Answer to Question #240224 in Mechanical Engineering for Blaze

Question

2.4 A force F is applied at point A of the block with edge lengths of a, b, c Determine (1) the projections of F on the x, y and z axes; (2) the moments of F about x, y and z axes; (3) the moment of F about OB axis.

Answer 1

(1)

$$F_x=-|\vec F|\sin \beta\cos \alpha$$

$$F_y=-|\vec F|\cos \beta\cos \alpha$$

$$F_z=|\vec F|\sin \alpha$$

(2)

$$\vec F=(-|\vec F|\sin \beta\sin \alpha, -|\vec F|\sin \beta\cos \alpha, |\vec F|\cos \beta)$$

$$\vec r=(a, b, c)$$

$$\vec M_O=\vec r \times\vec F$$

$$=\begin{vmatrix} \vec i & \vec j & \vec k \\ a & b & c\\ -|\vec F|\sin \beta\cos \alpha & -|\vec F|\cos \beta\cos \alpha & |\vec F|\sin \alpha\\ \end{vmatrix}$$

$$=( |\vec F|b\sin \alpha+|\vec F|c\cos \beta\cos \alpha)\vec i$$

$$+(- |\vec F|a\sin \alpha-|\vec F|c\sin \beta\cos \alpha)\vec j$$

$$+(- |\vec F|a\cos \alpha\cos \beta+|\vec F|b\sin \beta\cos \alpha)\vec k$$

$$M_x=\vec i\cdot(\vec r\times\vec F)=$$

$$= |\vec F|b\sin \alpha+|\vec F|c\cos \beta\cos \alpha$$

$$M_y=\vec j\cdot(\vec r\times\vec F)=$$

$$=- |\vec F|a\sin \alpha-|\vec F|c\sin \beta\cos \alpha$$

$$M_z=\vec k\cdot(\vec r\times\vec F)=$$

$$- |\vec F|a\cos \alpha\cos \beta+|\vec F|b\sin \beta\cos \alpha$$

(3)

$$\overrightarrow{OB}=(a, 0, c))$$

$$\vec u=(\dfrac{a}{\sqrt{a^2+b^2}}, 0, \dfrac{c}{\sqrt{a^2+c^2}})$$

$$M_{OB}=\vec u\cdot(\vec r\times\vec F)=$$

$$=\dfrac{ |\vec F|ab\sin \alpha+|\vec F|ac\cos \beta\cos \alpha}{\sqrt{a^2+b^2}}$$

$$+\dfrac{- |\vec F|ac\cos \alpha\cos \beta+|\vec F|bc\sin \beta\cos \alpha}{\sqrt{a^2+b^2}}$$

$$=\dfrac{ |\vec F|ab\sin \alpha+|\vec F|bc\sin \beta\cos \alpha}{\sqrt{a^2+b^2}}$$