A pair of stars rotates about a common centre of mass. One star has mass M = 2m and the separation of their centres is d. (a) Find the orbital period T. (b) Find Lₘ/ $L_{M}$ , the ratio of the angular momentum of the mass-m star to that of the mass-M star. (c) Find Kₘ/ $K_{M}$ , the corresponding kinetic-energy ratio.
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Question

A pair of stars rotates about a common centre of mass. One star has mass M = 2m and the separation of their centres is d. (a) Find the orbital period T. (b) Find Lₘ/ $L_{M}$ , the ratio of the angular momentum of the mass-m star to that of the mass-M star. (c) Find Kₘ/ $K_{M}$ , the corresponding kinetic-energy ratio.

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G1 Physics · Accepted Answer

Step 1: Treat the two-star system as circular motion about the common centre of mass. The separation d obeys ω² = G(M₁+M₂)/d³. Step 2: Use r₁:r₂ = M₂:M₁ for the individual orbital radii; energy and angular momentum follow from v=ωr. Step 3: Apply the stated mass/radius change and simplify. Final: A: (a) T = 2π√(d³/(3Gm)); (b) Lₘ/Lₘₐⱼ = 2; (c) Kₘ/Kₘₐⱼ = 2.