Find the magnitude of the two forces such that if they act at right angles, their resultant is √10 kg. But if they act at 60°, their resultant is √13 kg. (Agra University, l973)

Solution:

Assuming the two forces to be P and Q, when the actions take place at 90 degrees, the resultant force R = √10 kg
But since right angles the resultant force must be, R = √(P^2 + Q^2), gives
√(P^2 + Q^2) = √10, squaring the sides, provides,
P^2 + Q^2 = 10 --------------------------------------------------------(1)
Now at 60 degrees, the resultant force

R = √13 kg, which prompts us to use the relation,
R = √(P^2 + Q^2 + 2PQ cos ?),
Therefore √13 = √(P^2 + Q^2 + 2PQ cos ?), again squaring the terms, gives,
13 = P^2 + Q^2 + 2PQ cos ?,
The value of P^2 + Q^2 = 10 when substituted in the above equation gives,
13 = 10 + 2PQ × 0.5,
Therefore, PQ = 3------------------------------------------------------(2)
Also since (P + Q)^2 = P^2 + Q^2 + 2PQ = 10 + 6 = 16,
Therefore, P + Q = 4---------------------------------------------------(3)
Identically, P – Q = 2---------------------------------------------------(4)
Calculating the above two expressions gives P = 3 kg and Q = 1 kg

### Article Written By Swagatam

Swagatam is a blogger at Expertscolumn.com

Last updated on 25-07-2016 148