IB Physics HL D 1 Gravitational Fields Questions

Practise D.1 by using gravitational-field and potential models to analyse planetary orbits, satellite energy changes, equipotential diagrams and escape conditions.

Exam points

Apply Kepler’s laws to compare orbital periods and relate a planet’s period and orbital radius.
Calculate gravitational force with F=Gm₁m₂/r² and justify treating a spherical body as a point mass.
Calculate or compare field strength using g=F/m and g=GM/r² at a planet’s surface or altitude.
Sketch or interpret gravitational field lines and combine field-strength vectors from multiple masses.
Distinguish potential V=−GM/r from two-body potential energy Ep=−GMm/r, including their signs and units.
Read Vg against radius and use its gradient to determine the magnitude and direction of g.
Calculate work or energy change for a mass moved between gravitational potentials using ΔEp=mΔVg.
Draw equipotentials and infer field direction or strength from their orientation and spacing.
Derive or apply escape speed and compare launch energy with the energy needed to escape a planet.
Use vorb=√(GM/r) to derive orbital speed and compare satellites at different orbital radii.
Explain how atmospheric drag removes orbital energy, lowers a satellite’s orbit and can increase its speed.

Covered syllabus points

D.1.1—Kepler’s three laws of orbital motion D.1.2—Universal gravitation D.1.3—Point-mass approximation D.1.4—Gravitational field strength D.1.5—Gravitational field lines D.1.6 (HL)—Gravitational potential energy D.1.7 (HL)—Two-body potential energy D.1.8 (HL)—Gravitational potential D.1.9 (HL)—Potential gradient D.1.10 (HL)—Work in gravitational fields D.1.11 (HL)—Gravitational equipotentials D.1.12 (HL)—Equipotentials and field lines D.1.13 (HL)—Escape speed D.1.14 (HL)—Orbital speed D.1.15 (HL)—Atmospheric drag on orbits

[Maximum number: 13]

This question is in two parts. Part 1 is about gravitational force fields. Part 2 is about properties of a gas.

(a)

State Newton's universal law of gravitation.

[ 2 ]

(b)

A satellite of mass m orbits a planet of mass M. Derive the following relationship between the period of the satellite T and the radius of its orbit R (Kepler's third law).

T^{2}=\frac{4 \pi^{2} R^{3}}{G M}

[ 3 ]

$G \frac{M m}{R^{2}}=\frac{m v^{2}}{R}$ so $v^{2}=\frac{G m}{R}$ ;
$v=\frac{2 \pi R}{T} ;$ $\left(v^{2}=\right) \frac{4 \pi^{2} R^{2}}{T^{2}}=\frac{G m}{R} ;$
or
$G \frac{M m}{R^{2}}=m \omega^{2} R ;$ $\omega^{2}=\frac{4 \pi^{2}}{T^{2}} ;$ $\frac{4 \pi^{2}}{T^{2}}=\frac{G M}{R^{3}} ;$

Marking guidance:

Award [3] to a clear response with a missing step.

(c)

A polar orbiting satellite has an orbit which passes above both of the Earth's poles. One polar orbiting satellite used for Earth observation has an orbital period of $6.00 \times 10^{3} \mathrm{~s}$ .

\begin{array}{ll} \text { Mass of Earth } & =5.97 \times 10^{24} \mathrm{~kg} \\ \text { Average radius of Earth } & =6.37 \times 10^{6} \mathrm{~m} \end{array}

[ 8 ]

(i)

Using the relationship in (b), show that the average height above the surface of the Earth for this satellite is about 800 km .

[ 3 ]

$\quad R^{3}=\frac{6.67 \times 10^{-11} \times 5.97 \times 10^{24} \times 6000^{2}}{4 \times \pi^{2}}$ ;

\begin{aligned} & R=7.13 \times 10^{6}(\mathrm{~m}) \\ & h=\left(7.13 \times 10^{6}-6.37 \times 10^{6}\right)=760(\mathrm{~km}) \end{aligned}

Award [3] for an answer of 740 with $\pi$ taken as 3.14.

(ii)

The satellite moves from an orbit of radius 1200 km above the Earth to one of radius 2500 km . The mass of the satellite is 45 kg .

Calculate the change in the gravitational potential energy of the satellite.

[ 3 ]

clear use of $\Delta V=\frac{\Delta E}{m}$ and $V=-\frac{G m}{r}$ or $\Delta E=G M m\left(\frac{1}{r_{1}}-\frac{1}{r_{2}}\right)$ ;
one value of potential energy calculated $\left(2.37 \times 10^{9}\right.$ or $\left.2.02 \times 10^{9}\right)$ ;

Marking guidance:

$3.5\times10^{8}\ \mathrm{J}$

Award [3] for a bald correct answer.

(iii)

Explain whether the gravitational potential energy has increased, decreased or stayed the same when the orbit changes, as in (c)(ii).

[ 2 ]

[Maximum number: 9]

There is a proposal to place a satellite in orbit around planet Mars.

(a)

Outline what is meant by gravitational field strength at a point.

[ 2 ]

(b)

Newton's law of gravitation applies to point masses. Suggest why the law can be applied to a satellite orbiting Mars.

[ 2 ]

(c)

The satellite is to have an orbital time T equal to the length of a day on Mars. It can be shown that

T^{2}=k R^{3}

where R is the orbital radius of the satellite and k is a constant.

[ 5 ]

(i)

Mars has a mass of $6.4 \times 10^{23} \mathrm{~kg}$ . Show that, for Mars, k is about $9 \times 10^{-13} \mathrm{~s}^{2} \mathrm{~m}^{-3}$ .

[ 3 ]

$\frac{mv^2}{r}=\frac{GMm}{r^2}$ , hence $v=\sqrt{\frac{GM}{R}}$ . Also $v=\frac{2\pi R}{T}$ .

OR

$m\omega^2r=\frac{GMm}{r^2}$ , hence $\omega^2=\frac{GM}{R^3}$ .

uses either of the above to get $T^2=\frac{4\pi^2}{GM}R^3$

OR uses $k=\frac{4\pi^2}{GM}$ , $k=9.2\times10^{-13}\ /\ 9.3\times10^{-13}$

(ii)

The time taken for Mars to revolve on its axis is $8.9 \times 10^{4} \mathrm{~s}$ . Calculate, in $\mathrm{m} \mathrm{s}^{-1}$ , the orbital speed of the satellite.

[ 2 ]

$\begin{aligned} R^3 &= \frac{T^2}{k}=\frac{(8.9\times10^4)^2}{9.25\times10^{-13}}\\ R &= 2.04\times10^7\ \mathrm{m}\\ v &= \omega R=\frac{2\pi\times2.04\times10^7}{89000}=1.4\times10^3\ \mathrm{m\,s^{-1}} \end{aligned}$

OR

$v=\sqrt{\frac{6.67\times10^{-11}\times6.4\times10^{23}}{2.04\times10^7}}=1.4\times10^3\ \mathrm{m\,s^{-1}}$

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

Question (iii)

Question 2

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)