Suppose at t=0, the spheres are in equilibrium and at rest. Forces are balanced along vertical and horizontal.
T cos α = mg T sin α = q²/(4π∈ x²)
tan α = α = sin α = x/2l = q²/(4π∈ mg x²) as x << l
q² = (2π∈ mg/l) x³
q = √(2π∈mg/l) x^3/2
assume that this equation is valid, when the sphere are moving. Differentiate
dq/dt = √(2π∈mg/l) * 3/2 * √x * dx/dt
= - 3a √(π∈mg/2l) as dx/dt = v = - a/√x
If more accurate expression is to be derived :
You have to do by find the acceleration and equating net force = mass * acceleration. Then the value of q and dq/dt are nearly same expressions
compared to the above.
let c = 3a √(π∈mg/2l) and b = a²l/2g
q = c √[x³ - b]
dq/dt = - c √[x³/(x³-b)]
if b ≈ 0, then both methods give same answer.