# The cutting tool on the lathe exerts a force

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:

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:

$M_y=\begin{bmatrix}0&1&0\\30\cos40^0&0&30\sin40^0\\6&-4&-7\end{bmatrix}$

$M_y=276.57\,N\cdot mm$

$M_y=276.57\,N\cdot mm$