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 = √(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