In being positioned on a milling machine table, a device is given the following displacements: 5.0 cm at 0 degrees, 12.0 cm at 80 degrees, 7.0 at 110 degrees, 9.0 at 210 degrees. Find the magnitude and angle of the resultant displacement.
Please give the best detail you can :) thank!
Please give the best detail you can :) thank!
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You need to break it down into "horizontal and vertical" components for each excursion, using 0 degrees as being vertical. Using a vertical type reference avoids confusion re directions, as you can use compass directions. They are more positive for identification. North is vertical.
So the first displacement is north, 5cm.
The second displacement is to 80 degrees for 12cm. That's 10 degrees north, relative to east.
North component = (sin 10) x 12, = 2.0832cm.
East component = (cos 10) x 12, = 11.8176cm.
The third displacement is to 110 degrees, for 7cm. That's 20 degrees south, relative to east.
East component = (sin 20) x 7, = 2.394cm.
South component = (cos 20) x 7, = 6.5779cm.
The final displacement is to 210 degrees, for 9cm. That's 30 degrees west, relative to south.
West component = (sin 30) x 9, = 4.5cm.
South component = (cos 30) x 9, = 7.794cm.
Total north displacement = (5 + 2.0832) = 7.0832cm.
Total east component = (11.8176 + 2.394) = 14.2116cm.
Total south component = (6.5779 + 7.794) = 14.3719cm.
Total west component = 4.5cm.
Net south displacement = (14.3719 - 7.0832) = 7.2887cm. south of origin.
Net east displacement = (14.2116 - 4.5) = 9.7116cm. east of origin.
Sketch the right triangle resulting.
Total displacement = sqrt. (9.7116^2 + 7.2887^2), = 12.1425cm., and angle = atn. (7.2887/9.7116) = 36.88 degrees, to the south of east, = (90 + 36.88) = 126.88 degrees from origin.
So the first displacement is north, 5cm.
The second displacement is to 80 degrees for 12cm. That's 10 degrees north, relative to east.
North component = (sin 10) x 12, = 2.0832cm.
East component = (cos 10) x 12, = 11.8176cm.
The third displacement is to 110 degrees, for 7cm. That's 20 degrees south, relative to east.
East component = (sin 20) x 7, = 2.394cm.
South component = (cos 20) x 7, = 6.5779cm.
The final displacement is to 210 degrees, for 9cm. That's 30 degrees west, relative to south.
West component = (sin 30) x 9, = 4.5cm.
South component = (cos 30) x 9, = 7.794cm.
Total north displacement = (5 + 2.0832) = 7.0832cm.
Total east component = (11.8176 + 2.394) = 14.2116cm.
Total south component = (6.5779 + 7.794) = 14.3719cm.
Total west component = 4.5cm.
Net south displacement = (14.3719 - 7.0832) = 7.2887cm. south of origin.
Net east displacement = (14.2116 - 4.5) = 9.7116cm. east of origin.
Sketch the right triangle resulting.
Total displacement = sqrt. (9.7116^2 + 7.2887^2), = 12.1425cm., and angle = atn. (7.2887/9.7116) = 36.88 degrees, to the south of east, = (90 + 36.88) = 126.88 degrees from origin.