IB Questionbank

User interface language: English | Español

Date	May 2019	Marks available	2	Reference code	19M.3.HL.TZ2.22
Level	Higher level	Paper	Paper 3	Time zone	2
Command term	Show that	Question number	22	Adapted from	N/A

Question

The homogeneous model of the universe predicts that it may be considered as a spherical cloud of matter of radius r and uniform density ρ. Consider a particle of mass m at the edge of the universe moving with velocity v and obeying Hubble’s law.

Justify that the total energy of this particle is $E = \frac{1}{2}m{v^2} - \frac{4}{3}\pi G{\text{r}}{r^2}m$ .

[2]

At critical density there is zero total energy. Show that the critical density of the universe is: ${{\text{r}}_c} = \frac{{3H_0^2}}{{8\pi G}}$ .

[2]

The accepted value for the Hubble constant is 2.3 × 10⁻¹⁸ s⁻¹. Estimate the critical density of the universe.

[1]

Markscheme

total energy=kinetic energy+potential energy

total energy= $\frac{1}{2}m{v^2} - \frac{{GMm}}{r}$ ✔

substitution of M = $\frac{4}{3}\pi {r^3}\rho$ ✔

«Hence answer given»

Answer given so for MP2 look for clear evidence that M_Universe $\left( {\frac{4}{3}\pi {r^3}\rho } \right)$ is stated and substituted.

substitutes H₀r for v ✔

«total energy = 0»

$\frac{1}{2}mH_0^2{r^2} = \frac{4}{3}\pi G\rho {r^2}m$ ✔

«hence ρ_c= $\frac{{3H_0^2}}{{8\pi G}}$ »

Answer given, check working carefully.

9.5 × 10⁻²⁷ « kgm^–3» ✔

Examiners report

The vast majority of the candidates could state that the total energy is equal to the sum of the kinetic and potential energies but quite a few did not use the correct formula for the gravitational potential energy. The formula for the mass of the sun was usually correctly substituted.

This was a relatively easy demonstration given the equation in 22a. However many candidates did not show the process followed in a coherent manner that could be understood by examiners.

The question was well answered by many candidates.

Syllabus sections

Option D: Astrophysics » Option D: Astrophysics (Additional higher level option topics) » D.5 – Further cosmology (HL only)

Show 34 related questions

17M.3.HL.TZ1.17a:
The graph shows the variation with time t of the cosmic scale factor R in the flat model of the universe in which dark energy is ignored.

On the axes above draw a graph to show the variation of R with time, when dark energy is present.
16N.3.HL.TZ0.25b: Explain why the rotation curves are evidence for the existence of dark matter.
19M.3.HL.TZ2.22c: The accepted value for the Hubble constant is 2.3 × 10−18 s−1. Estimate the critical density...
17N.3.HL.TZ0.20b:
Suggest how fluctuations in the cosmic microwave background (CMB) radiation are linked to the observation that galaxies collide.
18N.3.HL.TZ0.20a: Outline what is meant by dark energy.
18N.3.HL.TZ0.21a:
Show that the temperature of the universe is inversely proportional to the cosmic scale factor.
18N.3.HL.TZ0.20b: State two candidates for dark matter.
19N.3.HL.TZ0.16b(iii): It is estimated that less than 20 % of the matter in the universe is observable. Discuss how...
17M.3.HL.TZ2.20c:
Curve A shows the actual rotation curve of a nearby galaxy. Curve B shows the predicted rotation curve based on the visible stars in the galaxy.

Explain how curve A provides evidence for dark matter.
18M.3.HL.TZ1.19a:
The mass of visible matter in the galaxy is M.

Show that for stars where r > R₀ the velocity of orbit is v = $\sqrt {\frac{{GM}}{r}}$ .
17M.3.HL.TZ2.20a:
Describe what is meant by dark matter.
17M.3.HL.TZ1.17b.i:
The density of the observable matter in the universe is only 0.05 ρ_c. Suggest how the remaining 0.27 ρ_c is accounted for.
19M.3.HL.TZ1.20b.ii:
Compare and contrast, the variation with time of the temperature of the cosmic background (CMB) radiation, for the two models from the present time onward.
18M.3.HL.TZ2.19a:
Explain the evidence that indicates the location of dark matter in galaxies.
17M.3.HL.TZ1.17b.ii:
The density of dark energy is ρ_Λc² where ρ_Λ = ρ_c – ρ_m. Calculate the amount of dark energy in 1 m³ of space.
19N.3.HL.TZ0.16b(i): State what is meant by the critical density.
16N.3.HL.TZ0.25a:
Calculate the rotation velocity of stars 4.0 kpc from the centre of the galaxy. The average density of the galaxy is 5.0 × 10^–21 kg m^–3.
18N.3.HL.TZ0.21b:
The present temperature of the cosmic microwave background (CMB) radiation is 3 K. Estimate the size of the universe relative to the present size of the universe when the temperature of the CMB was 300 K.
20N.3.HL.TZ0.21c: Hubble originally linked galactic redshift to a Doppler effect arising from galactic...
20N.3.HL.TZ0.24a: State the nature of the anisotropies observed in the CMB radiation.
19N.3.HL.TZ0.16b(ii):
Calculate the density of matter in the universe, using the Hubble constant 70 km s^–1Mpc^–1.
17M.3.HL.TZ2.19b:
State how the anisotropies in the CMB distribution are interpreted.
17N.3.HL.TZ0.20c:
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.
20N.3.HL.TZ0.24b:
Identify two possible causes of the anisotropies in (a).
18M.3.HL.TZ1.19b:
Draw on the axes the observed variation with r of the orbital speed v of stars in a galaxy.
17M.3.HL.TZ2.19a.i:
Derive, using the concept of the cosmological origin of redshift, the relation

T $\propto \frac{1}{R}$

between the temperature T of the cosmic microwave background (CMB) radiation and the cosmic scale factor R.
17M.3.HL.TZ2.19a.ii:
The present temperature of the CMB is 2.8 K. This radiation was emitted when the universe was smaller by a factor of 1100. Estimate the temperature of the CMB at the time of its emission.
19M.3.HL.TZ2.22a:
Justify that the total energy of this particle is $E = \frac{1}{2}m{v^2} - \frac{4}{3}\pi G{\text{r}}{r^2}m$ .
19M.3.HL.TZ1.20b.i: Draw, on the axes, a graph to show the variation with time of the cosmic scale factor R for...
19M.3.HL.TZ1.20a: Light from distant galaxies is redshifted. Explain the cosmological origin of this redshift.
18M.3.HL.TZ2.19b:
Outline why a hypothesis of dark energy has been developed.
17M.3.HL.TZ2.20b:
The distribution of mass in a spherical system is such that the density ρ varies with distance r from the centre as

ρ = $\frac{k}{{{r^2}}}$

where k is a constant.

Show that the rotation curve of this system is described by

v = constant.
18M.3.HL.TZ1.19c:
Explain, using the equation in (a) and the graphs, why the presence of visible matter alone cannot account for the velocity of stars when r > R₀.
21N.1.SL.TZ0.30: Which is correct for a black-body radiator? A. The power it emits from a unit surface area...

Hide related questions

Option D: Astrophysics » Option D: Astrophysics (Additional higher level option topics)

Option D: Astrophysics

Question

Markscheme

Examiners report

Syllabus sections

View options