D.1 Gravitational fields

A space probe moves in a circular orbit around Earth. The kinetic energy of the probe is $E$.

The probe will reach the escape speed when its kinetic energy is increased at least to:

A.  $\sqrt{2}E$

B.  $2E$

C.  $2\sqrt{2}E$

D.  $4E$

A space probe moves in a circular orbit around Earth. The kinetic energy of the probe is $E$.

The probe will reach the escape speed when its kinetic energy is increased at least to:

A.  $\sqrt{2}E$

B.  $2E$

C.  $2\sqrt{2}E$

D.  $4E$

What is the escape speed from the surface of a planet of radius $r$ that has an acceleration of gravity $g$ at its surface?

A.  $\sqrt{\frac{g}{r}}$

B.  $\sqrt{gr}$

C.  $\sqrt{\frac{2g}{r}}$

D.  $\sqrt{2gr}$

What is the escape speed from the surface of a planet of radius $r$ that has an acceleration of gravity $g$ at its surface?

A.  $\sqrt{\frac{g}{r}}$

B.  $\sqrt{gr}$

C.  $\sqrt{\frac{2g}{r}}$

D.  $\sqrt{2gr}$

Planets X and Y move in circular orbits around the same star.

The orbital period of planet Y is twice the orbital period of planet X. The orbital radius of planet X is $R$.

What is the orbital radius of planet Y?

A.  $\sqrt[3]{2}R$

B.  $\sqrt[3]{4}R$

C.  $2R$

D.  $4R$

Planets X and Y move in circular orbits around the same star.

The orbital period of planet Y is twice the orbital period of planet X. The orbital radius of planet X is $R$.

What is the orbital radius of planet Y?

A.  $\sqrt[3]{2}R$

B.  $\sqrt[3]{4}R$

C.  $2R$

D.  $4R$

Kepler’s Third law relates the orbital period $T$ of a planet about its sun to its orbital radius $r$. The mass of the Sun is $M$.

What is a correct algebraic form of the law?

A.  $T=\frac{2\mathrm{\pi }{r}^{1.5}}{{\left(GM\right)}^{0.5}}$

B.  $T=\frac{2\mathrm{\pi }{r}^{1.5}}{GM}$

C.  $T=\frac{4\mathrm{\pi }{r}^{0.67}}{{\left(GM\right)}^2}$

D.  $T=\frac{4\mathrm{\pi }{r}^{0.67}}{GM}$

Kepler’s Third law relates the orbital period $T$ of a planet about its sun to its orbital radius $r$. The mass of the Sun is $M$.

What is a correct algebraic form of the law?

A.  $T=\frac{2\mathrm{\pi }{r}^{1.5}}{{\left(GM\right)}^{0.5}}$

B.  $T=\frac{2\mathrm{\pi }{r}^{1.5}}{GM}$

C.  $T=\frac{4\mathrm{\pi }{r}^{0.67}}{{\left(GM\right)}^2}$

D.  $T=\frac{4\mathrm{\pi }{r}^{0.67}}{GM}$

orbital speed;

orbital speed;

The radius of the dwarf planet Pluto is 1.19 x 10⁶ m. The acceleration due to gravity at its surface is 0.617 m s⁻².

Determine the escape speed for an object at the surface of Pluto.

The radius of the dwarf planet Pluto is 1.19 x 10⁶ m. The acceleration due to gravity at its surface is 0.617 m s⁻².

Determine the escape speed for an object at the surface of Pluto.

The radius of the dwarf planet Pluto is 1.19 x 10⁶ m. The acceleration due to gravity at its surface is 0.617 m s⁻².

Determine the escape speed for an object at the surface of Pluto.

Pluto rotates about an axis through its centre. Its rotation is in the opposite sense to that of the Earth, i.e. from east to west.

Explain the advantage of an object launching from the equator of Pluto and travelling to the west.

Pluto rotates about an axis through its centre. Its rotation is in the opposite sense to that of the Earth, i.e. from east to west.

Explain the advantage of an object launching from the equator of Pluto and travelling to the west.

Pluto rotates about an axis through its centre. Its rotation is in the opposite sense to that of the Earth, i.e. from east to west.

Explain the advantage of an object launching from the equator of Pluto and travelling to the west.

The charges Q are replaced by neutral masses M and the charge q by a neutral mass m. The mass m is displaced away from C by a small distance $x$ and released. Discuss whether the motion of m will be the same as that of q.

The charges Q are replaced by neutral masses M and the charge q by a neutral mass m. The mass m is displaced away from C by a small distance $x$ and released. Discuss whether the motion of m will be the same as that of q.

The charges Q are replaced by neutral masses M and the charge q by a neutral mass m. The mass m is displaced away from C by a small distance $x$ and released. Discuss whether the motion of m will be the same as that of q.

Outline why the graph between P and O is negative.

Outline why the graph between P and O is negative.

Outline why the graph between P and O is negative.

Determine $\frac{g_{\mathrm{M}}}{g_{\mathrm{P}}}$.

Determine $\frac{g_{\mathrm{M}}}{g_{\mathrm{P}}}$.

