Two forces F̅₁ and F̅₂ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between F⃗₁ and F⃗₂ is cos ⁻¹(1/n). The value of |n| is _______.

Solution

Physics sir · 4 steps

Call the forces F and 3F, set the resultant equal to 3F, and the law of addition hands you cos θ.

1. Name the forces. Let the smaller force be F and the larger be 3F, with angle θ between them. The resultant equals the larger force, so R = 3F.

   R² = F₁² + F₂² + 2F₁F₂cos θ

2. Put in values. 9F² = F² + 9F² + 2(F)(3F)cos θ. The 9F² cancels on both sides, leaving 0 = F² + 6F²cos θ.
3. Solve for cos θ. cos θ = -1/6, so θ = cos⁻¹(-1/6). The negative sign tells you the angle is obtuse.
4. So. Comparing with cos⁻¹(1/n) gives n = -6, and |n| = 6.

The common slip: 3. Writing the cross term as F₁F₂cos θ instead of 2F₁F₂cos θ gives cos θ = -1/3 and |n| = 3, not 6.

The idea to keep. Whenever the resultant equals one of the forces, that force's square cancels in R² = F₁² + F₂² + 2F₁F₂cos θ. What is left fixes cos θ in one line.

Answer from the NTA final answer key. Solution by PrepWiser.