Problem 4 - An Introduction to the Mechanics of Solids in SI Units

Question

According to the distributive law for vector cross products $$ \mathbf{r} \times \mathbf{F}_{1}+\mathbf{r} \times \mathbf{F}_{2}=\mathbf{r} \times\left(\mathbf{F}_{1}+\mathbf{F}_{2}\right) $$ This states that the sum of the moments of two concurrent forces about a point is equal to the moment of the vector sum of the forces about the same point. Verify this by simple geometry in the special case where \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\) lie in the \(x y\) plane and intersect at the point \(P\), and \(\mathbf{r}\) is the displacement vector \(O P\).

Step-by-Step Solution

Answer

With the diagram and the steps followed, it is visually and arithmetically demonstrated the validity of the distributive law for vector cross products in the context of moments of forces about a point.

Step 1: Set up the scenario using a diagram

Draw a Cartesian coordinate system in three dimensions. On the xy plane, define two vectors \(\mathbf{F}_1\) and \(\mathbf{F}_2\) that intersect at point P. Also, draw the displacement vector \(\mathbf{r}\) from origin O to point P.

Step 2: Calculate the individual cross products

Calculate \(\mathbf{r} \times \(\mathbf{F}_1\) and \(\mathbf{r} \times \(\mathbf{F}_2\). In a cross product, the resultant vector is orthogonal (perpendicular) to the plane containing the original vectors. Therefore, these cross products yield vectors that are perpendicular to the xy plane.

Step 3: Compute the vector sum of the two forces and its cross product with \(\mathbf{r}\)

Next, add together the two force vectors \(\mathbf{F}_1\) and \(\mathbf{F}_2\) to get the vector sum \(\mathbf{F}_1 + \mathbf{F}_2\). Then, calculate the cross product \(\mathbf{r} \times (\mathbf{F}_1 + \mathbf{F}_2)\) using the same principles described in Step 2.

Step 4: Compare the results

The final step is to compare the results of Step 2 and Step 3. According to the given distributive law, the sum of the moments (which are the cross products) of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) about point P should be equal to the moment of the vector sum of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) about the same point. If the vectors obtained in Step 2 and Step 3 are equivalent, this verifies the law.