Question

Two forces with magnitudes of 150 and 100 pounds act on an object at angles of 40° and 170°, respectively. Find the direction and magnitude of the resultant force. Round to two decimal places in all intermediate steps and in your final answer.

Answer 1

Denote the two force vectors by $\vec F_1$ and $\vec F_2$, respectively.

Compute the horizontal and vertical components:

$F_{1,x}=(150\,\mathrm{lb})\cos40^\circ\approx114.91\,\mathrm{lb}$

$F_{1,y}=(150\,\mathrm{lb})\sin40^\circ\approx96.42\,\mathrm{lb}$

$F_{2,x}=(100\,\mathrm{lb})\cos170^\circ\approx-98.49\,\mathrm{lb}$

$F_{2,y}=(100\,\mathrm{lb})\sin170^\circ\approx17.37\,\mathrm{lb}$

The resultant force $\vec F=\vec F_1+\vec F_2$ has components equal to the sum of the corresponding components of $\vec F_1,\vec F_2$:

$F_x=F_{1,x}+F_{2,x}\approx16.43\,\mathrm{lb}$

$F_y=F_{1,y}+F_{2,y}\approx113.78\,\mathrm{lb}$

The resultant force then has magnitude

$\|\vec F\|=\sqrt{{F_x}^2+{F_y}^2}\approx\boxed{114.96\,\mathrm{lb}}$

and direction $\theta$, where

$\tan\theta=\dfrac{F_y}{F_x}\implies\boxed{\theta\approx81.79^\circ}$