the sum of the magnitude of two forces acting at a point is 18 and the magnitude of their resultant is 12.if the resultant is at 90 degree with the force of the smaller magnitude, then their magnitudes are:
F2^2 + 12^2 = F1^2 right triangle, F2 smaller
F1 = (18 - F2)
F2^2 + 144 = 324 -36 F2 + F2^2
36 F2 = 180
F2 = 5
then F1 = 18-5 = 13
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To solve this problem, we can use the principles of vector addition and trigonometry.
Let's denote the magnitude of the smaller force as F1 and the magnitude of the larger force as F2.
Given that the sum of the magnitudes of the two forces is 18, we can write the equation:
F1 + F2 = 18 (Equation 1)
We are also told that the magnitude of their resultant is 12 and is at a 90-degree angle with the smaller force. This situation indicates that the forces are acting perpendicularly, which means they can be represented by the sides of a right-angled triangle.
Using the Pythagorean theorem, we can establish the relationship:
F1² + F2² = 12² (Equation 2)
To solve this system of equations, we can use substitution. Rearrange Equation 1 to express F1 in terms of F2:
F1 = 18 - F2
Substitute this value of F1 into Equation 2:
(18 - F2)² + F2² = 12²
Expanding and simplifying:
324 - 36F2 + F2² + F2² = 144
Combine like terms:
2F2² - 36F2 + 324 - 144 = 0
2F2² - 36F2 + 180 = 0
Divide the entire equation by 2 to simplify:
F2² - 18F2 + 90 = 0
This quadratic equation can be factored as:
(F2 - 6)(F2 - 15) = 0
Setting each factor equal to zero:
F2 - 6 = 0 or F2 - 15 = 0
Solving for F2:
F2 = 6 or F2 = 15
Now that we have the values for F2, we can substitute them back into Equation 1 to find F1:
When F2 = 6: F1 = 18 - F2 = 18 - 6 = 12
When F2 = 15: F1 = 18 - F2 = 18 - 15 = 3
Therefore, the magnitudes of the forces are F1 = 12 and F2 = 6, or F1 = 3 and F2 = 15, depending on which force is smaller.