The cutting tool on the lathe exerts a force F on the shaft as shown. Determine the moment of this force about the y axis of the shaft.
Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.
Solution:
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The moment about the y-axis can be found by taking the dot product between the y-axis unit vector and the cross product of the position vector and force F. In other words:
M_y=u_y\cdot (r\times F)
The unit vector for the y-axis is: (0i+1j+0k)
The position vector is: (30\cos40^0i+0j+30\sin40^0k) mm
The force F is: (6i-4j-7k) kN
Thus, our moment is:
M_y=\begin{bmatrix}0&1&0\\30\cos40^0&0&30\sin40^0\\6&-4&-7\end{bmatrix}
M_y=276.57\,N\cdot mm
M_y=276.57\,N\cdot mm
Final Answer:
M_y=276.57\,N\cdot mm