Gravitational potential energy (general form) Knowledge-Based Download PDF

Questions

1. 1.

   A projectile of mass $m$ is launched vertically upward from the surface of a spherical, airless planet of mass $M$ and radius $R$ with an initial speed $v_0 = \sqrt{\frac{GM}{R}}$. What is the maximum distance from the center of the planet reached by the projectile?

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

   A particle of mass $m$ is released from rest at a distance $r$ from the center of a fixed, isolated planet of mass $M$. Neglecting any non-conservative forces, what is the kinetic energy of the particle when it reaches a distance of $\frac{r}{2}$ from the center of the planet?

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

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

   • A) $-\frac{G m^2}{a}$
   • B) $-\frac{3 G m^2}{2a}$
   • C) $-\frac{2 G m^2}{a}$
   • D) $-\frac{3 G m^2}{a}$
4. 4.

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

   • A) $-\frac{1}{2}$
   • B) $-1$
   • C) $-2$
   • D) $2$
5. 5.

   Taking the gravitational potential energy of two point masses, $M$ and $m$, to be zero at infinite separation, how does their gravitational potential energy change as their separation decreases to $r$, and what is its value at this separation?

   • A) It decreases to $-\frac{GMm}{r}$.
   • B) It increases to $+\frac{GMm}{r}$.
   • C) It decreases to $-\frac{GMm}{r^2}$.
   • D) It increases to $+\frac{GMm}{r^2}$.
6. 6.

   A satellite of mass $m$ is moved from a circular orbit of radius $r$ to a higher orbit of radius $3r$ around a planet of mass $M$. How much does the gravitational potential energy of the planet-satellite system change during this transfer?

   • A) $-\frac{2GMm}{3r}$
   • B) $-\frac{GMm}{3r}$
   • C) $+\frac{GMm}{3r}$
   • D) $+\frac{2GMm}{3r}$