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

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

[Maximum number: 3]

This question is in two parts

In a simple model of the hydrogen atom, the electron is regarded as being in a circular orbit about the proton. The magnitude of the electric field strength at the electron due to the proton is $E_{\mathrm{p}}$ . The magnitude of the gravitational field strength at the electron due to the proton is $g_{\mathrm{p}}$ .

Determine the order of magnitude of the ratio shown below.

\frac{E_{\mathrm{p}}}{g_{\mathrm{p}}}

$E_{\mathrm{p}}=\frac{e}{4 \pi \varepsilon_{0} r^{2}}$ and $g_{\mathrm{p}}=\frac{G m_{\mathrm{p}}}{r^{2}}$ ; (both needed)
$\frac{e}{4 \pi \varepsilon_{0} G m_{\mathrm{p}}}\left(=\frac{9 \times 10^{9} \times 1.6 \times 10^{-19}}{6.7 \times 10^{-11} \times 1.7 \times 10^{-27}}\right) ;$ $\approx 10^{28}$ ;

Marking guidance:

Award [2 max] if response calculates ratio of force as this is an ECF from the first marking point ( $10^{39}$ ).
Award [3] for solution that correctly evaluates field strengths separately and then divides.

[Maximum number: 3]

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

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.

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

[Maximum number: 2]

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

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

[Maximum number: 1]

The diagram shows the gravitational field lines of planet X.

Outline how this diagram shows that the gravitational field strength of planet X decreases with distance from the surface.

[Maximum number: 2]

This question is in two parts. Part 1 is about the use of renewable energy sources. Part 2 is about the gravitational potential of the Earth.

A satellite of mass 780 kg is put into circular orbit $4.0 \times 10^{3} \mathrm{~km}$ above the surface of the Earth. It is then moved to a higher orbit $3.2 \times 10^{4} \mathrm{~km}$ above the surface.

Calculate, for the satellite as it is moved from the lower orbit to the higher orbit, the change in

the gravitational potential energy.

change in gravitational potential $=(+) 28\left(\mathrm{MJ} \mathrm{kg}^{-1}\right)$ ;
change in gravitational potential energy $=\left(28 \times 10^{6} \times 780=\right) 2.2 \times 10^{10}(\mathrm{~J})$ ;

Marking guidance:

Award [2] for a bald correct answer.

[Maximum number: 2]

A satellite of mass m is in a circular orbit of radius r around Earth. Earth has mass M.

Show that the total energy of the satellite is given by $E_{\mathrm{T}}=-\frac{G M m}{2 r}$ .

Identifies orbital velocity as $v=\sqrt{\frac{G M}{r}}$ $E_{\kappa}=\frac{1}{2} m v^{2}=\frac{G M m}{2 r} \text { so } E_{\mathrm{T}}=\frac{G M m}{2 r}-\frac{G M m}{r} \text { Hence result }$

[Maximum number: 3]

Define gravitational potential at a point in a gravitational field.

[Maximum number: 2]

A uniform, spherical planet P has mass M and radius R.

The surface of P is an equipotential surface $E_{0}$ . The dotted circle labeled $E_{1}$ is an equipotential line at a distance $\frac{3 R}{2}$ from the centre of P . The potential difference between $E_{1}$ and $E_{0}$ is $2.2 \times 10^{6} \mathrm{Jkg}^{-1}$ .

The radius R of P is $8.5 \times 10^{6} \mathrm{~m}$ .

Estimate the magnitude of the gravitational field strength between $E_{1}$ and $E_{0}$ .

$\begin{aligned} g=\pi \frac{\Delta V_{\mathrm{g}}}{\Delta r}=>\frac{2.2 \times 10^{6}}{\frac{8.5 \times 10^{6}}{2}} g=0.52 \ll \mathrm{~N} \mathrm{~kg}^{-1} \gg \end{aligned}$

Ignore negative sign

[2]

[Maximum number: 2]

The graph shows how the gravitational potential V of Earth varies with distance R from the centre of Earth in the range $R=2.0 \times 10^{8} \mathrm{~m}$ to $R=5.0 \times 10^{8} \mathrm{~m}$ .

