Gravitation and Kepler's Laws — Practice Quiz

A Physics cheat sheet for Gravitation and Kepler's Laws — every key formula with its symbols defined — plus a medium-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. If the distance between two masses is doubled, how does the gravitational force between them change?

    • It is halved.
    • It is doubled.
    • It is quartered.
    • It is quadrupled.

    Answer: It is quartered.

  2. Two objects with masses $m_1 = 10 \text{ kg}$ and $m_2 = 20 \text{ kg}$ are separated by a distance of $r = 0.5 \text{ m}$. What is the gravitational force between them? Use $G = 6.674 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2$.

    • $5.34 \times 10^{-8} \text{ N}$
    • $2.67 \times 10^{-8} \text{ N}$
    • $1.07 \times 10^{-7} \text{ N}$
    • $1.33 \times 10^{-7} \text{ N}$

    Answer: $5.34 \times 10^{-8} \text{ N}$

  3. If a planet's radius is halved while its mass remains constant, how does the gravitational field strength at its surface change?

    • It is halved.
    • It is doubled.
    • It is quartered.
    • It is quadrupled.

    Answer: It is quadrupled.

  4. A satellite orbits a planet of mass $M = 5.97 \times 10^{24} \text{ kg}$ at an altitude of $3.6 \times 10^7 \text{ m}$ from the planet's surface. If the planet's radius is $R = 6.37 \times 10^6 \text{ m}$, what is the gravitational field strength at the satellite's position? Use $G = 6.674 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2$.

    • $9.81 \text{ N/kg}$
    • $0.22 \text{ N/kg}$
    • $0.05 \text{ N/kg}$
    • $1.62 \text{ N/kg}$

    Answer: $0.22 \text{ N/kg}$

  5. According to Kepler's Third Law, if a planet's average orbital radius around a star is increased by a factor of $4$, by what factor will its orbital period increase?

    • $2$
    • $4$
    • $8$
    • $16$

    Answer: $8$

  6. A satellite orbits Earth ($M = 5.97 \times 10^{24} \text{ kg}$) with an orbital period of $T = 86400 \text{ s}$ (1 day). Assuming a circular orbit, what is its orbital radius? Use $G = 6.674 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2$ and $\pi = 3.14159$.

    • $6.37 \times 10^6 \text{ m}$
    • $4.22 \times 10^7 \text{ m}$
    • $3.84 \times 10^8 \text{ m}$
    • $1.50 \times 10^{11} \text{ m}$

    Answer: $4.22 \times 10^7 \text{ m}$

  7. Which of the following statements about gravitational potential ($V$) is true?

    • Gravitational potential is always positive.
    • Gravitational potential increases as distance from the mass $M$ increases.
    • Gravitational potential is zero at the surface of a mass $M$.
    • Gravitational potential is independent of the mass $M$.

    Answer: Gravitational potential increases as distance from the mass $M$ increases.

  8. Two masses $M$ and $m$ are initially at an infinite distance from each other. As they are brought closer to a finite distance $r$, their gravitational potential energy ($PE$) becomes:

    • Positive and increasing.
    • Negative and increasing.
    • Positive and decreasing.
    • Negative and decreasing.

    Answer: Negative and decreasing.

  9. Consider two masses $M$ and $m$ separated by a distance $r$. If the distance $r$ is increased, what happens to the magnitude of the gravitational force ($F_g$) and the gravitational potential energy ($PE$)?

    • $F_g$ decreases, $PE$ increases.
    • $F_g$ increases, $PE$ decreases.
    • $F_g$ decreases, $PE$ decreases.
    • $F_g$ increases, $PE$ increases.

    Answer: $F_g$ decreases, $PE$ increases.

  10. Planet A has mass $M_A$ and radius $R_A$. Planet B has mass $M_B = 2M_A$ and radius $R_B = 2R_A$. How does the gravitational field strength ($g$) on the surface of Planet B compare to that on Planet A?

    • $g_B = \frac{1}{2} g_A$
    • $g_B = g_A$
    • $g_B = 2 g_A$
    • $g_B = 4 g_A$

    Answer: $g_B = \frac{1}{2} g_A$

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