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.

#### 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