Determine $\frac{g_{\mathrm{M}}}{g_{\mathrm{P}}}$.

State and explain the magnitude of the resultant gravitational field strength at O.

State and explain the magnitude of the resultant gravitational field strength at O.

State and explain the magnitude of the resultant gravitational field strength at O.

The gravitational potential due to the mass of M at the surface of P can be assumed to be negligible.

Estimate, using the graph, the gravitational potential at the surface of M due to the mass of M.

The gravitational potential due to the mass of M at the surface of P can be assumed to be negligible.

Estimate, using the graph, the gravitational potential at the surface of M due to the mass of M.

The gravitational potential due to the mass of M at the surface of P can be assumed to be negligible.

Estimate, using the graph, the gravitational potential at the surface of M due to the mass of M.

Show that the gravitational potential V_P at the surface of P due to the mass of P is given by V_P = −g_PR_P where R_P is the radius of the planet.

Show that the gravitational potential V_P at the surface of P due to the mass of P is given by V_P = −g_PR_P where R_P is the radius of the planet.

Show that the gravitational potential V_P at the surface of P due to the mass of P is given by V_P = −g_PR_P where R_P is the radius of the planet.

Draw on the axes the variation of gravitational potential between O and M.

Draw on the axes the variation of gravitational potential between O and M.

Draw on the axes the variation of gravitational potential between O and M.

An asteroid (minor planet) orbits the Sun in a circular orbit of radius 4.5 × 10⁸ km. The radius of Earth’s orbit is 1.5 × 10⁸ km. Calculate, in years, the orbital period of the asteroid.

An asteroid (minor planet) orbits the Sun in a circular orbit of radius 4.5 × 10⁸ km. The radius of Earth’s orbit is 1.5 × 10⁸ km. Calculate, in years, the orbital period of the asteroid.

An asteroid (minor planet) orbits the Sun in a circular orbit of radius 4.5 × 10⁸ km. The radius of Earth’s orbit is 1.5 × 10⁸ km. Calculate, in years, the orbital period of the asteroid.

Show that $k=\frac{1}{GM}$.

Show that $k=\frac{1}{GM}$.

Show that $k=\frac{1}{GM}$.

The table gives data relating to the two moons of Mars.

Determine r for Deimos.

The table gives data relating to the two moons of Mars.

Determine r for Deimos.

The table gives data relating to the two moons of Mars.

Determine r for Deimos.

 Determine the mass of Mars.

 Determine the mass of Mars.

 Determine the mass of Mars.

Show that $T\propto r^{\frac{3}{2}}$ for the planets in a solar system where $T$ is the orbital period of a planet and $r$ is the radius of circular orbit of planet about its sun.

Show that $T\propto r^{\frac{3}{2}}$ for the planets in a solar system where $T$ is the orbital period of a planet and $r$ is the radius of circular orbit of planet about its sun.

Show that $T\propto r^{\frac{3}{2}}$ for the planets in a solar system where $T$ is the orbital period of a planet and $r$ is the radius of circular orbit of planet about its sun.

Outline what is meant by one astronomical unit (1 AU)

Outline what is meant by one astronomical unit (1 AU)

Outline what is meant by one astronomical unit (1 AU)

Pluto is a dwarf planet of the Sun that orbits at a distance of 5.9 × 10⁹ km from the Sun. Determine, in years, the orbital period of Pluto.

Pluto is a dwarf planet of the Sun that orbits at a distance of 5.9 × 10⁹ km from the Sun. Determine, in years, the orbital period of Pluto.

Pluto is a dwarf planet of the Sun that orbits at a distance of 5.9 × 10⁹ km from the Sun. Determine, in years, the orbital period of Pluto.

escape speed from its orbit.

escape speed from its orbit.

The centre of the Earth and the Moon are a distance $D$ apart. There is a point X between them where their gravitational fields cancel out. The distance from the centre of the Earth to X is $d$. The mass of the Earth is $M_{\text{E}}$ and the mass of the Moon is $M_{\text{M}}$.

What is correct at X?

A.  $\frac{M_{\text{E}}}{d}=\frac{M_{\text{M}}}{D-d}$

B.  $\frac{M_{\text{E}}}{D-d}=\frac{M_{\text{M}}}{d}$

C.  $\frac{M_E}{d^2}=\frac{M_{\text{M}}}{{\left(D-d\right)}^2}$

D.  $\frac{M_E}{d^2}=\frac{M_{\text{M}}}{D^2-d^2}$

The centre of the Earth and the Moon are a distance $D$ apart. There is a point X between them where their gravitational fields cancel out. The distance from the centre of the Earth to X is $d$. The mass of the Earth is $M_{\text{E}}$ and the mass of the Moon is $M_{\text{M}}$.

What is correct at X?

