Question

The resultant of two forces P and Q is R. If Q is doubled, the new resultant is perpendicular to P. prove Q=R? [ do this question by triangle law ]

Answer 1

Resultant of two forces P and Q is R

R = P + Q

|R| = sqrt(P^2 + Q^2 + 2PQCosθ)

When Q is doubled new resultant R’ is

R’ = P + 2Q

Since it is perpendicular to P, dot product of R’ and P is zero

(R’).(P) = 0

(P + 2Q).(P) = 0

P^2 + 2PQCosθ = 0

(Dot product of P and P is P^2 and that of P and 2Q is 2PQCosθ)

Cosθ = -P/(2Q)

R = sqrt(P^2 + Q^2 + 2PQCosθ)

R = sqrt(P^2 + Q^2 + 2PQ*(-P/(2Q))

R = sqrt(P^2 + Q^2 - P^2)

R = Q