Problem structure
(a) 2, numerical, density/volume/mass relationship.
(b) 2, numerical, gravitational field strength calculation.
(c)(i) 4, numerical, Kepler's 3rd Law derivation/application.
(c)(ii) 2, numerical, orbital velocity calculation.
(d) 4, numerical, ratio reasoning and orbital geometry.
Plan your route before writing. Use equations, diagrams, units, and a clear final justification where needed.
A hypothetical "Nuclear Star" is a dense, spherical object composed entirely of neutrons. It has a radius $R$ of $12.0\text{ km}$ and a uniform density equal to that of an atomic nucleus, $\rho = 2.30 \times 10^{17}\text{ kg m}^{-3}$.
(a)
2 marks
Show that the mass $M$ of the Nuclear Star is approximately $1.7 \times 10^{30}\text{ kg}$.
(b)
2 marks
Calculate the gravitational field strength $g$ at the surface of the star.
(c)
2 marks
A small satellite is in a stable circular orbit at a height of $4.0\text{ km}$ above the surface of the star.
(i) Calculate the orbital period $T$ of the satellite. [4]
(ii) Calculate the orbital speed $v$ of the satellite.
(d)
A second satellite is placed in a higher circular orbit such that its orbital period is exactly double that of the first satellite. Determine the height $h_2$ of this second satellite above the surface of the star. [4]
Data:
$G = 6.67 \times 10^{-11}\text{ N m}^2\text{ kg}^{-2}$
