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=(F_y)/(F_x)\implies\boxed{\theta\approx81.79^\circ}`