Question

Which of the following concurrent forces cannot have a resultant of 4N?
1. 2 N and 4 N
2. 2 N and 6 N
3. 2 N and 8 N
4. All of these

Answer 1

Correct Answer - Option 3 : 2 N and 8 N

Concept:

Resultant force

\(F = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos θ} \)

Putting option (iii) and checking

F₁ = 2N, F₂ = 8 N

For maximum resultant force, cos θ = 1

F_max = \(F = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 } = \sqrt{(F_1 + F_2)^2} \)

F_max = F₁ + F₂

F_max = 2 + 8 = 10

For minimum resultant force, cos θ = -1

Fmax = \(F = \sqrt{F_1^2 + F_2^2 - 2 F_1 F_2 } = \sqrt{(F_1 - F_2)^2} \)

Fmax = F₁ - F2

Fmax = \(\sqrt{(2 - 8)^2} \) = 6 N

Here, we can see 4 N does not comes within the range of F_max and F_min

Similarly,

→ if we put option (i) F₁ = 2, F₂ = 4

F_max = F₁ + F₂ = 6 N

F_min = \(\sqrt{F_1 - F_2)^2} \) = 2N

Resultant 4 N is coming within the Range of F_max and F_min

→ if we put option (ii) F₁ = 2 N, F₂ = 6 N

Fmax = F1 + F2 = 8 N

Fmin = \(\sqrt{F_1 - F_2)^2} \) = 4 N

Resultant 4 N is coming within the Range of F_max and F_min