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Two forces of magnitudes 3P and 2P act on a particle, and their resultant has magnitude R. When the first force is doubled (to 6P) while its direction is kept the same, the resultant doubles to 2R. Find the angle between the original two forces.

1. 60 deg
2. 120 deg
3. 150 deg
4. 90 deg

Correct answer: 120 deg

Solution

Let theta be the angle between the two forces. Case 1: R² = 9P² + 4P² + 12P² cos(theta) = 13P² + 12P² cos(theta) Case 2: (2R)² = 36P² + 4P² + 24P² cos(theta) = 40P² + 24P² cos(theta) From Case 2: 4R² = 40P² + 24P² cos(theta) From Case 1 times 4: 4R² = 52P² + 48P² cos(theta) Setting equal: 52P² + 48P² cos(theta) = 40P² + 24P² cos(theta) 12P² = -24P² cos(theta) cos(theta) = -1/2 theta = 120 deg.

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