Newton's law of universal gravitation Transitional Reasoning Download PDF

Questions

1. 1.

   Two identical scientific satellites, Alpha and Beta, travel in stable circular orbits around a uniform spherical planet. Satellite Alpha moves at an orbital speed $v$ and experiences a gravitational force of magnitude $F$ from the planet. If Satellite Beta moves at an orbital speed of $2v$, what is the magnitude of the gravitational force exerted by the planet on Satellite Beta in terms of $F$?

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

   Two orbital research satellites, Satellite 1 and Satellite 2, travel in concentric circular orbits around a uniform spherical planet of mass $M$. Satellite 1 orbits at a distance $r$ from the planet's center, while Satellite 2 orbits at a distance of $2r$. Assuming gravity is the only force acting on the satellites, what is the ratio of the centripetal acceleration of Satellite 1 to that of Satellite 2, $a_1 / a_2$?

   • A) 2
   • B) 4
   • C) 1/2
   • D) 1/4
3. 3.

   An astronomical survey team discovers two exoplanets, Planet 1 and Planet 2, in stable circular orbits around a star of mass $M$. Planet 1 orbits at a distance $r$ with an orbital period $T$, while Planet 2 orbits at a distance of $4r$. Assuming that the gravitational attraction of the central star provides the necessary centripetal force and that interplanetary gravitational forces are negligible, what is the orbital period of Planet 2 in terms of $T$?

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

   An astronomical survey detects an isolated binary star system consisting of two identical stars, each of mass $M$, separated by a constant center-to-center distance $d$. Under the sole influence of their mutual gravitational attraction, the two stars travel in circular orbits around their common center of mass. Based on Newton's law of universal gravitation and circular motion principles, what is the orbital speed $v$ of each star?

   • A) $v = \sqrt{\frac{2GM}{d}}$
   • B) $v = \sqrt{\frac{GM}{d}}$
   • C) $v = \sqrt{\frac{GM}{4d}}$
   • D) $v = \sqrt{\frac{GM}{2d}}$
5. 5.

   A deep-space observatory deploys a formation of three identical spherical scientific probes, each of mass $m$, stationed at the vertices of an equilateral triangle with side length $L$. The mutual gravitational force between any pair of isolated probes has a magnitude of $F$. Under the influence of only their mutual gravitational attractions, what is the magnitude of the net gravitational force exerted on any one of the probes by the other two probes?

   • A) $F$
   • B) $\sqrt{3}F$
   • C) $2F$
   • D) $\sqrt{2}F$
6. 6.

   Two isolated spherical asteroids in deep space, Asteroid X of mass $M$ and Asteroid Y of mass $3M$, interact solely through their mutual gravitational attraction when separated by a center-to-center distance $d$. If both asteroids are released from rest in an inertial reference frame, how does the magnitude of the gravitational force exerted by Asteroid Y on Asteroid X compare to that exerted by Asteroid X on Asteroid Y, and what is the ratio of their initial accelerations, $a_X / a_Y$?

   • A) $F_{\text{Y on X}} = 3 F_{\text{X on Y}}$, and $a_X / a_Y = 3$
   • B) $F_{\text{Y on X}} = 3 F_{\text{X on Y}}$, and $a_X / a_Y = \frac{1}{3}$
   • C) $F_{\text{Y on X}} = F_{\text{X on Y}}$, and $a_X / a_Y = 3$
   • D) $F_{\text{Y on X}} = F_{\text{X on Y}}$, and $a_X / a_Y = 1$
7. 7.

   An autonomous exploration lander of mass $m$ is designed to operate on two newly surveyed, uniform spherical exoplanets, Kepler-Alpha and Kepler-Beta. Planetary data reveals that Kepler-Alpha has twice the radius of Kepler-Beta, yet both exoplanets share the exact same average mass density. If the magnitude of the gravitational force exerted by Kepler-Beta on the lander at its surface is $F$, what is the magnitude of the gravitational force exerted by Kepler-Alpha on the same lander at its surface?

   • A) $0.25F$
   • B) $0.50F$
   • C) $1.00F$
   • D) $2.00F$
8. 8.

   A communications satellite of mass $m$ orbits a uniform spherical planet of radius $R$ at an altitude equal to $R$ above the planet's surface, experiencing a gravitational force of magnitude $F$. Flight controllers activate thrusters to move the satellite into a higher circular orbit at an altitude equal to $3R$ above the surface. What is the magnitude of the gravitational force exerted by the planet on the satellite in this new orbit?

   • A) $\frac{1}{9}F$
   • B) $\frac{1}{3}F$
   • C) $\frac{1}{4}F$
   • D) $\frac{1}{16}F$
9. 9.

   An uncrewed scientific probe is positioned along the line directly connecting a dwarf planet of mass $M$ and its moon of mass $m$, where $M = 9m$. The center-to-center distance between the dwarf planet and the moon is fixed at $L$. At what distance from the center of the dwarf planet is the net gravitational force exerted on the probe by these two bodies equal to zero?

   • A) $\frac{1}{4}L$
   • B) $\frac{1}{10}L$
   • C) $\frac{9}{10}L$
   • D) $\frac{3}{4}L$
10. 10.

    A team of aerospace engineers measures a mutual gravitational attraction of magnitude $F$ between two spherical autonomous exploration modules separated by a center-to-center distance $r$. During a surface deployment, the mass of one module is quadrupled by loading rock core samples while the mass of the other remains unchanged, and their center-to-center distance is adjusted to $2r$. Assuming both modules behave as uniform spherical masses, what is the magnitude of the new gravitational force between them in terms of $F$?

    • A) $F$
    • B) $2F$
    • C) $F/2$
    • D) $4F$