Question

Two forces with magnitudes of 150 and 75 pounds act on an object at angles of 30° and 150° respectively. Find the direction and magnitude of the resultant force. Round to two decimal places in all intermediate steps and in your final answer.


Help me please show work please so that can make me more understand please

Answer 1

Answer on Question #42418 – Math – Analytical Geometry

Question.

Two forces with magnitudes of 150 and 75 pounds act on an object at angles of $30{}^{\circ}$ and $150{}^{\circ}$ respectively. Find the direction and magnitude of the resultant force. Round to two decimal places in all intermediate steps and in your final answer.

Given:

$F_{1} = 150$ pounds

$F_{2} = 75$ pounds

$\alpha_{1} = 30{}^{\circ}$

$\alpha_{2} = 150{}^{\circ}$

Find:

$F = ?$ is the magnitude of resultant force

$\alpha = ?$ is the direction of resultant force

Solution.

In the general case, the direction of the resultant force is determined by the parallelogram rule. But since $\alpha_{1} = \pi -\alpha_{2}$ we can project forces $F_{1}$ and $F_{2}$ on y-axis and it's will be resultant force:

$\alpha = \alpha_{1} + \frac{\alpha_{2} - \alpha_{1}}{2} = 30{}^{\circ} + 60{}^{\circ} = 90{}^{\circ}$

$F = F_{1y} + F_{2y} = F_{1}\cos \left(\alpha_{1} - \frac{\pi}{2}\right) + F_{2}\cos \left(\frac{\pi}{2} -\alpha_{2}\right)$

$F = F_{1}\cos \left(\alpha_{1} - \frac{\pi}{2}\right) + F_{2}\cos \left(\frac{\pi}{2} -\alpha_{2}\right) = F_{1}\cos (60{}^{\circ}) + F_{2}\cos (60{}^{\circ}) = (F_{1} + F_{2})\cos (60{}^{\circ})$

Calculate:

$F = (150 + 75)\cdot \frac{1}{2} = \frac{225}{2} = 112.5$ pounds

Answer.

The magnitude of resultant force is $F = 112.5$ pounds

The direction of resultant force $\alpha = 90{}^{\circ}$

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