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Date May 2017 Marks available 1 Reference code 17M.2.HL.TZ2.8
Level Higher level Paper Paper 2 Time zone Time zone 2
Command term Show that Question number 8 Adapted from N/A

Question

The gravitational potential due to the Sun at its surface is –1.9 x 1011 J kg–1. The following data are available.

Mass of Earth = 6.0 x 1024 kg
Distance from Earth to Sun = 1.5 x 1011 m
Radius of Sun = 7.0 x 108 m

Outline why the gravitational potential is negative.

[2]
a.

The gravitational potential due to the Sun at a distance r from its centre is VS. Show that

rVS = constant.

[1]
b.i.

Calculate the gravitational potential energy of the Earth in its orbit around the Sun. Give your answer to an appropriate number of significant figures.

[2]
b.ii.

Calculate the total energy of the Earth in its orbit.

[2]
b.iii.

An asteroid strikes the Earth and causes the orbital speed of the Earth to suddenly decrease. Suggest the ways in which the orbit of the Earth will change.

[2]
b.iv.

Outline, in terms of the force acting on it, why the Earth remains in a circular orbit around the Sun.

[2]
c.

Markscheme

potential is defined to be zero at infinity

so a positive amount of work needs to be supplied for a mass to reach infinity

a.

VS = \( - \frac{{GM}}{r}\) so r x VS «= –GM» = constant because G and M are constants

b.i.

GM = 1.33 x 1020 «J m kg–1»

GPE at Earth orbit «= –\(\frac{{1.33 \times {{10}^{20}} \times 6.0 \times {{10}^{24}}}}{{1.5 \times {{10}^{11}}}}\)» = «–» 5.3 x 1033 «J»

 

Award [1 max] unless answer is to 2 sf.

Ignore addition of Sun radius to radius of Earth orbit.

b.ii.

ALTERNATIVE 1
work leading to statement that kinetic energy \(\frac{{GMm}}{{2r}}\), AND kinetic energy evaluated to be «+» 2.7 x 1033 «J»

energy «= PE + KE = answer to (b)(ii) + 2.7 x 1033» = «–» 2.7 x 1033 «J»

 

ALTERNATIVE 2
statement that kinetic energy is \( =  - \frac{1}{2}\) gravitational potential energy in orbit

so energy «\( = \frac{{{\text{answer to (b)(ii)}}}}{2}\)» = «–» 2.7 x 1033 «J»

 

Various approaches possible.

b.iii.

«KE will initially decrease so» total energy decreases
OR
«KE will initially decrease so» total energy becomes more negative

Earth moves closer to Sun

new orbit with greater speed «but lower total energy»

changes ellipticity of orbit

b.iv.

centripetal force is required

and is provided by gravitational force between Earth and Sun

 

Award [1 max] for statement that there is a “centripetal force of gravity” without further qualification.

c.

Examiners report

[N/A]
a.
[N/A]
b.i.
[N/A]
b.ii.
[N/A]
b.iii.
[N/A]
b.iv.
[N/A]
c.

Syllabus sections

Additional higher level (AHL) » Topic 10: Fields » 10.2 – Fields at work
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