A.  $\frac{M_{\text{E}}}{d}=\frac{M_{\text{M}}}{D-d}$

B.  $\frac{M_{\text{E}}}{D-d}=\frac{M_{\text{M}}}{d}$

C.  $\frac{M_E}{d^2}=\frac{M_{\text{M}}}{{\left(D-d\right)}^2}$

D.  $\frac{M_E}{d^2}=\frac{M_{\text{M}}}{D^2-d^2}$

Determine the minimum energy required to launch the satellite. Ignore the original kinetic energy of the satellite due to Earth’s rotation.

Determine the minimum energy required to launch the satellite. Ignore the original kinetic energy of the satellite due to Earth’s rotation.

Determine the minimum energy required to launch the satellite. Ignore the original kinetic energy of the satellite due to Earth’s rotation.

Show that the kinetic energy of the satellite in orbit is about 2 × 10¹⁰ J.

Show that the kinetic energy of the satellite in orbit is about 2 × 10¹⁰ J.

Show that the kinetic energy of the satellite in orbit is about 2 × 10¹⁰ J.

Planets X and Y orbit the same star.

The average distance between planet X and the star is five times greater than the average distance between planet Y and the star.

What is $\frac{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{X}}{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{Y}}$?

A.  $\sqrt[3]{5}$

B.  $\sqrt{5}$

C.  $\sqrt[3]{5^2}$

D.  $\sqrt{5^3}$

Planets X and Y orbit the same star.

The average distance between planet X and the star is five times greater than the average distance between planet Y and the star.

What is $\frac{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{X}}{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{Y}}$?

A.  $\sqrt[3]{5}$

B.  $\sqrt{5}$

C.  $\sqrt[3]{5^2}$

D.  $\sqrt{5^3}$

Planets X and Y orbit the same star.

The average distance between planet X and the star is five times greater than the average distance between planet Y and the star.

What is $\frac{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{X}}{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{Y}}$?

A.  $\sqrt[3]{5}$

B.  $\sqrt{5}$

C.  $\sqrt[3]{5^2}$

D.  $\sqrt{5^3}$

Planets X and Y orbit the same star.

The average distance between planet X and the star is five times greater than the average distance between planet Y and the star.

What is $\frac{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{X}}{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{Y}}$?

A.  $\sqrt[3]{5}$

B.  $\sqrt{5}$

C.  $\sqrt[3]{5^2}$

D.  $\sqrt{5^3}$

Planets X and Y orbit the same star.

The average distance between planet X and the star is five times greater than the average distance between planet Y and the star.

What is $\frac{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{X}}{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{Y}}$?

A.  $\sqrt[3]{5}$

B.  $\sqrt{5}$

C.  $\sqrt[3]{5^2}$

D.  $\sqrt{5^3}$

Planets X and Y orbit the same star.

The average distance between planet X and the star is five times greater than the average distance between planet Y and the star.

What is $\frac{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{X}}{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{Y}}$?

A.  $\sqrt[3]{5}$

B.  $\sqrt{5}$

C.  $\sqrt[3]{5^2}$

D.  $\sqrt{5^3}$

Planets X and Y orbit the same star.

The average distance between planet X and the star is five times greater than the average distance between planet Y and the star.

What is $\frac{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{X}}{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{Y}}$?

A.  $\sqrt[3]{5}$

B.  $\sqrt{5}$

C.  $\sqrt[3]{5^2}$

D.  $\sqrt{5^3}$

Planets X and Y orbit the same star.

The average distance between planet X and the star is five times greater than the average distance between planet Y and the star.

What is $\frac{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{X}}{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{Y}}$?

A.  $\sqrt[3]{5}$

B.  $\sqrt{5}$

C.  $\sqrt[3]{5^2}$

D.  $\sqrt{5^3}$

Planets X and Y orbit the same star.

The average distance between planet X and the star is five times greater than the average distance between planet Y and the star.

What is $\frac{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{X}}{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{Y}}$?

A.  $\sqrt[3]{5}$

B.  $\sqrt{5}$

C.  $\sqrt[3]{5^2}$

D.  $\sqrt{5^3}$

Planets X and Y orbit the same star.

The average distance between planet X and the star is five times greater than the average distance between planet Y and the star.

What is $\frac{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{X}}{\mathrm{orbital} \mathrm{period} \mathrm{of} \mathrm{planet} \mathrm{Y}}$?

A.  $\sqrt[3]{5}$

B.  $\sqrt{5}$

C.  $\sqrt[3]{5^2}$

D.  $\sqrt{5^3}$

A satellite in a circular orbit around the Earth needs to reduce its orbital radius.

What is the work done by the satellite rocket engine and the change in kinetic energy resulting from this shift in orbital height?

A satellite in a circular orbit around the Earth needs to reduce its orbital radius.

What is the work done by the satellite rocket engine and the change in kinetic energy resulting from this shift in orbital height?

Satellite X is in orbit around the Earth. An identical satellite Y is in a higher orbit. What is correct for the total energy and the kinetic energy of the satellite Y compared with satellite X?

Satellite X is in orbit around the Earth. An identical satellite Y is in a higher orbit. What is correct for the total energy and the kinetic energy of the satellite Y compared with satellite X?

The escape speed from a planet of radius R is v_esc. A satellite orbits the planet at a distance R from the surface of the planet. What is the orbital speed of the satellite?

