Question

A mass M is broken into two parts of masses m₁ and m₂. For the force of attraction or gravitation to be maximum between them, the m₁ and m₂ have value

(a) m₁ = \(\frac{M}{3}\), m₂ = \(\frac{2M}{3}\)

(b) m₁ = \(\frac{2M}{3}\), m₂ = \(\frac{M}{3}\)

(c) m₁ = m₂ = \(\frac{M}{2}\)

(d) m₁ = \(\frac{M}{4}\), m₂ = \(\frac{3M}{4}\)

Answer 1

Answer : (c) m₁ = m₂ = \(\frac{M}{2}\)

Let m₁ = m and m₂ = M - m, 

Then,

F = \(\frac{Gm(M-m)}{r^2}\) 

= \(\frac{G}{r^2}\) (Mm - m²)

For maximum force,

\(\frac{dF}{dm}\) = 0

⇒ \(\frac{G}{r^2}\)(M - 2m) = 0

⇒ M = 2m or m = \(\frac{M}{2}\) 

∴ m₁ = m₂ = \(\frac{M}{2}\)