Gravitation and Kepler's Laws — Hard Practice Quiz

A Physics cheat sheet for Gravitation and Kepler's Laws — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Formulas & key concepts

Newton's Law of Universal Gravitation: The attractive force \(F_g\) between two masses \(m_1\), \(m_2\) separated by distance \(r\).

$$F_g = G \frac{m_1m_2}{r^2}$$

Where: \(G\) = Universal Gravitation Constant \((6.674 \times 10^{-11} N\cdot m^2/kg^2)\)

Gravitational Field Strength (Acceleration due to gravity): The field strength \(g\) at a distance \(r\) from a large mass \(M\).

$$g = G \frac{M}{r^2}$$

Where: \(G\) = Universal Gravitation Constant

Kepler's Third Law (Law of Harmonies): The square of the orbital period \(T\) is proportional to the cube of the average orbital radius (or semi-major axis) \(r\).

$$T^2 \propto r^3 \text{ or } T^2 = k r^3$$

Where: \(k\) = a constant that is the same for all objects orbiting the same central mass

Kepler's Third Law (General Form): The mathematical form derived from Newton's laws for an object of period \(T\) orbiting a mass \(M\) with radius \(r\) (for circular orbits).

$$T^2 = \frac{4\pi^2}{GM} r^3$$

Where: \(G\) = Universal Gravitation Constant; \(M\) = mass of the central body

Gravitational Potential: The potential energy per unit mass at a distance \(r\) from mass \(M\).

$$V = -\frac{GM}{r}$$

Where: \(G\) = Universal Gravitation Constant

Gravitational Potential Energy: The potential energy \(PE\) between two masses \(M\) and \(m\) separated by distance \(r\).

$$PE = -\frac{GMm}{r}$$

Where: \(G\) = Universal Gravitation Constant

Practice quiz

  1. A satellite orbits a planet at a constant radius $r$ with an orbital period $T$. If the gravitational field strength at this orbital radius is $g$, which of the following expressions correctly relates $T$, $r$, and $g$?

    • $T = 2\pi \sqrt{\frac{r}{g}}$
    • $T = \frac{2\pi r}{g}$
    • $T = \sqrt{\frac{g}{4\pi^2 r}}$
    • $T = \frac{4\pi^2 r}{g}$

    Answer: $T = 2\pi \sqrt{\frac{r}{g}}$

  2. Two point masses, $m_1$ and $m_2$, are initially separated by a distance $r$. The gravitational force between them is $F_g$ and their gravitational potential energy is $PE$. If the distance between the masses is doubled to $2r$, and simultaneously the mass $m_1$ is also doubled to $2m_1$, what are the new gravitational force $F_g'$ and gravitational potential energy $PE'$ in terms of $F_g$ and $PE$?

    • $F_g' = \frac{1}{2} F_g$, $PE' = PE$
    • $F_g' = \frac{1}{2} F_g$, $PE' = \frac{1}{2} PE$
    • $F_g' = F_g$, $PE' = PE$
    • $F_g' = 2 F_g$, $PE' = 2 PE$

    Answer: $F_g' = \frac{1}{2} F_g$, $PE' = PE$

  3. The gravitational potential at a distance $r$ from a central mass $M$ is given by $V$. Which of the following expressions correctly represents the magnitude of the gravitational field strength $g$ at the same distance $r$ in terms of $V$ and $r$?

    • $g = \frac{|V|}{r}$
    • $g = |V|r$
    • $g = \frac{|V|}{r^2}$
    • $g = \frac{r}{|V|}$

    Answer: $g = \frac{|V|}{r}$

  4. A satellite of mass $m$ is in a stable circular orbit around a planet of mass $M$ at a radius $r$. Which of the following expressions correctly gives the orbital speed $v$ of the satellite?

    • $v = \sqrt{\frac{GM}{r}}$
    • $v = \frac{GM}{r}$
    • $v = \sqrt{\frac{GMm}{r}}$
    • $v = \frac{2\pi r}{T}$

    Answer: $v = \sqrt{\frac{GM}{r}}$

  5. A small object of mass $m$ is initially at the surface of a planet of mass $M$ and radius $R$. How much external work must be done to move the object to an altitude equal to the planet's radius (i.e., to a distance $2R$ from the planet's center)?

    • $\frac{GMm}{2R}$
    • $\frac{GMm}{R}$
    • $-\frac{GMm}{2R}$
    • $0$

    Answer: $\frac{GMm}{2R}$

  6. Planet X orbits a star of mass $M_X$ with an orbital period $T_X$ and orbital radius $r_X$. Planet Y orbits a different star of mass $M_Y$ with an orbital period $T_Y$ and orbital radius $r_Y$. If $M_X = 4M_Y$ and $r_X = r_Y$, what is the ratio $\frac{T_X}{T_Y}$?

    • $\frac{1}{2}$
    • $2$
    • $\frac{1}{4}$
    • $4$

    Answer: $\frac{1}{2}$

  7. At what distance $r$ from a central mass $M$ would the magnitude of the gravitational potential $V$ be numerically equal to the gravitational field strength $g$? (Assume $r$ is in meters, $M$ in kilograms, $G$ in $\text{N} \cdot \text{m}^2/\text{kg}^2$, and ignore units for the numerical comparison).

    • $r = 1 \text{ m}$
    • $r = G \text{ m}$
    • $r = M \text{ m}$
    • This is never possible due to differing units.

    Answer: $r = 1 \text{ m}$

  8. A small satellite orbits a spherical planet of uniform density $\rho$ and radius $R$ in a circular orbit with an orbital radius approximately equal to the planet's radius $R$ (i.e., just above its surface). Which of the following expressions correctly gives the orbital period $T$ of the satellite?

    • $T = \sqrt{\frac{3\pi}{G\rho}}$
    • $T = 2\pi \sqrt{\frac{R^3}{GM}}$
    • $T = \sqrt{\frac{4\pi^2 R}{G\rho}}$
    • $T = \frac{2\pi R}{v}$

    Answer: $T = \sqrt{\frac{3\pi}{G\rho}}$

  9. A small mass $m$ is moved from an initial distance $r$ to a final distance $2r$ from a much larger central mass $M$. Let $\Delta PE$ be the change in gravitational potential energy of the system, and $W_g$ be the work done by the gravitational force during this displacement. Which of the following statements is true?

    • $\Delta PE = -W_g$
    • $\Delta PE = W_g$
    • $\Delta PE = 2W_g$
    • $\Delta PE = \frac{1}{2} W_g$

    Answer: $\Delta PE = -W_g$

  10. An object is at a distance $r$ from a central mass $M$, where the gravitational potential is $V$. Which of the following expressions correctly relates the escape velocity $v_{esc}$ from this distance to the gravitational potential $V$?

    • $v_{esc} = \sqrt{-2V}$
    • $v_{esc} = \sqrt{2V}$
    • $v_{esc} = -2V$
    • $v_{esc} = \frac{V}{r}$

    Answer: $v_{esc} = \sqrt{-2V}$

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