Two forces whose magnitudes are in the ratio 3 is to 5 give a resultant of 35N

Option 1 : F1 = 15N, F2 = 25N

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Option(1)

CONCEPT:

Force is a vector quantity that has both magnitude and direction, to calculate the magnitude of a vector quantity we apply vector law .
Triangle law of vector addition state that if two vectors can be represented both in magnitude and direction by the two sides of a triangle taken in the same order, then their resultant is represented completely, both in magnitude and direction, by the third side of the triangle taken in the opposite order.
- Vectors can be added geometrically by the vector law of addition.

Vector Law of Addition:

R = \(\sqrt{A^2+B^2 +2ABcosθ }\)

EXPLANATION:

If F1 and F2 are the two forces whose magnitude are in the ratio of 3 : 5 ,

Let assume F1 = 3x N and F2 = 5x Newton, R = 35, θ = 60°

we already know that R = \(\sqrt{A^2+B^2 +2ABcosθ }\)

35 = \(\sqrt{(3x)^2+(5x)^2 +2× 3x × 5x cosθ } \)

35 = 7x, or x = 35/7 = 5

hence F1 = 3x N = 3 × 5 = 15 N and F2 = 5x N = 5 × 5 = 25 N

Additional Information

Special case:

If two vectors \(\vec A\) and \(\vec B\) are acting along the same direction, θ = 0. Therefore the magnitude of the resultant is given by mathematically:
- \(R =\sqrt{A^2 + B^2 + 2ABcosθ }\) = \(\sqrt{A^2+ B^2 +2AB}\) = \(\sqrt{(A+ B)^2 } \)

R = A + B

If two vectors \(\vec A\) and \(\vec B\) are acting along the opposite direction, θ = 180°. Therefore the magnitude of the resultant is given by mathematically
- \(R =\sqrt{A^2 + B^2 + 2ABcosθ }\) = \(\sqrt{A^2+B^2+2ABcos180}=\sqrt{A^2+B^2-2AB}\)