A planet is in a circular orbit around a star. The speed of the planet is constant. The following data are given:

Mass of planet $=8\mathrm{.}0{10}^{24}kg$

Mass of star $=3\mathrm{.}2{10}^{30}kg$

Distance from the star to the planet $=4\mathrm{.}4{10}^{10}m$

a$.$ Explain why a centripetal force is needed for the planet to be in a circular orbit.

b$.$ Calculate the value of the centripetal force.

c$.$ A spacecraft is to be launched from the surface of the planet to escape from the star system. The radius of the planet is $9\mathrm{.}1{10}^{3}km.$

i$.$ Show that the gravitational potential due to the planet and the star at the surface of the planet is about $-5{10}^{9}Jk{g}^{-1}.$

ii$.$ Estimate the escape speed of the spacecraft from the planet$-$star system.

a$.$ The moon Phobos moves around the planet Mars in a circular orbit.

i$.$ Outline the origin of the force that acts on Phobos.

ii$.$ Outline why this force does no work on Phobos.

b$.$ 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.

i$.$ Show that $k=\frac{G}{4{}^2}.$

ii$.$ The following data for the Mars$-$Phobos system and the Earth$-$Moon system are available:

Mass of Earth $=5\mathrm{.}97{10}^{24}kg$

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.

c$.$ The graph shows the variation of the gravitational potential between Earth and the Moon with distance from the centre of Earth. The distance from the Earth is expressed as a fraction of the total distance between the centre of Earth and tha centre of the Moon.

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i$.$ Show that the gravitationet and the star at about $-5{10}^{9}Jg{g}^{-1}.$ from the planet$-$star system. i$.$ Outline the

ii$.$ Outline why this force does no work on Phobos.

and the Earth$-$Moon system are available:

Mass $of$ Earth $=5\mathrm{.}97{10}^{24}kg$

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.

Determine, using the graph, the mass of the $\frac{M}{o}O$

$3.$ An electron is placed at a distance of $0\mathrm{.}40m$ froma fixed point charge of $-6\mathrm{.}0mC.$

a$.$ Show that the electric field strength due to the point charge at the position of the electron is $3\mathrm{.}4{10}^{8}N{C}^{-1}.$

b$.$ i$.$ Calculate the magnitude of the initial acceleration of the electron.

ii$.$ Describe the subsequent motion of the electron.