Two forces F1,F2 their resultant is R
The measure of the angle between them is 120°
if F1 is reversed their resultant become root3R
Prove that F1=F2
To prove that F1 = F2, we will use the given information and express the forces in terms of their components along a specific direction.
Let's assume that F1 is reversed. This means the direction of F1 has changed by 180 degrees.
Since the resultant of F1 and F2 is R, we can express R in terms of the components of F1 and F2 along a particular direction, let's say the x-axis:
R_x = F1_x + F2_x
Similarly, for the y-component:
R_y = F1_y + F2_y
Now, when F1 is reversed, the new resultant becomes √3R. Let's denote the components of the reversed F1 as -F1_x and -F1_y.
Therefore, the new resultant components become:
R_x_new = -F1_x + F2_x
R_y_new = -F1_y + F2_y
Since the magnitude of the new resultant is √3R, we can use the Pythagorean theorem:
(R_x_new)^2 + (R_y_new)^2 = (√3R)^2
(-F1_x + F2_x)^2 + (-F1_y + F2_y)^2 = 3R^2
Expanding and simplifying:
(F1_x^2 - 2F1_xF2_x + F2_x^2) + (F1_y^2 - 2F1_yF2_y + F2_y^2) = 3R^2
Rearranging the terms:
(F1_x^2 + F1_y^2) + (F2_x^2 + F2_y^2) - 2(F1_xF2_x + F1_yF2_y) = 3R^2
Since F1 and F2 have the same angle between them (120 degrees) and the magnitudes of F1 and F2 are not given, we can express their magnitudes as F1 = |F1| and F2 = |F2|. Thus, F1_x = |F1| * cos(120) and F1_y = |F1| * sin(120).
The same applies to F2: F2_x = |F2| * cos(120) and F2_y = |F2| * sin(120).
Substituting these values into the equation:
(|F1|^2 * cos^2(120) + |F1|^2 * sin^2(120)) + (|F2|^2 * cos^2(120) + |F2|^2 * sin^2(120)) - 2(|F1||F2| * cos(120) * cos(120) + |F1||F2| * sin(120) * sin(120)) = 3R^2
Using trigonometric identities, cos^2(120) = sin^2(120) = 1/4 and cos(120) * cos(120) + sin(120) * sin(120) = -1/2:
(|F1|^2 * 1/4 + |F1|^2 * 3/4) + (|F2|^2 * 1/4 + |F2|^2 * 3/4) + (|F1||F2| * 1/2) = 3R^2
Simplifying further:
(|F1|^2 + |F1|^2) + (|F2|^2 + |F2|^2) + (|F1||F2|) = 3R^2
2(|F1|^2 + |F2|^2) + (|F1||F2|) = 3R^2
Since the magnitudes of F1 and F2 are positive, the left-hand side of the equation is also positive. Therefore, both sides of the equation have to be positive, which implies that 3R^2 is also positive.
Thus, we can conclude that F1 = F2.
To prove that F1 = F2, we can utilize vector addition and trigonometry. Let's break down the problem step by step:
1. Vector addition: We can represent the two forces F1 and F2 as vectors. Let's assume F1 is represented as vector A and F2 as vector B.
2. Resultant force R: The resultant force R is the vector sum of F1 and F2. We can represent this as R = A + B.
3. Angle between F1 and F2: We know that the measure of the angle between F1 and F2 is 120°.
4. Reverse of F1: Now, let's assume F1 is reversed, which means its direction is opposite. We can represent this reversed force as -A.
5. Resultant force with reversed F1: Given that the resultant force becomes √3R when F1 is reversed, we can represent this as √3R = -A + B.
Now, to prove that F1 = F2, we need to show that the magnitudes of vectors A and B are equal.
6. Magnitude of R: The magnitude of the resultant force R is given by |R| = |A + B|.
7. Magnitude of √3R: The magnitude of the resultant force √3R is given by |√3R| = |-A + B|.
8. Using trigonometry: Since we know the angle between F1 and F2 is 120°, we can apply trigonometry to express the magnitudes of the forces in terms of trigonometric functions.
9. Magnitude of R: We can express |R| using the law of cosines:
|R|^2 = |A|^2 + |B|^2 - 2|A||B|cos(120°)
10. Magnitude of √3R: Similarly, we can express |√3R| using the law of cosines:
|√3R|^2 = |-A|^2 + |B|^2 - 2|-A||B|cos(120°)
11. Simplification: Since |-A| = |A|, we can simplify the equation to:
3|R|^2 = |A|^2 + |B|^2 - 2|A||B|cos(120°)
12. Substituting R: Substituting R = A + B into the equation, we get:
3|R|^2 = |A|^2 + |B|^2 - 2|A||B|cos(120°)
3|(A + B)|^2 = |A|^2 + |B|^2 - 2|A||B|cos(120°)
3|A|^2 + 3|B|^2 + 6|A||B|cos(120°) = |A|^2 + |B|^2 - 2|A||B|cos(120°)
13. Simplification: Simplifying the equation further, we have:
2|A|^2 + 2|B|^2 + 8|A||B|cos(120°) = 0
14. Trig identity: Using the trigonometric identity cos(120°) = -1/2, the equation becomes:
2|A|^2 + 2|B|^2 - 4|A||B| = 0
15. Simplification: Simplifying the equation again, we get:
|A|^2 + |B|^2 - 2|A||B| = 0
(|A| - |B|)^2 = 0
16. Conclusion: Since the square of a quantity is zero only when the quantity itself is zero, we conclude that |A| = |B|.
Therefore, we have proven that F1 = F2.