Two forces P and Q acting at a point have a resultant R. If Q be doubled, R is doubled. Again if the direction of Q is reversed, then R is doubled, show that

Problem Statement

Two forces P and Q acting at a point have a resultant R. If Q be doubled, R is doubled. Again if the direction of Q is reversed, then R is doubled. Show that P : Q : R = √2 : √3 : √2.

Step-by-Step Solution

Step 1: Initial Equation

R² = P² + Q² + 2PQ cos θ …(i)

Step 2: When Q is Doubled

(2R)² = P² + (2Q)² + 2P(2Q) cos θ
4R² = P² + 4Q² + 4PQ cos θ …(ii)

Step 3: When Q is Reversed

(2R)² = P² + (-Q)² + 2P(-Q) cos θ
4R² = P² + Q² – 2PQ cos θ …(iii)

Key Calculations

Adding (i) and (iii):

5R² = 2P² + 2Q² …(iv)

Multiplying (iii) by 2 and adding to (ii):

4R² = P² + 2Q² …(v)

Final Derivation

Subtracting (v) from (iv):

R² = P² ⇒ R = P

Substituting in (v):

3P² = 2Q² ⇒ Q = √(3/2)P

Final Ratio

P : Q : R = 1 : √3/2 : 1 = √2 : √3 : √2

Detailed Explanation

The solution uses the law of cosines for resultant forces. Through systematic elimination:

Three scenarios give different equations for the resultant
Combining equations eliminates θ and relates P, Q, R
Final substitution reveals the proportional relationship
Rationalization gives the ratio in simplest radical form

This demonstrates how force relationships can be solved through algebraic manipulation of vector equations.