The pole is subjected to the force F


The pole is subjected to the force F, which has components acting along the x, y, z axes as shown. If the magnitude of F is 3 kN, β = 30°, and γ = 75°, determine the magnitudes of its three components.

The pole is subjected to the force F

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.

Solution:

Let us first substitute the values given in the question into our diagram.

The pole is subjected to the force F

To calculate the value of \alpha we have to remember the following formula.

\text{cos}^2\alpha +\text{cos}^2\beta +\text{cos}^2\gamma =1

 

Let’s substitute the values we know. Thus, we have:

\text{cos}^2\alpha +\text{cos}^{2}(30^0)+\text{cos}^{2}(75^0) =1

Now we will solve for \alpha by isolating for \alpha:

\text{cos}^2\alpha=1-\text{cos}^{2}(30^0)-\text{cos}^{2}(75^0)

\text{cos}^2\alpha=0.183

(Take the square root of both sides)

\text{cos}\,\alpha=0.4278

\alpha=\text{cos}^{-1}(0.4278)

\alpha=64.67^0

To find the magnitude of the components, we simply substitute the angle values like so:

 

F_x=3\text{cos}\,64.67^0\,=\,1.28 kN

F_y=3\text{cos}\,30^0\,=\,2.60 kN

F_z=3\text{cos}\,75^0\,=\,0.766 kN

This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 2, question 2-84.

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