Gravitational Fields: Physics 9702 (Cambridge International AS & A Level)

The dwarf planet Piri, a small spherical body being studied by a robotic probe, has mass $3.20\times10^{20}\text{ kg}$ and radius $4.00\times10^5\text{ m}$. Piri may be treated as a uniform sphere, so for a point outside it, its mass acts as a point mass located at its centre.

Take $G = 6.67\times10^{-11}\text{ N m}^2\text{ kg}^{-2}$.

What is the gravitational field strength at the surface of Piri?

The dwarf planet Kerrigan has mass $1.80\times10^{21}\text{ kg}$. Kerrigan is a uniform sphere, so for any point outside it, its mass may be treated as a point mass located at its centre.

Take $G = 6.67\times10^{-11}\text{ N m}^2\text{ kg}^{-2}$.

(a) State Newton's law of gravitation for the force between two point masses, giving it as an equation and defining each symbol used. [2]

(b) Calculate the gravitational field strength at a point $2.50\times10^6\text{ m}$ from the centre of Kerrigan. [2]

(c) A space probe of mass $420\text{ kg}$ is at the point described in (b). Calculate the magnitude of the gravitational force acting on the probe. [2]

(d) An identical probe is instead placed at a point $5.00\times10^6\text{ m}$ from the centre of Kerrigan, twice the distance used in (b). By considering how gravitational field strength depends on distance from a point mass, determine the new field strength at this point without recalculating $GM/r^2$ from scratch. [2]

A space agency probe is exploring the region around the exoplanet Virellon, which has mass $4.20\times10^{25}\text{ kg}$. Virellon may be treated as a uniform sphere, so its mass acts as a point mass at its centre.

Take $G = 6.67\times10^{-11}\text{ N m}^2\text{ kg}^{-2}$.

(a) Define gravitational potential at a point, and explain why the gravitational potential due to Virellon is negative at every point a finite distance from it. [3]

(b) Calculate the gravitational potential at a point $8.00\times10^6\text{ m}$ from the centre of Virellon. [2]

(c) The probe, of mass $250\text{ kg}$, moves from the point in (b) out to a point $1.60\times10^7\text{ m}$ from the centre of Virellon. Calculate the change in the gravitational potential energy of the probe, stating whether this is an increase or a decrease. [3]

(d) State one similarity and one difference between the shape of the graph of gravitational field strength $g$ against distance $r$ from the centre of Virellon, and the shape of the graph of gravitational potential $\phi$ against distance $r$, for points outside Virellon. [2]

A newly discovered exoplanet, Draymoor, has mass $8.10\times10^{24}\text{ kg}$. A small moon orbits Draymoor in a circular orbit of radius $5.00\times10^8\text{ m}$.

Take $G = 6.67\times10^{-11}\text{ N m}^2\text{ kg}^{-2}$.

(a) By equating the gravitational force on the moon to the centripetal force required for its circular motion, show that the orbital speed $v$ of the moon is given by $v = \sqrt{\frac{GM}{r}}$ where $M$ is the mass of Draymoor and $r$ is the orbital radius. [3]

(b) Calculate the orbital speed of the moon. [2]

(c) Hence calculate the orbital period of the moon, giving your answer in days. [2]

(d) A second moon orbits Draymoor with an orbital period exactly twice that of the first moon. Using the relationship between orbital period and orbital radius for objects orbiting the same planet, determine the radius of the second moon's orbit. [2]

An exoplanet named Halvane has mass $5.40\times10^{24}\text{ kg}$ and rotates on its axis once every $30.0$ hours. A communications satellite is to be placed in a geostationary orbit around Halvane, so that it remains above the same point on Halvane's surface at all times.

Take $G = 6.67\times10^{-11}\text{ N m}^2\text{ kg}^{-2}$.

