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๐ด๐ต๐ถ๐ท is a square with sides of length 28 cm. Two forces of magnitude 117 N act along ๐ด๐ต and ๐ถ๐ท, and two other forces of magnitude 177 N act along ๐ด๐ท and ๐ถ๐ต as shown in the figure. Find the algebraic measure of the moment of the resultant couple.

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Video Transcript

๐ด๐ต๐ถ๐ท is a square with sides of length 28 centimeters. Two forces of magnitude 117 newtons act along ๐ด๐ต and ๐ถ๐ท. And two other forces of magnitude 177 newtons act along ๐ด๐ท and ๐ถ๐ต, as shown in the figure. Find the algebraic measure of the moment of the resultant couple.

Given the four forces that are acting, one on each side of this square, we want to solve for the moment of the resultant couple. Weโ€™ll call that moment capital ๐‘€. To start solving for the moment ๐‘€, weโ€™ll consider these pairs of forces as pointing to the corners of our square, ๐ต and ๐ท. Regardless of which of these two corners we look at, we see thereโ€™s an opposition of forces going on. In each case, thereโ€™s a force of 177 newtons that tends to make our entire square rotate clockwise. And there are opposing forces of 117 newtons which tend to make the square rotate counterclockwise.

If the opposing forces were equal in magnitude, we would experience no net moment for this object. But because they are not equal, we do have a net moment ๐‘€. At both corners, ๐ต and ๐ท, the net force resulting from the opposition of the forces pointing into that corner is equal to 177 newtons minus 117 newtons, or 60 newtons. We could redraw our diagram to reflect this net force reality, where now weโ€™ve eliminated the 117 newton forces acting counterclockwise and only show the remaining net forces that act clockwise.

To solve for the moment ๐‘€, we want to take our net force and multiply it by a distance ๐‘‘, where ๐‘‘ is the perpendicular distance between the lines of action of our net or resultant forces. In our exercise setup, weโ€™re told that the length of the sides of the square is 28 centimeters. Thatโ€™s ๐‘‘. So weโ€™re now ready to plug in for ๐น net and ๐‘‘ and solve for ๐‘€. There is one last factor to include. And thatโ€™s the fact that weโ€™re told in our diagram that counterclockwise rotation of our shape is considered rotation in the positive direction. Our net forces, however, tend to rotation in the clockwise direction. That means weโ€™ll experience rotation in the negative direction. So weโ€™ll include a minus sign in our calculation of ๐‘€.

When we multiply these two numbers together, we find that ๐‘€ is negative 1680 newton centimeters. Thatโ€™s the moment of this force couple.

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