Gravitational potential energy (general form) Knowledge-Based Practice

Questions

1. 1.

   An object of mass $m$ is moved radially away from the center of a planet of mass $M$. If $\Delta U_1$ is the change in gravitational potential energy when moving from radial distance $r$ to $2r$, and $\Delta U_2$ is the change when moving from $2r$ to $3r$, what is the ratio $\frac{\Delta U_1}{\Delta U_2}$?

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

   If the zero-reference point for gravitational potential energy is chosen to be at the surface of a spherical planet of mass $M$ and radius $R$, which expression gives the gravitational potential energy $U(r)$ of an object of mass $m$ at a radial distance $r \ge R$ from the center of the planet?

   • A) $U(r) = -GMm\left(\frac{1}{R} - \frac{1}{r}\right)$
   • B) $U(r) = -\frac{GMm}{r - R}$
   • C) $U(r) = GMm\left(\frac{1}{r} - \frac{1}{R}\right)$
   • D) $U(r) = GMm\left(\frac{1}{R} - \frac{1}{r}\right)$
3. 3.

   Which of the following expressions represents the work done by the gravitational force on an object of mass $m$ as it moves radially from an initial distance $r_1$ to a final distance $r_2$ from the center of a spherical planet of mass $M$?

   • A) $W = GMm\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$
   • B) $W = GMm\left(\frac{1}{r_1^2} - \frac{1}{r_2^2}\right)$
   • C) $W = GMm\left(\frac{1}{r_2} - \frac{1}{r_1}\right)$
   • D) $W = -GMm\left(\frac{1}{r_1} + \frac{1}{r_2}\right)$
4. 4.

   An object of mass $m$ is raised from the surface of a spherical planet of radius $R$ and mass $M$ to a height $h$ above the surface. Expressed in terms of the surface gravitational acceleration $g = \frac{GM}{R^2}$, what is the exact change in gravitational potential energy $\Delta U$ of the object-planet system?

   • A) $\Delta U = mgh\left(1 + \frac{h}{R}\right)$
   • B) $\Delta U = \frac{mgh}{\left(1 + \frac{h}{R}\right)^2}$
   • C) $\Delta U = \frac{mgh}{1 + \frac{h}{R}}$
   • D) $\Delta U = \frac{mgh}{1 - \frac{h}{R}}$
5. 5.

   A satellite of mass $m$ moves in a circular orbit of radius $r$ around a planet of mass $M$. Taking the gravitational potential energy of the system to be zero at infinite separation, what is the satellite's kinetic energy expressed in terms of the system's gravitational potential energy $U$?

   • A) $-\frac{1}{2}U$
   • B) $-U$
   • C) $\frac{1}{2}U$
   • D) $-2U$
6. 6.

   Three identical particles, each of mass $m$, are situated at the vertices of an equilateral triangle of side length $L$. Taking the gravitational potential energy to be zero at infinite separation, what is the total gravitational potential energy of this three-particle system?

   • A) $-\frac{3Gm^2}{L}$
   • B) $-\frac{Gm^2}{L}$
   • C) $-\frac{3Gm^2}{2L}$
   • D) $-\frac{Gm^2}{3L}$
7. 7.

   An object of mass $m$ is released from rest at a distance of $4R$ from the center of a spherical planet of mass $M$ and radius $R$. Neglecting any atmospheric resistance, what is the speed of the object immediately before it reaches the surface of the planet?

   • A) $\sqrt{\frac{3GM}{4R}}$
   • B) $\sqrt{\frac{3GM}{2R}}$
   • C) $\sqrt{\frac{GM}{2R}}$
   • D) $\sqrt{\frac{2GM}{R}}$
8. 8.

   What is the minimum work that an external agent must perform to move a body of mass $m$ from the surface of a spherical planet of mass $M$ and radius $R$ to an infinite distance away, bringing it to rest?

   • A) $\frac{GMm}{R}$
   • B) -$\frac{GMm}{R}$
   • C) $\frac{GMm}{2R}$
   • D) $\frac{2GMm}{R}$
9. 9.

   Taking the gravitational potential energy of a two-body system to be zero at infinite separation, the gravitational potential energy at a finite separation $r$ is given by $U(r) = -\frac{GMm}{r}$. What does the negative sign of $U(r)$ physically indicate about the system?

   • A) The gravitational force between the two masses is repulsive at all finite separations.
   • B) The total mechanical energy of the two-body system must always be strictly zero.
   • C) Energy is released by the system when the two masses are separated from distance $r$ to infinity.
   • D) The system is bound, meaning positive external work is required to separate the masses to infinity.
10. 10.

    A spacecraft of mass $m$ is moved from an initial orbital radius of $R$ to a new orbital radius of $3R$ around a planet of mass $M$. What is the change in gravitational potential energy, $\Delta U = U_{\text{final}} - U_{\text{initial}}$, of the planet-spacecraft system?

    • A) $-\frac{2GMm}{3R}$
    • B) $-\frac{GMm}{3R}$
    • C) $\frac{2GMm}{3R}$
    • D) $\frac{4GMm}{3R}$

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