(a) Besides having an orbital period equal to Halvane's rotation period, state two further conditions that the satellite's orbit must satisfy for it to remain geostationary. [2]

(b) By equating the gravitational force on the satellite to the centripetal force required for circular motion, and writing the centripetal force in terms of the orbital period $T$, show that the orbital radius $r$ of a geostationary orbit is given by $r = \left(\frac{GMT^2}{4\pi^2}\right)^{1/3}$ [3]

(c) Calculate the radius of the geostationary orbit around Halvane. [3]

(d) Calculate the orbital speed of the satellite in this geostationary orbit. [2]

Two asteroids in a binary system, P and Q, may be treated as point masses. Asteroid P has mass $5.00\times10^{13}\text{ kg}$ and asteroid Q has mass $2.00\times10^{10}\text{ kg}$. The distance between their centres is $4.00\times10^4\text{ m}$.

Take $G = 6.67\times10^{-11}\text{ N m}^2\text{ kg}^{-2}$.

What is the magnitude of the gravitational force of attraction between P and Q?

Two planets, A and B, may be treated as point masses. Planet A has mass $4.00\times10^{24}\text{ kg}$ and planet B has mass $1.00\times10^{24}\text{ kg}$. The distance between their centres is $6.00\times10^8\text{ m}$. A null point is a point at which the resultant gravitational field strength due to A and B is zero.

Take $G = 6.67\times10^{-11}\text{ N m}^2\text{ kg}^{-2}$.

(a) State the condition, in terms of the magnitudes and directions of the two individual gravitational field contributions, that must be satisfied at a null point. Explain why this point must lie on the line joining the centres of A and B, somewhere between them. [2]

(b) The null point lies on the line joining the centres of A and B, at a distance $x$ from the centre of A (so at a distance $(d-x)$ from the centre of B, where $d$ is the separation of A and B). Show that $\frac{x^2}{(d-x)^2} = \frac{M_A}{M_B}$ and hence calculate $x$. [5]

(c) Calculate the magnitude of the gravitational field strength due to A alone at the null point, and use it to confirm that the resultant field strength there is indeed zero. [3]

Two point masses, C and D, are fixed in space. Mass C is $3.00\times10^{23}\text{ kg}$ and mass D is $7.00\times10^{23}\text{ kg}$. The distance between C and D is $5.00\times10^7\text{ m}$.

Take $G = 6.67\times10^{-11}\text{ N m}^2\text{ kg}^{-2}$.

What is the gravitational potential at the midpoint of the line joining C and D (a distance of $2.50\times10^7\text{ m}$ from each mass)?

A satellite of mass $m$ orbits a planet of mass $M$ in a circular orbit of radius $r$.

(a) By equating the gravitational force on the satellite to the centripetal force required for its circular motion, show that the kinetic energy of the satellite in orbit is given by $E_k = \frac{GMm}{2r}$ [3]

(b) State the expression for the gravitational potential energy $E_p$ of the satellite, and hence show that the total energy of the satellite in its orbit is $E = -\frac{GMm}{2r}$ [2]

(c) A satellite of mass $600\text{ kg}$ orbits a planet of mass $5.00\times10^{24}\text{ kg}$ in a circular orbit of radius $7.00\times10^6\text{ m}$. Take $G=6.67\times10^{-11}\text{ N m}^2\text{ kg}^{-2}$. Calculate the total energy of the satellite in this orbit. [3]

(d) The satellite is moved to a new circular orbit of radius $1.40\times10^7\text{ m}$, i.e. twice the radius in (c). By considering how the total energy depends on orbital radius, determine the new total energy without recalculating $GMm/2r$ from scratch, and state whether the total energy of the satellite has increased or decreased. [2]

A planet named Vexis has mass $6.40\times10^{23}\text{ kg}$. Two space probes orbit Vexis in circular orbits: Probe 1 at radius $1.00\times10^7\text{ m}$, and Probe 2 at radius $4.00\times10^7\text{ m}$ (four times the radius of Probe 1's orbit).

Take $G = 6.67\times10^{-11}\text{ N m}^2\text{ kg}^{-2}$.

(a) Calculate the orbital speed of Probe 1. [2]

(b) Calculate the orbital period of Probe 1, giving your answer in hours. [2]

(c) Using Kepler's third law, and without recalculating $GM/r$ from first principles, calculate the orbital period of Probe 2. [3]

(d) State and explain how the orbital speed of Probe 2 compares with that of Probe 1 (i.e. whether it is faster, slower or the same, and by what factor). [2]