A. $\frac{1}{2}{v}_{\text{esc}}$

B. $\frac{\sqrt{2}}{2}{v}_{\text{esc}}$

C. $\sqrt{2}{v}_{\text{esc}}$

D. $2v_{\text{esc}}$

The escape speed from a planet of radius R is v_esc. A satellite orbits the planet at a distance R from the surface of the planet. What is the orbital speed of the satellite?

A. $\frac{1}{2}{v}_{\text{esc}}$

B. $\frac{\sqrt{2}}{2}{v}_{\text{esc}}$

C. $\sqrt{2}{v}_{\text{esc}}$

D. $2v_{\text{esc}}$

Satellite X orbits a planet with orbital radius R. Satellite Y orbits the same planet with orbital radius 2R. Satellites X and Y have the same mass.

What is the ratio $\frac{\text{centripetal acceleration of X}}{\text{centripetal acceleration of Y}}$?

A. $\frac{1}{4}$

B. $\frac{1}{2}$

C. 2

D. 4

Satellite X orbits a planet with orbital radius R. Satellite Y orbits the same planet with orbital radius 2R. Satellites X and Y have the same mass.

What is the ratio $\frac{\text{centripetal acceleration of X}}{\text{centripetal acceleration of Y}}$?

A. $\frac{1}{4}$

B. $\frac{1}{2}$

C. 2

D. 4

The orbital period T of a moon orbiting a planet of mass M is given by

$$\frac{R^3}{T^2}=kM$$

where R is the average distance between the centre of the planet and the centre of the moon.

Show that $k=\frac{G}{4{\pi }^2}$

The orbital period T of a moon orbiting a planet of mass M is given by

$$\frac{R^3}{T^2}=kM$$

where R is the average distance between the centre of the planet and the centre of the moon.

Show that $k=\frac{G}{4{\pi }^2}$

The orbital period T of a moon orbiting a planet of mass M is given by

$$\frac{R^3}{T^2}=kM$$

where R is the average distance between the centre of the planet and the centre of the moon.

Show that $k=\frac{G}{4{\pi }^2}$

The following data for the Mars–Phobos system and the Earth–Moon system are available:

Mass of Earth = 5.97 × 10²⁴ kg

The Earth–Moon distance is 41 times the Mars–Phobos distance.

The orbital period of the Moon is 86 times the orbital period of Phobos.

Calculate, in kg, the mass of Mars.

The following data for the Mars–Phobos system and the Earth–Moon system are available:

Mass of Earth = 5.97 × 10²⁴ kg

The Earth–Moon distance is 41 times the Mars–Phobos distance.

The orbital period of the Moon is 86 times the orbital period of Phobos.

Calculate, in kg, the mass of Mars.

The following data for the Mars–Phobos system and the Earth–Moon system are available:

Mass of Earth = 5.97 × 10²⁴ kg

The Earth–Moon distance is 41 times the Mars–Phobos distance.

The orbital period of the Moon is 86 times the orbital period of Phobos.

Calculate, in kg, the mass of Mars.

Show that the total energy of the planet is given by the equation shown.

$E=\frac{1}{2}mV$

Show that the total energy of the planet is given by the equation shown.

$E=\frac{1}{2}mV$

Show that the total energy of the planet is given by the equation shown.

$E=\frac{1}{2}mV$

Suppose the star could contract to half its original radius without any loss of mass. Discuss the effect, if any, this has on the total energy of the planet.

Suppose the star could contract to half its original radius without any loss of mass. Discuss the effect, if any, this has on the total energy of the planet.

Suppose the star could contract to half its original radius without any loss of mass. Discuss the effect, if any, this has on the total energy of the planet.

The gravitational potential is $V$ at a distance $R$ above the surface of a spherical planet of radius $R$ and uniform density. What is the gravitational potential a distance $2R$ above the surface of the planet?

A. $\frac{V}{4}$

B. $\frac{4V}{9}$

C. $\frac{V}{2}$

D. $\frac{2V}{3}$

The gravitational potential is $V$ at a distance $R$ above the surface of a spherical planet of radius $R$ and uniform density. What is the gravitational potential a distance $2R$ above the surface of the planet?

A. $\frac{V}{4}$

B. $\frac{4V}{9}$

C. $\frac{V}{2}$

D. $\frac{2V}{3}$

Planet X has a gravitational field strength of $18 \mathrm{N} {\mathrm{kg}}^{-1}$ at its surface. Planet Y has the same density as X but three times the radius of X. What is the gravitational field strength at the surface of Y?

A.  $6 \mathrm{m} {\mathrm{s}}^{-2}$

B.  $18 \mathrm{m} {\mathrm{s}}^{-2}$

C.  $54 \mathrm{m} {\mathrm{s}}^{-2}$

D.  $162 \mathrm{m} {\mathrm{s}}^{-2}$

Planet X has a gravitational field strength of $18 \mathrm{N} {\mathrm{kg}}^{-1}$ at its surface. Planet Y has the same density as X but three times the radius of X. What is the gravitational field strength at the surface of Y?

