(i) Define gravitational field strength.

(ii) State the SI unit for gravitational field strength.

A planet orbits the Sun in a circular orbit with orbital period T and orbital radius R. The mass of the Sun is M.

(i) Show that \(T = \sqrt {\frac{{4{\pi ^2}{R^3}}}{{GM}}} \).

(ii) The Earth’s orbit around the Sun is almost circular with radius 1.5×10¹¹ m. Estimate the mass of the Sun.

(i)  «gravitational» force per unit mass on a «small or test» mass 

(ii)  N kg^–1

Award mark if N kg^-1 is seen, treating any further work as neutral.
Do not accept bald m s^–2

i
clear evidence that v in \({v^2} = \frac{{4{\pi ^2}{R^2}}}{{{T^2}}}\) is equated to orbital speed \(\sqrt {\frac{{GM}}{R}} \)
OR
clear evidence that centripetal force is equated to gravitational force
OR
clear evidence that a in \(a = \frac{{{v^2}}}{R}\) etc is equated to g in \(g = \frac{{GM}}{{{R^2}}}\) with consistent use of symbols
Minimum is a statement that \(\sqrt {\frac{{GM}}{R}} \) is the orbital speed which is then used in \(v = \frac{{2\pi R}}{T}\)
Minimum is F_c = F_g ignore any signs.
Minimum is g = a.

substitutes and re-arranges to obtain result
Allow any legitimate method not identified here.
Do not allow spurious methods involving equations of shm etc

\( \ll T = \sqrt {\frac{{4{\pi ^2}R}}{{\left( {\frac{{GM}}{{{R^2}}}} \right)}}}  = \sqrt {\frac{{4{\pi ^2}{R^3}}}{{GM}}}  \gg \)

ii
«T = 365 × 24 × 60 × 60 = 3.15 × 10⁷ s»

\(M = \, \ll \frac{{4{\pi ^2}{R^3}}}{{G{T^2}}} =  \gg \,\, = \frac{{4 \times {{3.14}^2} \times {{\left( {1.5 \times {{10}^{11}}} \right)}^3}}}{{6.67 \times {{10}^{ - 11}} \times {{\left( {3.15 \times {{10}^7}} \right)}^2}}}\)
2×10³⁰«kg»

Allow use of 3.16 x 10⁷ s for year length (quoted elsewhere in paper).
Condone error in power of ten in MP1.
Award [1 max] if incorrect time used (24 h is sometimes seen, leading to 2.66 x 10³⁵ kg).
Units are not required, but if not given assume kg and mark POT accordingly if power wrong.
Award [2] for a bald correct answer.
No sf penalty here.