Gravitational field strength Transitional Reasoning Download PDF

Questions

1. 1.

   A robotic lander on an airless, spherical planet of radius $R$ measures the local surface gravitational field strength by dropping a small test mass from rest from a height $h \ll R$, recording a fall time $t$. An orbiter is simultaneously placed into a circular orbit at an altitude equal to the planet's radius $R$. In terms of $R$, $h$, and $t$, what is the orbital speed of the orbiter?

   • A) $\frac{\sqrt{2Rh}}{t}$
   • B) $\frac{\sqrt{Rh}}{t}$
   • C) $\frac{2\sqrt{Rh}}{t}$
   • D) $\frac{\sqrt{Rh}}{2t}$
2. 2.

   An exploration mission deploys orbiters around two newly discovered airless, spherical planets, Planet X and Planet Y. Flight data shows that a satellite in a low-altitude circular orbit skimming the surface of Planet X moves at twice the orbital speed of an equivalent surface-skimming satellite orbiting Planet Y ($v_X = 2v_Y$). If the radius of Planet X is half the radius of Planet Y ($R_X = \frac{1}{2}R_Y$), what is the ratio of the surface gravitational field strength of Planet X to that of Planet Y, $\frac{g_X}{g_Y}$?

   • A) 2
   • B) 4
   • C) 8
   • D) 16
3. 3.

   To map the gravitational environment of an uncharted exoplanet, an automated scientific probe enters a stable circular orbit. Navigational sensors aboard the probe record that it maintains a constant orbital speed $v$ and completes one full revolution in a period $T$. Assuming the planet's gravitational attraction is the only force acting on the craft, which expression correctly gives the magnitude of the gravitational field strength $g$ at the probe's orbit?

   • A) $\frac{v}{T}$
   • B) $\frac{\pi v^2}{T}$
   • C) $\frac{4\pi^2 v}{T^2}$
   • D) $\frac{2\pi v}{T}$
4. 4.

   A planetary lander touches down on the equator of a uniform, spherical planet of radius $R$ that rotates uniformly with a rotational period $T$. Due to the planet's rotation, a sensitive force sensor measures an object's apparent weight to be exactly $\frac{3}{4}$ of the true gravitational force exerted on it. In terms of $R$ and $T$, what is the gravitational field strength $g$ at the planet's equator?

   • A) $\frac{4\pi^2 R}{T^2}$
   • B) $\frac{16\pi^2 R}{T^2}$
   • C) $\frac{16\pi^2 R}{3T^2}$
   • D) $\frac{64\pi^2 R}{3T^2}$
5. 5.

   A deep-space survey identifies two identical, spherically symmetric stars, each of mass $M$, separated by a fixed center-to-center distance of $2D$. A scientific probe is positioned on the perpendicular bisector of the line connecting their centers, at a distance $D$ from the midpoint of that line. Assuming these two stars are the only significant sources of gravitational attraction in the region, what is the magnitude of the net gravitational field strength at the position of the probe?

   • A) $\frac{GM}{2D^2}$
   • B) $\frac{GM}{D^2}$
   • C) $\frac{\sqrt{2}GM}{2D^2}$
   • D) $\frac{\sqrt{2}GM}{4D^2}$
6. 6.

   A research station orbits a spherical planet in a stable circular path of radius $r$ with an orbital period $T_0$. A deep-space observatory is placed in a wider circular orbit around the same planet where the gravitational field strength is precisely $\frac{1}{4}$ of its value at the station's orbit. Assuming the planet is the only significant source of gravity, what is the orbital period of the observatory in terms of $T_0$?

   • A) $2T_0$
   • B) $4T_0$
   • C) $8T_0$
   • D) $2\sqrt{2}T_0$
7. 7.

   An autonomous scientific satellite orbits an uncharted spherical moon of radius $R$ in a stable circular orbit at an orbital radius of $1.5R$. Instruments aboard the satellite record its constant orbital speed as $v$. Assuming the moon has a uniform mass distribution and no atmosphere, which of the following expressions represents the gravitational field strength at the surface of the moon?

   • A) $\frac{9v^2}{4R}$
   • B) $\frac{3v^2}{2R}$
   • C) $\frac{2v^2}{3R}$
   • D) $\frac{4v^2}{9R}$
8. 8.

   A deep-space probe travels along the straight line connecting the centers of a planet of mass $M$ and its moon of mass $m$, which are separated by a center-to-center distance $d$. If the mass of the planet is 16 times the mass of the moon ($M = 16m$), at what distance from the center of the planet is the net gravitational field strength acting on the probe equal to zero?

   • A) $0.80d$
   • B) $0.94d$
   • C) $0.75d$
   • D) $0.20d$
9. 9.

   An interplanetary probe analyzes two newly surveyed spherical moons, Moon X and Moon Y, which both have uniform internal compositions. Moon X has half the radius of Moon Y, but its mean density is three times that of Moon Y. What is the ratio of the surface gravitational field strength of Moon X to that of Moon Y, $\frac{g_X}{g_Y}$?

   • A) $\frac{2}{3}$
   • B) $\frac{3}{2}$
   • C) $\frac{3}{4}$
   • D) 6
10. 10.

    An astronomical survey team discovers a spherical exoplanet with 8 times the mass of Earth and 2 times the radius of Earth. A research satellite is placed into orbit at an altitude above the exoplanet's surface that is equal to the exoplanet's radius. If the gravitational field strength at Earth's surface is $g$, what is the gravitational field strength experienced by the satellite at this orbital location?

    • A) $2.0g$
    • B) $1.0g$
    • C) $0.5g$
    • D) $0.25g$