DP Physics (last assessment 2024)

Question 17N.3.HL.TZ0.c

[Maximum mark: 3]

17N.3.HL.TZ0.c

Show that the critical density of the universe is

$\frac{{3{H^2}}}{{8\pi G}}$

where H is the Hubble parameter and G is the gravitational constant.

[3]

Markscheme

ALTERNATIVE 1

kinetic energy of galaxy $\frac{1}{2}$ mv² = $\frac{1}{2}$ mH²r² «uses Hubble’s law»

potential energy = $\frac{{GMm}}{r}$ = G $\frac{4}{3}$ $\pi$ r³ $\rho \frac{m}{r}$ «introduces density»

KE=PE to get expression for critical $\rho$

ALTERNATIVE 2

escape velocity of distant galaxy v = $\sqrt {\frac{{2GM}}{r}}$

where H₀r = $\sqrt {\frac{{2GM}}{r}}$

substitutes M = $\frac{4}{3}$ $\pi$ r³ $\rho$ to get result

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