 = 8$

$𝐚 \times 𝐛 = - 5 𝐢 + 10 𝐣 - 𝐤$

4. Show that the vector product can be written as $𝐚 \times 𝐛 = | \begin{matrix} 𝐢 & 𝐣 & 𝐤 \\ a_{x} & a_{y} & a_{z} \\ b_{x} & b_{y} & b_{z} \end{matrix} |$

5. Given two vectors $𝐚 = 3 𝐢 + 2 𝐣 + 𝐤$ $𝐛 = 𝐢 + 3 𝐣 + 2 𝐤$ Find the angle between these vectors, using the definition of a scalar product.

answer

38° 11’

6. Find the moment about the point $𝐢 + 3 𝐣 + 2 𝐤$ of a force represented by the vector from the point $2 𝐢 + 𝐣 - 𝐤$ to the point $2 𝐢 - 𝐤$ at a scale of 10 lb per ft.

answer

$- 10 𝐤 + 30 𝐢$

7. A body is rotating about an axis with an angular speed of $ω$ radians per second. In vector notation $𝝎$ designates the angular velocity with magnitude $ω$ and with direction of the axis of rotation given by the right-hand screw rule. If $𝐯$ is the linear velocity of a point $A$ in the above-mentioned body and $𝐫$ is the position vector of the point $A$ with respect to any point of the rotation axis, show that, $𝐯 = 𝝎 \times 𝐫$

8. A body is rotating with an angular speed of 2 radians per second about an axis parallel to $3 𝐣 - 4 𝐤$ passing through the point $2 𝐢 + 𝐤$ ft. Referring to the result of the preceding problem, express vectorially the velocity of the point $3 𝐢 + 2 𝐤$ ft.

1.10.1 PROBLEMS

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