Two forces F1 and F2 are acting at a point having resultant F if F2 is doubled F is also doubled if F2 is reversed then also F is doubled then find F1:F2:F
Answers
wo forces F1 and F2 are acting at a point having resultant F if F2 is doubled F is also doubled if F2 is reversed then also F is doubled then
F₁ : F₂ : F = (1/√3) : 1/2 : (1/√3)
F² = F₁² + F₂² + 2F₁F₂Cosθ - eq 1
θ = Angle between F₁ & F₂
if F2 is doubled F is also doubled
(2F)² = F₁² + (2F₂)² + 2F₁(2F₂)Cosθ
4F² = F₁² + 4F₂² + 4F₁F₂Cosθ - eq2
F2 is reversed then also F is doubled
(2F)² = F₁² + (-F₂)² + 2F₁(-F₂)Cosθ
4F² = F₁² + F₂² - 2F₁F₂Cosθ - eq 3
Adding eq 1 & eq 3
5F² = 2F₁² + 2F₂²
Equating 2 & 3
F₁² + 4F₂² + 4F₁F₂Cosθ = F₁² + F₂² - 2F₁F₂Cosθ
=> 3F₂² = -6F₁F₂Cosθ
=> F₂ = -2F₁Cosθ
=> Cosθ = -F₂/2F₁
putting these values in equations
F² = F₁² + F₂² + 2F₁F₂( -F₂/2F₁)
F² = F₁²
F = F₁
5F² = 2F₁² + 2F₂²
=> 5F₁² = 2F₁² + 2F₂²
=> 3F₁² = 2F₂²
=> √3 F₁ = √2 F₂
=> F₁ = √2/√3 F₂
F = F₁ = √2/√3 F₂
=> √3F = √3F₁ = √2F₂
=> √3F₁ = √2F₂ = √3F
Diving each by √3 *√2
=> 3F₁/√2 = F₂/√3 = F√2
F₁ : F₂ : F = √2 : √3 : √2
Answer:
hope this helps u....,
