Question

Find the force F⁴ , so as to give the resultant of the force as shown in the figure given below. 

Given : Forces and their resultant
To find : Force F⁴

Answer 1

 

Assume that force F₄ acts at an angle θ 
Taking forces having direction towards right as positive and forces having direction upwards as positive.

Resolving forces along X direction :

`-F_1sin30 – F_2cos30 + F_3cos45 + F_4cosθ = Rcos50`

`-500sin30 – 300cos30 + 400cos45 + F_4cosθ = 800cos50`

F₄cosθ = 741.195 …………(1) 
Resolving forces along Y direction : 

`-F_1cos30 + F_2sin30 + F_3sin45 + F_4sinθ = -Rsin50 ` `-500cos30 + 300sin30 + 400sin45 +F_4sin θ = -800sin50 ` F₄sin θ = -612.6656 ……….(2)

Squaring and adding (1) and (2)

`(F_4sinθ)^2 + (F_4cos θ)^2 = (-612.6656)^2 + (741.195)^2`

`F_4^2(sin^2 θ + cos^2 θ) = 924729.1173`
`F_4 = 961.6284 N`

Dividing (2) by (1)

`(F_4sinθ)/(F_4cosθ) = (−612.6656)/(741.195) `

tan θ = -0.8266

θ = 39.5769^o (in fourth quadrant)  

F₄ = 961.6284 N ( at an angle 39.5769^o in fourth quadrant )