Forces \(F_{1} (18N, 330°), F_{2} (10N, 090°)\) and \(F_{3} (25N, 180°)\) act on a body at rest. Find, correct to one decimal place, the magnitude and direc...
Question 1 Report
Forces \(F_{1} (18N, 330°), F_{2} (10N, 090°)\) and \(F_{3} (25N, 180°)\) act on a body at rest. Find, correct to one decimal place, the magnitude and direction of the resultant force.
Answer Details
Resolve each force into components (angles measured anticlockwise from the positive \(x\)-axis).
\(F_{1}=18\,\text{N}\) at \(330^{\circ}\): \(x=18\cos330^{\circ}=15.59\), \(y=18\sin330^{\circ}=-9.00\).
\(F_{2}=10\,\text{N}\) at \(090^{\circ}\): \(x=0\), \(y=10\).
\(F_{3}=25\,\text{N}\) at \(180^{\circ}\): \(x=-25\), \(y=0\).
Sum the components:
\[R_{x}=15.59+0-25=-9.41,\qquad R_{y}=-9+10+0=1.00\]
Magnitude:
\[|R|=\sqrt{(-9.41)^{2}+(1.00)^{2}}=\sqrt{89.55}\approx9.5\ \text{N}\]
Direction: \(R\) lies in the second quadrant. The reference angle is \(\tan^{-1}\dfrac{1}{9.41}=6.1^{\circ}\), so
\[\theta=180^{\circ}-6.1^{\circ}=173.9^{\circ}\ \text{(anticlockwise from the positive } x\text{-axis)}\]
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