Question 2.160: The z component of the force F is 80 lb. (a) Express F in te......
The z component of the force F is 80 lb. (a) Express F in terms of components. (b) what are the angles θ_x,~θ_y,~and~θ_z between F and the positive coordinate axes?
The z component of the force F is 80 lb.
Step 1:
Find the magnitude of the force. We are given the z component of the force as 80 lb. Using this information, we can use the equation |F| cos 20° cos 60° = 80 lb to solve for the magnitude of the force, denoted as |F|. By rearranging the equation, we get |F| = 170 lb.
Step 2:
Find the direction of the force. To find the direction of the force, we need to determine the direction cosines. The direction cosines can be calculated using the equation θ = cos^(-1)(component/magnitude). We can calculate the direction cosines for each component of the force vector.
For the x component: θ_x = cos^(-1)(139/170) = 35.5°
For the y component: θ_y = cos^(-1)(58.2/170) = 70.0°
For the z component: θ_z = cos^(-1)(80/170) = 62.0°
Step 3:
Summarize the results. The force vector can be expressed as F = (139i + 58.2j + 80k) lb. The magnitude of the force is |F| = 170 lb. The direction of the force can be described by the direction cosines: θ_x = 35.5°, θ_y = 70.0°, θ_z = 62.0°.
By following these steps, we have determined the magnitude and direction of the given force vector.
Final Answer
We can write the force as
F =| F |\left(\cos 20^{\circ} \sin 60^{\circ} i +\sin 20^{\circ} j +\cos 20^{\circ} \cos 60^{\circ} k \right)We know that the z component is 80 lb. Therefore
|F| cos 20° cos 60° = 80 lb ⇒ |F| = 170 lb
(a) F = (139i + 58.2j + 80k) lb
(b) The direction cosines can be found:
\begin{aligned}& \theta_x=\cos ^{-1}\left(\frac{139}{170}\right)=35.5^{\circ} \\& \theta_y=\cos ^{-1}\left(\frac{58.2}{170}\right)=70.0^{\circ} \\& \theta_z=\cos ^{-1}\left(\frac{80}{170}\right)=62.0^{\circ} \\& \theta_x=35.5^{\circ}, \theta_y=70.0^{\circ}, \theta_z=62.0^{\circ}\end{aligned}Related Answered Questions
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