Gravitational field strength Transitional Reasoning Practice

Questions

1. 1.

   A deep-space tracking station monitors two identical spherical research outposts, each of mass $M$, that are held stationary at a separation distance of $2d$. A navigation beacon is positioned along the perpendicular bisector of the line connecting their centers, at a distance $d$ from the midpoint between them. Assuming the two outposts can be treated as point masses, what is the magnitude of the net gravitational field strength at the beacon?

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

   An exploration lander rests on the equator of a uniform spherical planet of radius $R$, where the local gravitational field strength is $g$. Due to the planet's rotation, a calibrated scale beneath a test object measures an apparent weight equal to 75% of the true gravitational force exerted by the planet. What is the linear speed $v$ of the lander at the equator due to this rotation?

   • A) $\frac{\sqrt{3}}{2}\sqrt{gR}$
   • B) $\frac{1}{4}\sqrt{gR}$
   • C) $\frac{1}{2}\sqrt{gR}$
   • D) $\sqrt{gR}$
3. 3.

   Two scientific satellites, Alpha and Beta, travel in concentric circular orbits around a spherical planet. Satellite Alpha orbits at a distance where the planet's gravitational field strength is $16\text{ N/kg}$, while Satellite Beta orbits farther away where the gravitational field strength is $1.0\text{ N/kg}$. What is the ratio of Satellite Alpha's orbital speed to Satellite Beta's orbital speed, $\frac{v_{\alpha}}{v_{\beta}}$?

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

   An exploratory lander on a newly discovered spherical exoplanet releases a test probe from rest near the planet's surface. The probe falls vertically through a distance of 18 m in 3.0 s in the absence of atmospheric resistance. Given that the exoplanet's radius is half of Earth's radius and Earth's surface gravitational field strength is 10 m/s2, what is the mass of the exoplanet in terms of Earth's mass, $M_{\text{Earth}}$?

   • A) $0.80 M_{\text{Earth}}$
   • B) $0.40 M_{\text{Earth}}$
   • C) $0.20 M_{\text{Earth}}$
   • D) $0.10 M_{\text{Earth}}$
5. 5.

   A scientific satellite orbits a planet in a circular path of radius $r$ with an orbital period $T$. Flight controllers command the satellite to transition into a higher circular orbit where the planet's gravitational field strength is reduced to one-fourth of its initial value. What is the satellite's new orbital period in terms of $T$?

   • A) $2\sqrt{2}T$
   • B) $2T$
   • C) $4T$
   • D) $8T$
6. 6.

   A stationary robotic lander on an asteroid measures the gravitational force exerted on a 0.50 kg calibration mass at its landing site to be 0.15 N. When the lander replaces the calibration mass with a 2.0 kg rock sample at the exact same position, how do the gravitational force acting on the sample and the asteroid's local gravitational field strength compare to the initial values?

   • A) The gravitational force increases to 0.60 N, while the gravitational field strength remains 0.30 N/kg.
   • B) Both the gravitational force and the gravitational field strength quadruple to 0.60 N and 1.20 N/kg, respectively.
   • C) The gravitational force remains 0.15 N, while the gravitational field strength decreases to 0.075 N/kg.
   • D) The gravitational force increases to 0.60 N, while the gravitational field strength decreases to 0.075 N/kg.
7. 7.

   A deep-space probe enters a stable circular orbit around an unknown spherical asteroid at a radial distance $r$ from its center, maintaining a constant orbital speed $v$. Flight engineers must determine the asteroid's gravitational field strength at a scientific monitoring station situated at a radial distance of $2r$ from the asteroid's center. Assuming the asteroid behaves as a uniform spherical mass, what is the magnitude of the asteroid's gravitational field strength at the monitoring station in terms of $v$ and $r$?

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

   A scientific probe travels along the straight line connecting the centers of a spherical planet of mass $16M$ and its moon of mass $M$, which are separated by a constant distance $d$. As the probe moves between the two bodies, the gravitational fields exerted by the planet and the moon act in opposite directions. At what distance from the center of the planet is the magnitude of the net gravitational field strength equal to zero?

   • A) $0.20d$
   • B) $0.25d$
   • C) $0.75d$
   • D) $0.80d$
9. 9.

   Astronomers analyzing two spherical exoplanets, Planet A and Planet B, determine that both have uniform density throughout their volumes. Planet B has an average density equal to one-third that of Planet A, but its radius is 1.5 times the radius of Planet A. Based on these measurements, what is the gravitational field strength at the surface of Planet B in terms of the surface gravitational field strength of Planet A, $g_A$?

   • A) $0.25 g_A$
   • B) $0.50 g_A$
   • C) $1.50 g_A$
   • D) $4.50 g_A$
10. 10.

    An automated research satellite orbits a spherical planet of radius $R$ at an initial altitude equal to $R$ above the surface, recording a gravitational field strength of $g_{\text{orbit}}$. Flight controllers later command the satellite to boost into a higher circular orbit at an altitude of $3R$ above the surface. Assuming the planet has a uniform spherical mass distribution, what is the gravitational field strength at this new orbital altitude?

    • A) $0.11 g_{\text{orbit}}$
    • B) $0.33 g_{\text{orbit}}$
    • C) $0.50 g_{\text{orbit}}$
    • D) $0.25 g_{\text{orbit}}$

Answers and step-by-step solutions appear after you finish the test.