The chisel exerts a force of 20 lb 1


The chisel exerts a force of 20 lb on the wood dowel rod which is turning in a lathe. Resolve this force into components acting (a) along the n and t axes and (b) along the x and y axes.

The chisel exerts a force of 20 lb

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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We will first draw the components of the 20 lb force along the n and t axes.

The chisel exerts a force of 20 lb

Using the sine and cosine functions, we can figure out the values F_n and F_t. Remember that sine is opposite over hypotneuse, while cosine is adjacent over hypotneuse.

\cos45^0\,=\,\dfrac{-F_n}{20}

F_n\,=\,-14.1 lb

(Note how the value is negative, this is because force F_n is heading towards the negative n-axis)

 

\sin45^0\,=\,\dfrac{F_t}{20}

F_t\,=\,14.1 lb

 

Now, we can focus on part (b) of the question. This time, we will draw our vector components along the x and y axes.

The chisel exerts a force of 20 lb on the wood

(Notice how we extended the 20 lb force. This visual representation makes it easier for us to calculate the values required)

The 15^0 angle can be found by realizing that if we extend the x-axis as shown in our diagram, it creates a semi-circle, which has an angle of 180^0. Our force therefore must be applied at an angle of 15^0 because (45+60+30+60)-180^0=15^0

As before, we can use sine and cosine functions to figure out the components along the x and y-axes.

\cos15^0\,=\,\dfrac{F_x}{20}

F_x\,=\,19.3 lb

 

\sin15^0\,=\,\dfrac{F_y}{20}

F_y\,=\,5.18 lb

 

Final Answers:

a)

F_n\,=\,-14.1 lb

F_t\,=\,14.1 lb

 

b)

F_x\,=\,19.3 lb

F_y\,=\,5.18 lb

 

This question can be found in Engineering Mechanics: Statics, 13th edition, chapter 2, question 2-25.

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