A.  $6 \mathrm{m} {\mathrm{s}}^{-2}$

B.  $18 \mathrm{m} {\mathrm{s}}^{-2}$

C.  $54 \mathrm{m} {\mathrm{s}}^{-2}$

D.  $162 \mathrm{m} {\mathrm{s}}^{-2}$

An object of mass $m$ released from rest near the surface of a planet has an initial acceleration $z$. What is the gravitational field strength near the surface of the planet?

A.  $z$

B.  $\frac{z}{m}$

C.  $mz$

D.  $\frac{m}{z}$

An object of mass $m$ released from rest near the surface of a planet has an initial acceleration $z$. What is the gravitational field strength near the surface of the planet?

A.  $z$

B.  $\frac{z}{m}$

C.  $mz$

D.  $\frac{m}{z}$

A satellite orbits planet $\text{X}$ with a speed $v_{\text{X}}$ at a distance $\text{r}$ from the centre of planet $\text{X}$. Another satellite orbits planet $\text{Y}$ at a speed of $v_{\text{y}}$ at a distance $\text{r}$ from the centre of planet $\text{Y}$. The mass of planet $\text{X}$ is $M$ and the mass of planet $\text{Y}$ is $4M$. What is the ratio of $\frac{v_{\text{X}}}{v_{\text{y}}}$?

A.  0.25

B.  0.5

C.  2.0

D.  4.0

A satellite orbits planet $\text{X}$ with a speed $v_{\text{X}}$ at a distance $\text{r}$ from the centre of planet $\text{X}$. Another satellite orbits planet $\text{Y}$ at a speed of $v_{\text{y}}$ at a distance $\text{r}$ from the centre of planet $\text{Y}$. The mass of planet $\text{X}$ is $M$ and the mass of planet $\text{Y}$ is $4M$. What is the ratio of $\frac{v_{\text{X}}}{v_{\text{y}}}$?

A.  0.25

B.  0.5

C.  2.0

D.  4.0

Which is the definition of gravitational field strength at a point?

A. The sum of the gravitational fields created by all masses around the point

B. The gravitational force per unit mass experienced by a small point mass at that point

C. $G\frac{M}{r^2}$, where $M$ is the mass of a planet and $r$ is the distance from the planet to the point

D. The resultant force of gravitational attraction on a mass at that point

Which is the definition of gravitational field strength at a point?

A. The sum of the gravitational fields created by all masses around the point

B. The gravitational force per unit mass experienced by a small point mass at that point

C. $G\frac{M}{r^2}$, where $M$ is the mass of a planet and $r$ is the distance from the planet to the point

D. The resultant force of gravitational attraction on a mass at that point

Show that the gravitational potential due to the planet and the star at the surface of the planet is about −5 × 10⁹J kg⁻¹.

Show that the gravitational potential due to the planet and the star at the surface of the planet is about −5 × 10⁹J kg⁻¹.

Show that the gravitational potential due to the planet and the star at the surface of the planet is about −5 × 10⁹J kg⁻¹.

Estimate the escape speed of the spacecraft from the planet–star system.

Estimate the escape speed of the spacecraft from the planet–star system.

Estimate the escape speed of the spacecraft from the planet–star system.

Calculate, for the surface of $\text{Io}$, the gravitational field strength g_Io due to the mass of $\text{Io}$. State an appropriate unit for your answer.

Calculate, for the surface of $\text{Io}$, the gravitational field strength g_Io due to the mass of $\text{Io}$. State an appropriate unit for your answer.

Calculate, for the surface of $\text{Io}$, the gravitational field strength g_Io due to the mass of $\text{Io}$. State an appropriate unit for your answer.

Show that the $\frac{\mathrm{gravitational} \mathrm{potential} \mathrm{due} \mathrm{to} \mathrm{Jupiter} \mathrm{at} \mathrm{the} \mathrm{orbit} \mathrm{of} \mathrm{Io}}{ \mathrm{gravitational} \mathrm{potential} \mathrm{due} \mathrm{to} \mathrm{Io} \mathrm{at} \mathrm{the} \mathrm{surface} \mathrm{of} \mathrm{Io}}$ is about 80.

Show that the $\frac{\mathrm{gravitational} \mathrm{potential} \mathrm{due} \mathrm{to} \mathrm{Jupiter} \mathrm{at} \mathrm{the} \mathrm{orbit} \mathrm{of} \mathrm{Io}}{ \mathrm{gravitational} \mathrm{potential} \mathrm{due} \mathrm{to} \mathrm{Io} \mathrm{at} \mathrm{the} \mathrm{surface} \mathrm{of} \mathrm{Io}}$ is about 80.

Show that the $\frac{\mathrm{gravitational} \mathrm{potential} \mathrm{due} \mathrm{to} \mathrm{Jupiter} \mathrm{at} \mathrm{the} \mathrm{orbit} \mathrm{of} \mathrm{Io}}{ \mathrm{gravitational} \mathrm{potential} \mathrm{due} \mathrm{to} \mathrm{Io} \mathrm{at} \mathrm{the} \mathrm{surface} \mathrm{of} \mathrm{Io}}$ is about 80.

