Question

Two small balls of mass m each are suspended side by side by two equal threads of length L as shown in the figure. If the distance between the upper ends of the threads be a, the angle θ that the threads will make with the vertical due to attraction between the balls is

Answer 1

the answer is in tan inverse

Answer 2

The threads will make with the vertical due to attraction between the balls is  tan⁻¹ ((G × m)/(g × (a - x)²))

Explanation:

The force between the masses is given by the formula:

F = (G × m × m)/(a - x)²

⇒ F = (G × m²)/(a - x)² → (Equation 1)

Now,

ΣFx = 0

Tsin θ = F → (Equation 2)

ΣFy = 0

Tcos θ = mg → (Equation 3)

Now, the equation (2) becomes,

T = F/sin θ

On substituting equation (1) in above equation, we get,

T = ((G × m²)/(a - x)²)/sin θ = (G × m²)/((a - x)² × sin θ)

On substituting above equation in equation (3), we get,

(G × m²)/((a - x)² × sin θ) × cos θ = mg

cot θ × (G × m²)/(a - x)² = mg

cot θ × (G × m)/(a - x)² = g

cot θ = g × (a - x)²/(G × m)

tan θ = (G × m)/(g × (a - x)²)

∴ θ = tan⁻¹ ((G × m)/(g × (a - x)²))