Four forces are acting at a point P in equilibrium as shown in figure. The ratio of force F₁ to F₂ is 1: x where x= _____________.

Solution

Physics sir · 4 steps

Resolve the two slanted forces along x and y. In equilibrium each direction must balance on its own.

1. Resolve the slanted forces. Both slanted forces make 45° with the vertical, so each has equal horizontal and vertical parts. The 1 N force gives 1/√2 N to the right and 1/√2 N up. The 2 N force gives 2/√2 N to the left and 2/√2 N up.
2. Balance horizontally. F₁ and the 1 N part point right, and the 2 N part points left. So F₁ + 1/√2 = 2/√2, which gives F₁ = 1/√2 N.
3. Balance vertically. Both upward parts add, and only F₂ points down. So F₂ = 1/√2 + 2/√2 = 3/√2 N.

   F₁/F₂ = (1/√2)/(3/√2) = 1/3

4. So. F₁ : F₂ = 1 : 3, so x = 3.

The common slip: 1. Adding the horizontal parts of the 1 N and 2 N forces instead of subtracting them makes F₁ = 3/√2 N as well, so the ratio comes out 1 : 1 and x = 1.

The idea to keep. For a point in equilibrium, set the sum of x-components to zero and the sum of y-components to zero separately. Watch the sign of each component: left and right parts cancel, while parts in the same direction add.

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