Outline, using (b)(i), why it is not correct to use the equation $\sqrt{\frac{2G\times \text{mass of Io}}{\text{radius of Io}}}$ to calculate the speed required for the spacecraft to reach infinity from the surface of $\text{Io}$.

Outline, using (b)(i), why it is not correct to use the equation $\sqrt{\frac{2G\times \text{mass of Io}}{\text{radius of Io}}}$ to calculate the speed required for the spacecraft to reach infinity from the surface of $\text{Io}$.

Outline, using (b)(i), why it is not correct to use the equation $\sqrt{\frac{2G\times \text{mass of Io}}{\text{radius of Io}}}$ to calculate the speed required for the spacecraft to reach infinity from the surface of $\text{Io}$.

An engineer needs to move a space probe of mass 3600 kg from Ganymede to Callisto. Calculate the energy required to move the probe from the orbital radius of Ganymede to the orbital radius of Callisto. Ignore the mass of the moons in your calculation. 

An engineer needs to move a space probe of mass 3600 kg from Ganymede to Callisto. Calculate the energy required to move the probe from the orbital radius of Ganymede to the orbital radius of Callisto. Ignore the mass of the moons in your calculation. 

An engineer needs to move a space probe of mass 3600 kg from Ganymede to Callisto. Calculate the energy required to move the probe from the orbital radius of Ganymede to the orbital radius of Callisto. Ignore the mass of the moons in your calculation. 

A satellite of mass $m$ orbits a planet of mass $M$ in a circular orbit of radius $r$. What is the work that must be done on the satellite to increase its orbital radius to $2r$?

A.  $\frac{GMm}{r}$

B.  $\frac{GMm}{2r}$

C.  $\frac{GMm}{4r}$

D.  $\frac{GMm}{8r}$

A satellite of mass $m$ orbits a planet of mass $M$ in a circular orbit of radius $r$. What is the work that must be done on the satellite to increase its orbital radius to $2r$?

A.  $\frac{GMm}{r}$

B.  $\frac{GMm}{2r}$

C.  $\frac{GMm}{4r}$

D.  $\frac{GMm}{8r}$

The gravitational field strength at the surface of a planet of radius R is $g$. A satellite is moving in a circular orbit a distance R above the surface of the planet. What is the magnitude of the acceleration of the satellite?

A.  $0$

B.  $\frac{g}{4}$

C.  $\frac{g}{2}$

D.  $g$

The gravitational field strength at the surface of a planet of radius R is $g$. A satellite is moving in a circular orbit a distance R above the surface of the planet. What is the magnitude of the acceleration of the satellite?

A.  $0$

B.  $\frac{g}{4}$

C.  $\frac{g}{2}$

D.  $g$

The orbital radius of Titan around Saturn is $R$ and the period of revolution is $T$.

Show that $T^2=\frac{4{\mathrm{\pi }}^2{R}^{ 3}}{GM}$ where $M$ is the mass of Saturn.

The orbital radius of Titan around Saturn is $R$ and the period of revolution is $T$.

Show that $T^2=\frac{4{\mathrm{\pi }}^2{R}^{ 3}}{GM}$ where $M$ is the mass of Saturn.

The orbital radius of Titan around Saturn is $R$ and the period of revolution is $T$.

Show that $T^2=\frac{4{\mathrm{\pi }}^2{R}^{ 3}}{GM}$ where $M$ is the mass of Saturn.

The orbital radius of Titan around Saturn is 1.2 × 10⁹m and the orbital period is 15.9 days. Estimate the mass of Saturn.

The orbital radius of Titan around Saturn is 1.2 × 10⁹m and the orbital period is 15.9 days. Estimate the mass of Saturn.

The orbital radius of Titan around Saturn is 1.2 × 10⁹m and the orbital period is 15.9 days. Estimate the mass of Saturn.

P and Q are two moons of equal densities orbiting a planet. The orbital radius of P is twice the orbital radius of Q. The volume of P is half that of Q. The force exerted by the planet on P is F. What is the force exerted by the planet on Q?

A.  F

B.  2F

C.  4F

D.  8F

P and Q are two moons of equal densities orbiting a planet. The orbital radius of P is twice the orbital radius of Q. The volume of P is half that of Q. The force exerted by the planet on P is F. What is the force exerted by the planet on Q?

A.  F

B.  2F

C.  4F

D.  8F

An object of mass $m$ is launched from the surface of the Earth. The Earth has a mass $M$ and radius $r$. The acceleration due to gravity at the surface of the Earth is $g$. What is the escape speed of the object from the surface of the Earth?

A.  $\sqrt{gr}$

B.  $\sqrt{2gr}$

C.  $\sqrt{2Mgr}$

D.  $\sqrt{2mgr}$

An object of mass $m$ is launched from the surface of the Earth. The Earth has a mass $M$ and radius $r$. The acceleration due to gravity at the surface of the Earth is $g$. What is the escape speed of the object from the surface of the Earth?

