Gravitation

An exoplanet is discovered by the Kepler mission. It has a mass $M$ with angular velocity $\omega$. A small moon of mass $m$ and radius $a$ orbits the planet at a centre to centre distance of $r$. What is the condition for this circular orbit?

If $R$ is the reaction force on a loose rock on the moon's surface, write down the equation for the 'equilibrium' of the rock on the moon's surface. Assume that the moon orbits the planet always keeping the same face towards the planet. Deduce the condition, independent of $\omega$, to be satisfied by $M / m$, for the rock to be lifted off the moon by the planet's gravitational attraction. [8]

Estimate the mean density of the Earth, in $\mathrm{g} \mathrm{cm}^{-3}$, assuming that the radius of the Earth $R_{E}=6.38 \times 10^{3} \mathrm{~km}$ and the value of $\mathrm{g}=9.81 \mathrm{~ms}^{-2}$. State any assumptions made. [5]

(i) A satellite is in orbit just above a spherical planet of radius $R$ and uniform density $\rho$. If the periodic time for each orbit is $T$, find an expression for $\rho T^{2}$. Comment on the result.

(ii) A man with a mass of $75 \mathrm{~kg}$ stands at the end of a diving board, depressing it by $0.30 \mathrm{~m}$. What would be the period of his motion if he was to jump lightly in rhythm with the harmonic motion of the diving board? [6]

The largest moon of Jupiter, Ganymede, revolves around the planet in a circular orbit of radius $1.07 \times 10^{6} \mathrm{~km}$ and period $7.16$ days. Determine the mass of Jupiter, $M_{\jmath}$, in terms of the mass of the Earth, $M_{E}$. [5]

$$ \text { The radius of the Earth } R_{E}=6.38 \times 10^{6} \mathrm{~m} $$

Assuming the Earth is a homogeneous sphere, calculate the fractional difference between the acceleration due to free fall at the Earth's equator and at the poles, indicating which is the greater. You may not use a value of $g$ given in the constants table as that is an average value, neither correct at the poles nor at the equator. [8]

Mass of the $M_{\mathrm{E}}=5.98 \times 10^{24} \mathrm{~kg} \quad$ Radius of the Earth $R_{\mathrm{E}}=6.38 \times 10^{6} \mathrm{~m}$

Two spheres, of uniform density, one of mass $m_{1}$ and radius $r_{1}$ and the other of mass $m_{2}$ and radius $r_{2}$, attract each other gravitationally. What is their relative speed at the instant of collision if they are released from rest when a great distance apart? [7]

A person might reasonably expect to jump a height of $1 \mathrm{~m}$ on Earth. On a planet with a density two thirds that of Earth, and radius twice that of the Earth, to what height might the person jump? Assume that they supply the same energy to make the jump. [4]

Io and Europa are both moons of Jupiter. Europa takes twice as long as Io to complete an orbit. What is the ratio of the centripetal acceleration of Io and Europa, $\frac{a_{\mathrm{Io}}}{a_{\text {Europa }}}$ ?

You may use the result that for this gravitational system, $\omega^{2} r^{3}=$ constant, where $\omega$ is the angular velocity and $r$ is the radius of the orbit. [2]

The Earth orbits the Sun at a distance of $8.0$ light minutes away. The Sun lies at a position in the Milky Way galaxy at about $8.3\,\text{kpc}$ from the centre. The galaxy rotates about the galactic centre with an orbital period of $240$ million years. What is the value of the orbital speed of the Solar System in the galaxy divided by the orbital speed of the Earth about the Sun? $$1.00\,\text{pc (parsec)} = 3.26\ \text{light years}$$ [3]