Answer: P = 20/3 N.
Explanation:
Let, A = P, B = 2P
After the influence of two external forces,
A′ = P + 20, B′ = 2(2P) = 4P
There maybe a change in the resultant force, but the direction between the vectors or forces remain same. (ATQ.)
Angle between two vectors is given by:
tanα = B sinx/(A + B cosx)
In former case,
tanα = (2P sinx)/(P + 2P cosx) __(1)
After external forces,
tanα = (4P sinx)/(P + 20 + 4P cosx) __(2)
From equation 1 & 2:
(2P sinx)/(P + 2P cosx) = (4P sinx)/(P + 20 + 4P cosx)
=> 1/(P + 2P cosx) = 2/(P + 20 + 4P cosx)
=> 2(P + 2P cosx) = P + 20 + 4P cosx
=> 4P + 4P cosx = P + 20 + 4P cosx
=> 4P - P = 20
=> P = 20/3 N.
Therefore, P = 20/3 N.