If θ = 60 ° \theta = 60 \degree θ = 60° and F = 450 N \textbf{F} = 450 \ \text{N} F = 450 N , determine the magnitude of the resultant force and its direction, measured counterclockwise from the positive x axis.
Engineering Mechanics: Statics 14th Edition by RC Hibbeler, Problem 2-1
Solution:
The parallelogram law and the triangulation rule are shown in the figures below.
(a) Parallelogram Law
(b) Triangulation Rule
Considering figure (b), we can solve for the magnitude of F R \textbf{F}_R F R using the cosine law.
F R = 70 0 2 + 45 0 2 − 2 ( 700 ) ( 450 ) cos 4 5 ∘ = 497.01 N = 497 N \begin{align*}
\textbf{F}_R & = \sqrt{700^2+450^2-2\left( 700 \right)\left( 450 \right)\cos45^{\circ}}\\
& = 497.01 \ \text{N}\\
& = 497 \ \text{N}
\end{align*} F R = 70 0 2 + 45 0 2 − 2 ( 700 ) ( 450 ) cos 4 5 ∘ = 497.01 N = 497 N
Then we use the sine law to solve for the interior angle θ \theta θ .
sin θ 700 = sin 4 5 ∘ 497.01 sin θ = 700 sin 4 5 ∘ 497.01 θ = sin − 1 ( 700 sin 4 5 ∘ 497.01 ) This is an ambiguous case θ = 84.8 1 ∘ o r θ = 95.1 9 ∘ \begin{align*}
\frac{\sin \theta}{700} & = \frac{\sin 45^{\circ}}{497.01}\\
\sin \theta & =\frac{700\ \sin 45^{\circ }}{497.01}\\
\theta & = \sin^{-1} \left( \frac{700\ \sin 45^{\circ }}{497.01} \right)\\
& \text{This is an ambiguous case }\\
\theta & = 84.81^\circ \ or \ \theta =95.19^\circ \\
\end{align*} 700 sin θ sin θ θ θ = 497.01 sin 4 5 ∘ = 497.01 700 sin 4 5 ∘ = sin − 1 ( 497.01 700 sin 4 5 ∘ ) This is an ambiguous case = 84.8 1 ∘ or θ = 95.1 9 ∘
In here, the correct angle measurement is θ = 95.1 9 ∘ \theta = 95.19^{\circ} θ = 95.1 9 ∘ .
Thus, the direction angle ϕ \phi ϕ of F R \textbf{F}_R F R measured counterclockwise from the positive x-axis, is
ϕ = θ + 6 0 ∘ = 95.1 9 ∘ + 6 0 ∘ = 15 5 ∘ \begin{align*}
\phi & = \theta +60^\circ \\
& = 95.19^\circ +60^\circ \\
& = 155^\circ
\end{align*} ϕ = θ + 6 0 ∘ = 95.1 9 ∘ + 6 0 ∘ = 15 5 ∘
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