The Moon is at a distance of $4.0 \times 10^{8} \mathrm{~m}$ from the centre of Earth. At some time in the past it was at a distance of $2.7 \times 10^{8} \mathrm{~m}$ from the centre of Earth.

Use the graph opposite to determine

by how much the potential energy of the Moon has changed as a result of moving from $R=2.7 \times 10^{8} \mathrm{~m}$ to $R=4.0 \times 10^{8} \mathrm{~m}$ . The mass of the Moon is $7.4 \times 10^{22} \mathrm{~kg}$ .

change in $V=0.45 \times 10^{6}-0.50 \times 10^{6} \mathrm{~J} \mathrm{~kg}^{-1}$ ;
change in $\mathrm{PE}=\left(0.5 \times 10^{6} \times 7.4 \times 10^{22}=\right) 3.3-3.7 \times 10^{28} \mathrm{~J}$ ;

[Maximum number: 2]

The diagram shows the gravitational field lines of planet X.

The diagram shows part of the surface of planet X. The gravitational potential at the surface of planet X is -3 V and the gravitational potential at point Y is -V .

Sketch on the grid the equipotential surface corresponding to a gravitational potential of -2 V.

[Maximum number: 1]

The diagram shows 5 gravitational equipotential lines. The gravitational potential on each line is indicated. A point mass m is placed on the middle line and is then released. Values given in $\mathrm{MJ} \mathrm{kg}^{-1}$ .

Which is correct about the direction of motion and the acceleration of the point mass?

Direction

Acceleration

to the right

decreasing

to the right

increasing

to the left

decreasing

to the left

increasing

[Maximum number: 2]

A uniform, spherical planet P has mass M and radius R.

A projectile of mass m is launched with kinetic energy $\frac{2 G M m}{3 R}$ from the surface of P.

Determine whether the projectile will escape the gravitational field of P.

ALTERNATE 1
$\begin{aligned} & E_{\mathrm{T}}=\lessdot \frac{2 G M m}{3 R}-\frac{G M m}{R}=»-\frac{1}{3} \frac{G M m}{R} & E_{\mathrm{T}}<0 \text { hence it will not escape } \end{aligned}$

ALTERNATE 2
$\frac{1}{2} m v^{2}=\frac{2 G M m}{3 R} \Rightarrow v=\sqrt{\frac{4 G M}{3 R}}$ $v<v_{\text {esc }}$ hence it will not escape

Argument necessary to get second MP. Do not accept a bald statement that it will not escape.

Allow ECF from MP1.

[2]

[Maximum number: 2]

The graph shows the variation with time of the height of the International Space Station (ISS) during a 30-day period.

The following data are available:

\begin{aligned} \text { Mass of Earth } & =5.98 \times 10^{24} \mathrm{~kg} \\ \text { Mass of the ISS } & =4.20 \times 10^{5} \mathrm{~kg} \\ \text { Radius of the Earth } & =6.38 \times 10^{6} \mathrm{~m} \end{aligned}

For the 30-day period indicated in the graph the loss of total energy of the ISS is $4.5 \times 10^{9} \mathrm{~J}$ .

Show, using the average height during the 30-day period, that the speed of the ISS is about $8 \times 10^{3} \mathrm{~ms}^{-1}$ .

Average height 417.5 km
$v=\sqrt{\frac{6.67 \times 10^{-11} \times 5.98 \times 10^{24}}{(6.38+0.4175) \times 10^{6}}} \text { OR } 7.7 \times 10^{3} \mathrm{~ms}^{-1}$

Accept $417.5 \pm 0.3$ for MP1

[Maximum number: 4]

A satellite of mass m is in a circular orbit of radius r around Earth. Earth has mass M.

The satellite experiences a drag force due to the atmosphere of Earth. With reference to the results in (a) and (b)(i), state and explain the likely fate of this satellite.

The total energy is reduced OR the satellite gets closer to the surface
«From $v=\sqrt{\frac{G M}{r}}$ » the satellite moves faster
«From $\rho S v^{2}$ » the opposing force increases « because v and $\rho$ increase »
The satellite burns / crashes / falls to Earth

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)

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16