NEET Gravitation — Set 11

Question 1

A satellite orbits Earth at $ 17 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 = 17 R_E $, $ v = \sqrt{\frac{9.8 \times 6.4 \times 10^6}{17}} $. $ v = \sqrt{3.694 \times 10^6} \approx 1.92 \times 10^3 \, \text{m/s} $.

Question 2

A planet has an orbital period of 4 years around the Sun. If Earth’s orbital radius is $ 1.5 \times 10^{11} \, \text{m} $, what is the planet’s semi-major axis? ($ 1 \, \text{year} = 3.156 \times 10^7 \, \text{s} $)

Accepted Answer

Kepler’s third law: $ T^2 \propto a^3 $, $ \frac{T_p^2}{T_E^2} = \frac{a_p^3}{a_E^3} $. $ T_E = 1 \, \text{year} $, $ T_p = 4 \, \text{years} $, $ a_E = 1.5 \times 10^{11} \, \text{m} $. $ \frac{4^2}{1^2} = \frac{a_p^3}{(1.5 \times 10^{11})^3} $. $ 16 = \frac{a_p^3}{3.375 \times 10^{33}} $. $ a_p^3 = 16 \times 3.375 \times 10^{33} = 5.4 \times 10^{34} $. $ a_p = (5.4 \times 10^{34})^{1/3} \approx 3.78 \times 10^{11} \, \text{m} $.

Question 3

A body weighs $ 245 \, \text{N} $ on Earth’s surface. What is its weight at a depth $ d = R_E/5 $? ($ g = 9.8 \, \text{m/s}^2 $)

Accepted Answer

$ g(d) = g (1 - d/R_E) $. $ d = R_E/5 $, $ g(d) = 9.8 (1 - 1/5) = 9.8 \times 4/5 = 7.84 \, \text{m/s}^2 $. Mass: $ m = 245/9.8 = 25 \, \text{kg} $. Weight: $ W = 25 \times 7.84 = 196 \, \text{N} $.

NEET Physics: Gravitation — Practice Set 11

Q1. A satellite orbits Earth at \( 17 R_E \) from the center. What is its orbital speed? (\( g = 9.8 \, \text{m/s}^2, R_E = 6.4 \times 10^6 \, \text{m} \))

Q2. A planet has an orbital period of 4 years around the Sun. If Earth’s orbital radius is \( 1.5 \times 10^{11} \, \text{m} \), what is the planet’s semi-major axis? (\( 1 \, \text{year} = 3.156 \times 10^7 \, \text{s} \))

Q3. A body weighs \( 245 \, \text{N} \) on Earth’s surface. What is its weight at a depth \( d = R_E/5 \)? (\( g = 9.8 \, \text{m/s}^2 \))

Q4. A satellite near Earth has a period of 90 minutes. What is its period at \( h = 3 R_E \)? (\( R_E = 6.4 \times 10^6 \, \text{m} \))

Q5. A planet orbits the Sun with a period of 6 years. If Earth’s orbital radius is \( 1.5 \times 10^{11} \, \text{m} \), what is its semi-major axis?

Q6. Three masses of \( 18 \, \text{kg} \) each form an equilateral triangle with side \( 12 \, \text{m} \). What is the net force on one mass? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q7. What is the kinetic energy of a \( 800 \, \text{kg} \) satellite at \( 10 R_E \) from Earth’s center? (\( M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q8. A satellite orbits a planet at \( 2.5 \times 10^7 \, \text{m} \) from its center with a period of 5 hours. What is the planet’s mass? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q9. What role does the Sun play in planetary motion according to Kepler’s laws?

Q10. Which of Kepler’s laws implies that the gravitational force acting on a planet is a central force?