A.  $\sqrt{gr}$

B.  $\sqrt{2gr}$

C.  $\sqrt{2Mgr}$

D.  $\sqrt{2mgr}$

A simple pendulum has a time period $T$ on the Earth. The pendulum is taken to the Moon where the gravitational field strength is $\frac{1}{6}$ that of the Earth.

What is the time period of the pendulum on the Moon?

A.  $T\sqrt{6}$

B.  $T$

C.  $\frac{\sqrt{6}}{6}T$

D.  $\frac{T}{6}$

A simple pendulum has a time period $T$ on the Earth. The pendulum is taken to the Moon where the gravitational field strength is $\frac{1}{6}$ that of the Earth.

What is the time period of the pendulum on the Moon?

A.  $T\sqrt{6}$

B.  $T$

C.  $\frac{\sqrt{6}}{6}T$

D.  $\frac{T}{6}$

A satellite is orbiting Earth in a circular path at constant speed. Three statements about the resultant force on the satellite are:

I.   It is equal to the gravitational force of attraction on the satellite.
II.  It is equal to the mass of the satellite multiplied by its acceleration.
III. It is equal to the centripetal force on the satellite.

Which combination of statements is correct?

A.  I and II only

B.  I and III only

C.  II and III only

D.  I, II and III

A satellite is orbiting Earth in a circular path at constant speed. Three statements about the resultant force on the satellite are:

I.   It is equal to the gravitational force of attraction on the satellite.
II.  It is equal to the mass of the satellite multiplied by its acceleration.
III. It is equal to the centripetal force on the satellite.

Which combination of statements is correct?

A.  I and II only

B.  I and III only

C.  II and III only

D.  I, II and III

Two isolated point masses, P of mass m and Q of mass 2m, are separated by a distance 3d. X is a point a distance d from P and 2d from Q.

What is the net gravitational field strength at X and the net gravitational potential at X?

Two isolated point masses, P of mass m and Q of mass 2m, are separated by a distance 3d. X is a point a distance d from P and 2d from Q.

What is the net gravitational field strength at X and the net gravitational potential at X?

Two isolated point masses, P of mass m and Q of mass 2m, are separated by a distance 3d. X is a point a distance d from P and 2d from Q.

What is the net gravitational field strength at X and the net gravitational potential at X?

Two isolated point masses, P of mass m and Q of mass 2m, are separated by a distance 3d. X is a point a distance d from P and 2d from Q.

What is the net gravitational field strength at X and the net gravitational potential at X?

The escape speed from the surface of earth is v_esc. The radius of earth is R. A satellite of mass m is in orbit at a height $\frac{R}{4}$ above the surface of the Earth. What is the energy required to move the satellite to infinity?

A.  $\frac{\mathrm{m}{v^2}_{\mathrm{esc}}}{5}$

B.  $\frac{2\mathrm{m}{v^2}_{\mathrm{esc}}}{5}$

C.  $\mathrm{m}{v^2}_{\mathrm{esc}}$

D.  $2\mathrm{m}{v^2}_{\mathrm{esc}}$

The escape speed from the surface of earth is v_esc. The radius of earth is R. A satellite of mass m is in orbit at a height $\frac{R}{4}$ above the surface of the Earth. What is the energy required to move the satellite to infinity?

A.  $\frac{\mathrm{m}{v^2}_{\mathrm{esc}}}{5}$

B.  $\frac{2\mathrm{m}{v^2}_{\mathrm{esc}}}{5}$

C.  $\mathrm{m}{v^2}_{\mathrm{esc}}$

D.  $2\mathrm{m}{v^2}_{\mathrm{esc}}$

The escape speed from the surface of earth is v_esc. The radius of earth is R. A satellite of mass m is in orbit at a height $\frac{R}{4}$ above the surface of the Earth. What is the energy required to move the satellite to infinity?

A.  $\frac{\mathrm{m}{v^2}_{\mathrm{esc}}}{5}$

B.  $\frac{2\mathrm{m}{v^2}_{\mathrm{esc}}}{5}$

C.  $\mathrm{m}{v^2}_{\mathrm{esc}}$

D.  $2\mathrm{m}{v^2}_{\mathrm{esc}}$

The escape speed from the surface of earth is v_esc. The radius of earth is R. A satellite of mass m is in orbit at a height $\frac{R}{4}$ above the surface of the Earth. What is the energy required to move the satellite to infinity?

A.  $\frac{\mathrm{m}{v^2}_{\mathrm{esc}}}{5}$

B.  $\frac{2\mathrm{m}{v^2}_{\mathrm{esc}}}{5}$

C.  $\mathrm{m}{v^2}_{\mathrm{esc}}$

D.  $2\mathrm{m}{v^2}_{\mathrm{esc}}$

The mass of Mars is about ten times that of the Moon. The radius of Mars is about twice that of the Moon.

What is the $\frac{\mathrm{escape} \mathrm{speed} \mathrm{from} \mathrm{Mars}}{\mathrm{Moon}}$?

