Solved examples of Newton's Law of Gravitation involving orbital motion

📚 Quick Study Guide

• 🌍 Newton's Law of Gravitation: The gravitational force between two objects is $F = G \frac{m_1 m_2}{r^2}$, where $G$ is the gravitational constant ($6.674 × 10^{-11} N(m/kg)^2$), $m_1$ and $m_2$ are the masses, and $r$ is the distance between their centers.
• 🛰️ Orbital Motion: An object in orbit experiences a centripetal force provided by gravity. Thus, $G \frac{Mm}{r^2} = m \frac{v^2}{r}$, where $M$ is the mass of the central body, $m$ is the mass of the orbiting object, $v$ is the orbital speed, and $r$ is the orbital radius.
• ⏱️ Orbital Period: The period $T$ of an orbit is related to the orbital radius $r$ and speed $v$ by $T = \frac{2\pi r}{v}$. It can also be expressed as $T = 2\pi \sqrt{\frac{r^3}{GM}}$.
• 📏 Kepler's Third Law: The square of the orbital period is proportional to the cube of the semi-major axis of the orbit: $T^2 \propto r^3$.
• 💡 Escape Velocity: The minimum speed required for an object to escape the gravitational pull of a planet is $v_e = \sqrt{\frac{2GM}{r}}$.

🧪 Practice Quiz

1. Question 1: A satellite orbits Earth at a distance of 2 times Earth's radius. What is its orbital speed, given Earth's mass $M$ and radius $R$?
   1. $v = \sqrt{\frac{GM}{R}}$
   2. $v = \sqrt{\frac{GM}{2R}}$
   3. $v = \frac{1}{2} \sqrt{\frac{GM}{R}}$
   4. $v = 2\sqrt{\frac{GM}{R}}$
2. Question 2: Two stars of equal mass $M$ are in a circular orbit around their center of mass. The distance between the stars is $2R$. What is the orbital speed of each star?
   1. $v = \sqrt{\frac{GM}{R}}$
   2. $v = \sqrt{\frac{GM}{2R}}$
   3. $v = \sqrt{\frac{2GM}{R}}$
   4. $v = \frac{1}{2}\sqrt{\frac{GM}{R}}$
3. Question 3: A planet has a mass twice that of Earth and a radius twice that of Earth. What is the escape velocity of this planet compared to Earth's escape velocity ($v_E$)?
   1. $v = 0.5 v_E$
   2. $v = v_E$
   3. $v = 2 v_E$
   4. $v = 4 v_E$
4. Question 4: A satellite's orbital period around a planet is $T$. If the orbital radius is increased by a factor of 4, what is the new orbital period?
   1. $2T$
   2. $4T$
   3. $8T$
   4. $16T$
5. Question 5: What is the gravitational force between two 1000 kg masses separated by a distance of 1 meter? (Use $G = 6.674 × 10^{-11} N(m/kg)^2$)
   1. $6.674 × 10^{-5} N$
   2. $6.674 × 10^{-8} N$
   3. $6.674 × 10^{-11} N$
   4. $6.674 × 10^{-14} N$
6. Question 6: A satellite is moved from an orbit with radius $R$ to an orbit with radius $4R$ around a planet of mass $M$. How does the gravitational potential energy change?
   1. Increases by a factor of 4
   2. Decreases by a factor of 4
   3. Increases by a factor of 1/4
   4. Decreases by a factor of 1/4
7. Question 7: A planet has a radius R and gravitational acceleration g at its surface. What is the escape velocity from the surface of the planet?
   1. $\sqrt{gR}$
   2. $\sqrt{2gR}$
   3. $gR$
   4. $2gR$