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.
Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.
Solution:
We will first draw the components of the 20 lb force along the n and t axes.
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.
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.
(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.
F_x\,=\,19.3 lb
\sin15^0\,=\,\dfrac{F_y}{20}
F_y\,=\,5.18 lb
Final Answers:
F_n\,=\,-14.1 lb
F_t\,=\,14.1 lb
b)
F_x\,=\,19.3 lb
F_y\,=\,5.18 lb
good