Newton's law of universal gravitation Transitional Reasoning Practice

Questions

1. 1.

   Astronomers observing two newly discovered exoplanetary systems detect a single natural satellite in a stable circular orbit around each exoplanet. The satellite of Exoplanet X orbits at a distance $r$ from the planet's center with an orbital period $T$, while the satellite of Exoplanet Y orbits at a distance $2r$ with an orbital period $2T$. Assuming each satellite has negligible mass compared to its host planet and gravitational attraction provides the required centripetal force, what is the ratio of the mass of Exoplanet Y to that of Exoplanet X, $\frac{M_Y}{M_X}$?

   • A) 1
   • B) 2
   • C) 4
   • D) 8
2. 2.

   To calibrate a sensitive laboratory instrument, researchers investigate the mutual gravitational attraction between two spherical masses made of the same uniform metal alloy. In the initial setup, two identical solid spheres of radius $R$ are placed in direct contact with each other, resulting in a mutual gravitational force of magnitude $F_1$. In a modified setup, two identical solid spheres of radius $2R$ made of the same alloy are placed in direct contact with each other, producing a mutual gravitational force of magnitude $F_2$. What is the ratio $\frac{F_2}{F_1}$?

   • A) 4
   • B) 16
   • C) 32
   • D) 64
3. 3.

   A robotic exploration probe of constant mass descends radially toward the surface of a spherically symmetric planet of radius $R$. At an altitude of $R$ above the planet's surface, the probe measures a gravitational force of magnitude $F_1$ acting on it. Later in the descent, at an altitude of $\frac{1}{3}R$ above the surface, the probe measures a gravitational force of magnitude $F_2$. Assuming the planet has a uniform mass distribution and no other celestial bodies affect the probe, what is the ratio $\frac{F_2}{F_1}$?

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

   During a deep-space formation-flying test, three identical autonomous probes, each of mass $m$, are positioned at the vertices of an equilateral triangle with side length $L$. The mutual gravitational attraction between any pair of probes has a magnitude $F_0$. Considering the vector nature of Newton's law of universal gravitation, what is the magnitude of the net gravitational force exerted on any one of the probes by the other two?

   • A) $F_0$
   • B) $\sqrt{3}F_0$
   • C) $2F_0$
   • D) $\sqrt{2}F_0$
5. 5.

   Two scientific satellites, Satellite P and Satellite Q, travel in concentric circular orbits around an isolated spherical planet. Tracking data reveals that the orbital period of Satellite Q is 8 times the orbital period of Satellite P. Based on Newton's law of universal gravitation, what is the ratio of the planet's gravitational acceleration at the orbit of Satellite Q to that at the orbit of Satellite P, $\frac{a_Q}{a_P}$?

   • A) $\frac{1}{16}$
   • B) $\frac{1}{8}$
   • C) $\frac{1}{4}$
   • D) $\frac{1}{64}$
6. 6.

   Two scientific observation satellites, Satellite X with mass $m$ and Satellite Y with mass $3m$, orbit a spherical planet of mass $M$ in concentric circular paths. Satellite X maintains an orbital radius $r$, while Satellite Y maintains an orbital radius $4r$. If gravitational attraction from the planet provides the necessary centripetal force for each satellite, what is the ratio of the orbital speed of Satellite X to that of Satellite Y, $\frac{v_X}{v_Y}$?

   • A) 1/2
   • B) 2
   • C) 4
   • D) 12
7. 7.

   Astronomers analyzing two newly discovered, spherically symmetric exoplanets determine their physical properties for an upcoming exploration mission. Planet X has a radius $R$ and a uniform average density $\rho$, while Planet Y has a radius $2R$ and a uniform average density $3\rho$. If an identical robotic lander of mass $m$ is placed on the surface of each planet, what is the ratio of the gravitational force exerted by Planet Y on the lander to that exerted by Planet X, $\frac{F_Y}{F_X}$?

   • A) 24
   • B) 12
   • C) 6
   • D) 3
8. 8.

   A binary star system consists of Star Alpha with mass $M_1 = 4M$ and Star Beta with mass $M_2 = M$, whose centers are separated by a distance $d$. Let $F_{\alpha \beta}$ be the magnitude of the gravitational force exerted by Star Alpha on Star Beta, and $F_{\beta \alpha}$ be the magnitude of the gravitational force exerted by Star Beta on Star Alpha. If $a_\alpha$ and $a_\beta$ represent the magnitudes of the resulting gravitational accelerations of Star Alpha and Star Beta, respectively, which relationship correctly compares these forces and accelerations?

   • A) $F_{\alpha \beta} = 4F_{\beta \alpha}$ and $a_\alpha = a_\beta$
   • B) $F_{\alpha \beta} = 4F_{\beta \alpha}$ and $a_\alpha = \frac{1}{4}a_\beta$
   • C) $F_{\alpha \beta} = F_{\beta \alpha}$ and $a_\alpha = 4a_\beta$
   • D) $F_{\alpha \beta} = F_{\beta \alpha}$ and $a_\alpha = \frac{1}{4}a_\beta$
9. 9.

   A deep-space probe travels along the line connecting two isolated, spherically symmetric moons, Moon A of mass $M$ and Moon B of mass $9M$, whose centers are separated by a distance $D$. As the probe moves between the moons, it passes through a point where the gravitational pull of Moon A exactly balances the gravitational pull of Moon B. What is the distance of this equilibrium point from the center of Moon A?

   • A) $\frac{D}{10}$
   • B) $\frac{D}{5}$
   • C) $\frac{D}{4}$
   • D) $\frac{D}{3}$
10. 10.

    An uncrewed spacecraft of mass $m$ orbits a spherically symmetric asteroid of mass $M$ at a distance $r$ from its center, experiencing a gravitational force of magnitude $F$. During an orbital reconfiguration maneuver, the spacecraft fires thrusters and releases a spent equipment module, reducing its total mass to $\frac{1}{2}m$, while establishing a new orbit at a distance of $3r$ from the asteroid's center. Assuming the asteroid's mass remains unchanged, what is the magnitude of the gravitational force that the asteroid exerts on the spacecraft at this new location?

    • A) $\frac{1}{6}F$
    • B) $\frac{1}{9}F$
    • C) $\frac{2}{9}F$
    • D) $\frac{1}{18}F$

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