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The sum of the magnitudes of two forces acting at a point is 16 N the resultant of these forces is perpendicular to the smaller force has a magnitude of 8 N of the smaller force is magnitude x find the value of x?
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The sum of the magnitudes of two forces acting at a point is 16 N the ...
Given information:
- Sum of the magnitudes of two forces = 16 N
- Resultant of these forces is perpendicular to the smaller force
- Magnitude of the resultant force = 8 N
- Magnitude of the smaller force = x

To find:
The value of x.

Solution:

Let's assume that the magnitudes of the two forces are F1 and F2, where F1 is the smaller force.

Step 1: Express the given information mathematically:
- F1 + F2 = 16 N (Sum of the magnitudes of the two forces)
- F1^2 + F2^2 = 8^2 (Magnitude of the resultant force)

Step 2: Express one variable in terms of the other:
From the first equation, we can express F2 in terms of F1:
F2 = 16 N - F1

Step 3: Substitute the value of F2 in the second equation:
(F1)^2 + (16 N - F1)^2 = 8^2

Step 4: Simplify the equation:
Expanding and simplifying the equation:
F1^2 + (256 N^2 - 32 N F1 + F1^2) = 64
2F1^2 - 32 N F1 + 256 N^2 - 64 = 0

Step 5: Solve the quadratic equation:
Using the quadratic formula:
F1 = [32 N ± √((32 N)^2 - 4(2)(256 N^2 - 64))] / (2(2))
F1 = [32 N ± √(1024 N^2 - 2048 N^2 + 512)] / 4
F1 = [32 N ± √(-1024 N^2 + 512)] / 4

Since we are dealing with magnitudes, the value inside the square root cannot be negative. Therefore, we can disregard the negative sign:
F1 = [32 N + √(-1024 N^2 + 512)] / 4

Step 6: Simplify the expression:
F1 = [8 N + √(-256 N^2 + 128)] / 1
F1 = 8 N + √(-256 N^2 + 128)

Step 7: Substitute the given information to find the value of x:
Since F1 is the magnitude of the smaller force, we can equate F1 to x:
x = 8 N + √(-256 N^2 + 128)

Conclusion:
The value of x is 8 N + √(-256 N^2 + 128).
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The sum of the magnitudes of two forces acting at a point is 16 N the ...
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