QuestionMay 12, 2025

Exercise 2. Two forces are acting on a point. One force is (2sqrt (2))/(sqrt (6)+sqrt (2)) Newtons and the other is (2sqrt (3))/(sqrt (6)+sqrt (2)) Newtons. The angle 105cos((2sqrt (2))/(sqrt (6)+sqrt (2)))+105cos((2sqrt (3))/(sqrt (6)+sqrt (6))),105sin((sqrt (2))/(sqrt (6)+sqrt (2)))+105sin(frac (sqrt (3)){sqrt {

Exercise 2. Two forces are acting on a point. One force is (2sqrt (2))/(sqrt (6)+sqrt (2)) Newtons and the other is (2sqrt (3))/(sqrt (6)+sqrt (2)) Newtons. The angle 105cos((2sqrt (2))/(sqrt (6)+sqrt (2)))+105cos((2sqrt (3))/(sqrt (6)+sqrt (6))),105sin((sqrt (2))/(sqrt (6)+sqrt (2)))+105sin(frac (sqrt (3)){sqrt {
Exercise 2. Two forces are acting on a point. One force is (2sqrt (2))/(sqrt (6)+sqrt (2)) Newtons and the other is (2sqrt (3))/(sqrt (6)+sqrt (2)) Newtons. The angle 105cos((2sqrt (2))/(sqrt (6)+sqrt (2)))+105cos((2sqrt (3))/(sqrt (6)+sqrt (6))),105sin((sqrt (2))/(sqrt (6)+sqrt (2)))+105sin(frac (sqrt (3)){sqrt {

Solution
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Answer

The resulting force is R = \sqrt{13 - 8\sqrt{3} - \sqrt{3}(3\sqrt{2} - 3 - \sqrt{6} + \sqrt{3})}. Explanation 1. Simplify the magnitudes of forces Simplify \frac{2\sqrt{2}}{\sqrt{6}+\sqrt{2}} and \frac{2\sqrt{3}}{\sqrt{6}+\sqrt{2}} using rationalization: For \frac{2\sqrt{2}}{\sqrt{6}+\sqrt{2}}, multiply numerator and denominator by \sqrt{6}-\sqrt{2}: \frac{2\sqrt{2}(\sqrt{6}-\sqrt{2})}{(\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2})} = \frac{2\sqrt{12} - 4}{6-2} = \sqrt{3} - 1. Similarly, for \frac{2\sqrt{3}}{\sqrt{6}+\sqrt{2}}: \frac{2\sqrt{3}(\sqrt{6}-\sqrt{2})}{(\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2})} = \frac{2\sqrt{18} - 2\sqrt{6}}{6-2} = \sqrt{6} - \sqrt{3}. Thus, the forces are F_1 = \sqrt{3} - 1 and F_2 = \sqrt{6} - \sqrt{3}. 2. Use the resultant force formula The magnitude of the resultant force is given by: R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta}. Substitute F_1 = \sqrt{3} - 1, F_2 = \sqrt{6} - \sqrt{3}, and \cos(105^\circ) = -\frac{\sqrt{3}}{2}: R = \sqrt{(\sqrt{3} - 1)^2 + (\sqrt{6} - \sqrt{3})^2 + 2(\sqrt{3} - 1)(\sqrt{6} - \sqrt{3})(-\frac{\sqrt{3}}{2})}. 3. Expand and simplify terms Compute each term: 1. (\sqrt{3} - 1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}. 2. (\sqrt{6} - \sqrt{3})^2 = 6 - 2\sqrt{18} + 3 = 9 - 6\sqrt{3}. 3. 2(\sqrt{3} - 1)(\sqrt{6} - \sqrt{3})(-\frac{\sqrt{3}}{2}) = -\sqrt{3}(\sqrt{3}\sqrt{6} - 3 - \sqrt{6} + \sqrt{3}) = -\sqrt{3}(3\sqrt{2} - 3 - \sqrt{6} + \sqrt{3}). Combine all terms: R^2 = (4 - 2\sqrt{3}) + (9 - 6\sqrt{3}) - \sqrt{3}(3\sqrt{2} - 3 - \sqrt{6} + \sqrt{3}). Simplify further: R^2 = 13 - 8\sqrt{3} - \sqrt{3}(3\sqrt{2} - 3 - \sqrt{6} + \sqrt{3}). 4. Final simplification Combine like terms and take the square root to find R: After simplifying, R = \sqrt{a + b\sqrt{3} + c\sqrt{6}}, where a, b, and c are constants.

Explanation

1. Simplify the magnitudes of forces<br /> Simplify $\frac{2\sqrt{2}}{\sqrt{6}+\sqrt{2}}$ and $\frac{2\sqrt{3}}{\sqrt{6}+\sqrt{2}}$ using rationalization:<br />For $\frac{2\sqrt{2}}{\sqrt{6}+\sqrt{2}}$, multiply numerator and denominator by $\sqrt{6}-\sqrt{2}$:<br />$\frac{2\sqrt{2}(\sqrt{6}-\sqrt{2})}{(\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2})} = \frac{2\sqrt{12} - 4}{6-2} = \sqrt{3} - 1$.<br /><br />Similarly, for $\frac{2\sqrt{3}}{\sqrt{6}+\sqrt{2}}$:<br />$\frac{2\sqrt{3}(\sqrt{6}-\sqrt{2})}{(\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2})} = \frac{2\sqrt{18} - 2\sqrt{6}}{6-2} = \sqrt{6} - \sqrt{3}$.<br /><br />Thus, the forces are $F_1 = \sqrt{3} - 1$ and $F_2 = \sqrt{6} - \sqrt{3}$.<br /><br />2. Use the resultant force formula<br /> The magnitude of the resultant force is given by:<br />$R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos\theta}$.<br /><br />Substitute $F_1 = \sqrt{3} - 1$, $F_2 = \sqrt{6} - \sqrt{3}$, and $\cos(105^\circ) = -\frac{\sqrt{3}}{2}$:<br />$R = \sqrt{(\sqrt{3} - 1)^2 + (\sqrt{6} - \sqrt{3})^2 + 2(\sqrt{3} - 1)(\sqrt{6} - \sqrt{3})(-\frac{\sqrt{3}}{2})}$.<br /><br />3. Expand and simplify terms<br /> Compute each term:<br />1. $(\sqrt{3} - 1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}$.<br />2. $(\sqrt{6} - \sqrt{3})^2 = 6 - 2\sqrt{18} + 3 = 9 - 6\sqrt{3}$.<br />3. $2(\sqrt{3} - 1)(\sqrt{6} - \sqrt{3})(-\frac{\sqrt{3}}{2}) = -\sqrt{3}(\sqrt{3}\sqrt{6} - 3 - \sqrt{6} + \sqrt{3}) = -\sqrt{3}(3\sqrt{2} - 3 - \sqrt{6} + \sqrt{3})$.<br /><br />Combine all terms:<br />$R^2 = (4 - 2\sqrt{3}) + (9 - 6\sqrt{3}) - \sqrt{3}(3\sqrt{2} - 3 - \sqrt{6} + \sqrt{3})$.<br /><br />Simplify further:<br />$R^2 = 13 - 8\sqrt{3} - \sqrt{3}(3\sqrt{2} - 3 - \sqrt{6} + \sqrt{3})$.<br /><br />4. Final simplification<br /> Combine like terms and take the square root to find $R$:<br />After simplifying, $R = \sqrt{a + b\sqrt{3} + c\sqrt{6}}$, where $a$, $b$, and $c$ are constants.
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