NEET Gravitation — Set 12

Question 1

What does Kepler’s third law imply about the relationship between orbital period and distance?

Accepted Answer

Kepler’s third law: the square of the orbital period ($ T^2 $) is proportional to the cube of the semi-major axis ($ a^3 $), i.e., $ T^2 \propto a^3 $. This relates time and distance for planetary motion.

Question 2

A moon orbits a planet with a period of 9 days and radius $ 6 \times 10^8 \, \text{m} $. What is the planet’s mass? ($ G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2, 1 \, \text{day} = 86400 \, \text{s} $)

Accepted Answer

$ M = \frac{4\pi^2 r^3}{G T^2} $. $ T = 9 \times 86400 = 7.776 \times 10^5 \, \text{s} $. $ T^2 = 6.046 \times 10^{11} \, \text{s}^2 $. $ r^3 = (6 \times 10^8)^3 = 2.16 \times 10^{26} \, \text{m}^3 $. $ M = \frac{4 \times (3.14)^2 \times 2.16 \times 10^{26}}{6.67 \times 10^{-11} \times 6.046 \times 10^{11}} $. $ M = \frac{8.51 \times 10^{26}}{4.033 \times 10^1} \approx 2.11 \times 10^{25} \, \text{kg} $.

Question 3

A satellite orbits Earth at $ 7 R_E $ from the center. What is its orbital speed? ($ g = 9.8 \, \text{m/s}^2, R_E = 6.4 \times 10^6 \, \text{m} $)

Accepted Answer

$ v = \sqrt{\frac{g R_E^2}{r}} $. $ r = 7 R_E $, $ v = \sqrt{\frac{9.8 \times 6.4 \times 10^6}{7}} $. $ v = \sqrt{8.966 \times 10^6} \approx 2.99 \times 10^3 \, \text{m/s} $.

NEET Physics: Gravitation — Practice Set 12

Q1. What does Kepler’s third law imply about the relationship between orbital period and distance?

Q2. A moon orbits a planet with a period of 9 days and radius \( 6 \times 10^8 \, \text{m} \). What is the planet’s mass? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2, 1 \, \text{day} = 86400 \, \text{s} \))

Q3. A satellite orbits Earth at \( 7 R_E \) from the center. What is its orbital speed? (\( g = 9.8 \, \text{m/s}^2, R_E = 6.4 \times 10^6 \, \text{m} \))

Q4. A planet orbits the Sun with a period of 7 years. If Earth’s orbital radius is \( 1.5 \times 10^{11} \, \text{m} \), what is its semi-major axis?

Q5. What is the gravitational force on a \( 12 \, \text{kg} \) mass \( 10 \, \text{m} \) from the center of a spherical shell of mass \( 600 \, \text{kg} \) and radius \( 8 \, \text{m} \)? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q6. Four \( 4 \, \text{kg} \) masses form a square of side \( 5 \, \text{m} \). What is the potential energy of the system? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q7. What is the gravitational potential due to Earth at \( 1.92 \times 10^7 \, \text{m} \) from its center? (\( M_E = 6 \times 10^{24} \, \text{kg}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q8. What does the heliocentric model, supported by Kepler’s laws, improve over the geocentric model?

Q9. Four \( 5 \, \text{kg} \) masses form a square of side \( 6 \, \text{m} \). What is the potential energy of the system? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q10. A projectile launched at \( 10 \, \text{km/s} \) from Earth reaches a maximum distance of? (Escape speed = \( 11.2 \, \text{km/s}, R_E = 6.4 \times 10^6 \, \text{m} \))