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.

The cutting tool on the lathe exerts a force

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

 

Final Answer:

M_y=276.57\,N\cdot mm

 

This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 4, question 4-56.

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