A.  $\sqrt{5}$

B.  2$\sqrt{5}$

C.  5

D.  25

The mass of Mars is about ten times that of the Moon. The radius of Mars is about twice that of the Moon.

What is the $\frac{\mathrm{escape} \mathrm{speed} \mathrm{from} \mathrm{Mars}}{\mathrm{Moon}}$?

A.  $\sqrt{5}$

B.  2$\sqrt{5}$

C.  5

D.  25

The mass of Mars is about ten times that of the Moon. The radius of Mars is about twice that of the Moon.

What is the $\frac{\mathrm{escape} \mathrm{speed} \mathrm{from} \mathrm{Mars}}{\mathrm{Moon}}$?

A.  $\sqrt{5}$

B.  2$\sqrt{5}$

C.  5

D.  25

The mass of Mars is about ten times that of the Moon. The radius of Mars is about twice that of the Moon.

What is the $\frac{\mathrm{escape} \mathrm{speed} \mathrm{from} \mathrm{Mars}}{\mathrm{Moon}}$?

A.  $\sqrt{5}$

B.  2$\sqrt{5}$

C.  5

D.  25

The radius of the Earth is R. A satellite is launched to a height h = $\frac{R}{4}$ above the Earth’s surface.

What is $\frac{\mathrm{gravitational} \mathrm{force} \mathrm{on} \mathrm{satellite} \mathrm{at} \mathrm{the} \mathrm{surface}}{\mathrm{gravitational} \mathrm{force} \mathrm{on} \mathrm{satellite} \mathrm{at} \mathrm{height} h}$?

A.  $\frac{4}{5}$

B.  $\frac{16}{25}$

C.  $\frac{25}{16}$

D.  $\frac{5}{4}$

The radius of the Earth is R. A satellite is launched to a height h = $\frac{R}{4}$ above the Earth’s surface.

What is $\frac{\mathrm{gravitational} \mathrm{force} \mathrm{on} \mathrm{satellite} \mathrm{at} \mathrm{the} \mathrm{surface}}{\mathrm{gravitational} \mathrm{force} \mathrm{on} \mathrm{satellite} \mathrm{at} \mathrm{height} h}$?

A.  $\frac{4}{5}$

B.  $\frac{16}{25}$

C.  $\frac{25}{16}$

D.  $\frac{5}{4}$

The radius of the Earth is R. A satellite is launched to a height h = $\frac{R}{4}$ above the Earth’s surface.

What is $\frac{\mathrm{gravitational} \mathrm{force} \mathrm{on} \mathrm{satellite} \mathrm{at} \mathrm{the} \mathrm{surface}}{\mathrm{gravitational} \mathrm{force} \mathrm{on} \mathrm{satellite} \mathrm{at} \mathrm{height} h}$?

A.  $\frac{4}{5}$

B.  $\frac{16}{25}$

C.  $\frac{25}{16}$

D.  $\frac{5}{4}$

The radius of the Earth is R. A satellite is launched to a height h = $\frac{R}{4}$ above the Earth’s surface.

What is $\frac{\mathrm{gravitational} \mathrm{force} \mathrm{on} \mathrm{satellite} \mathrm{at} \mathrm{the} \mathrm{surface}}{\mathrm{gravitational} \mathrm{force} \mathrm{on} \mathrm{satellite} \mathrm{at} \mathrm{height} h}$?

A.  $\frac{4}{5}$

B.  $\frac{16}{25}$

C.  $\frac{25}{16}$

D.  $\frac{5}{4}$

Determine $\frac{g_{\mathrm{M}}}{g_{\mathrm{P}}}$.

Determine $\frac{g_{\mathrm{M}}}{g_{\mathrm{P}}}$.

Determine $\frac{g_{\mathrm{M}}}{g_{\mathrm{P}}}$.

State and explain the magnitude of the resultant gravitational field strength at O.

State and explain the magnitude of the resultant gravitational field strength at O.

State and explain the magnitude of the resultant gravitational field strength at O.

Outline why the graph between P and O is negative.

Outline why the graph between P and O is negative.

Outline why the graph between P and O is negative.

Show that the gravitational potential V_P at the surface of P due to the mass of P is given by V_P = −g_PR_P where R_P is the radius of the planet.

Show that the gravitational potential V_P at the surface of P due to the mass of P is given by V_P = −g_PR_P where R_P is the radius of the planet.

Show that the gravitational potential V_P at the surface of P due to the mass of P is given by V_P = −g_PR_P where R_P is the radius of the planet.

The gravitational potential due to the mass of M at the surface of P can be assumed to be negligible.

Estimate, using the graph, the gravitational potential at the surface of M due to the mass of M.

The gravitational potential due to the mass of M at the surface of P can be assumed to be negligible.

Estimate, using the graph, the gravitational potential at the surface of M due to the mass of M.

The gravitational potential due to the mass of M at the surface of P can be assumed to be negligible.

Estimate, using the graph, the gravitational potential at the surface of M due to the mass of M.

Draw on the axes the variation of gravitational potential between O and M.

Draw on the axes the variation of gravitational potential between O and M.

Draw on the axes the variation of gravitational potential between O and M.