NEET Gravitation — Set 23

Question 1

A planet moves in an elliptical orbit around the Sun with a semi-major axis of $ 2.25 \times 10^{11} \, \text{m} $. If its orbital period is 2 years, what is the mass of the Sun? (Take $ G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 $, $ 1 \, \text{year} = 3.156 \times 10^7 \, \text{s} $)

Accepted Answer

Using Kepler’s third law: $ T^2 = \frac{4\pi^2}{G M_s} a^3 $. Rearrange for $ M_s $: $ M_s = \frac{4\pi^2 a^3}{G T^2} $. $ T = 2 \times 3.156 \times 10^7 = 6.312 \times 10^7 \, \text{s} $. $ a = 2.25 \times 10^{11} \, \text{m} $. $ T^2 = (6.312 \times 10^7)^2 = 3.984 \times 10^{15} \, \text{s}^2 $. $ a^3 = (2.25 \times 10^{11})^3 = 1.139 \times 10^{33} \, \text{m}^3 $. $ M_s = \frac{4 \times (3.14)^2 \times 1.139 \times 10^{33}}{6.67 \times 10^{-11} \times 3.984 \times 10^{15}} $. $ M_s = \frac{4.49 \times 10^{34}}{2.657 \times 10^5} \approx 1.69 \times 10^{30} \, \text{kg} $.

Question 2

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

Accepted Answer

Kepler’s third law: $ \frac{T_p^2}{T_E^2} = \frac{a_p^3}{a_E^3} $. $ T_E = 1 \, \text{year} $, $ T_p = 10 \, \text{years} $, $ a_E = 1.5 \times 10^{11} \, \text{m} $. $ \frac{10^2}{1^2} = \frac{a_p^3}{(1.5 \times 10^{11})^3} $. $ 100 = \frac{a_p^3}{3.375 \times 10^{33}} $. $ a_p^3 = 100 \times 3.375 \times 10^{33} = 3.375 \times 10^{35} $. $ a_p = (3.375 \times 10^{35})^{1/3} \approx 6.96 \times 10^{11} \, \text{m} $.

Question 3

Three masses of $ 4 \, \text{kg} $ each are at the vertices of an equilateral triangle with side $ 3 \, \text{m} $. What is the net force on one mass? ($ G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 $)

Accepted Answer

Force between two masses: $ F = G \frac{m_1 m_2}{r^2} = 6.67 \times 10^{-11} \frac{4 \times 4}{3^2} = 1.185 \times 10^{-10} \, \text{N} $. Two forces act at 60°: $ F_R = \sqrt{F^2 + F^2 + 2 F^2 \cos 60^\circ} $. $ F_R = \sqrt{(1.185 \times 10^{-10})^2 (1 + 1 + 1)} = 1.185 \times 10^{-10} \sqrt{3} $. $ F_R \approx 2.05 \times 10^{-10} \, \text{N} $.

NEET Physics: Gravitation — Practice Set 23

Q2. A planet orbits the Sun with a period of 10 years. If Earth’s orbital radius is \( 1.5 \times 10^{11} \, \text{m} \), what is its semi-major axis?

Q3. Three masses of \( 4 \, \text{kg} \) each are at the vertices of an equilateral triangle with side \( 3 \, \text{m} \). What is the net force on one mass? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q4. A satellite near Earth has a period of 86 minutes. What is its period at \( h = 5 R_E \)? (\( R_E = 6.4 \times 10^6 \, \text{m} \))

Q5. What is the primary source of centripetal force for an Earth satellite?

Q6. What is the kinetic energy of a \( 150 \, \text{kg} \) satellite at \( 3 R_E \) from Earth’s center? (\( M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q7. What is the escape speed from a planet with mass \( 1.8 \times 10^{24} \, \text{kg} \) and radius \( 3.5 \times 10^6 \, \text{m} \)? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q8. What condition must an object’s energy satisfy to remain bound to Earth?

Q9. What is the kinetic energy of a \( 500 \, \text{kg} \) satellite at \( 7 R_E \) from Earth’s center? (\( M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q10. What does the negative total energy of a bound system indicate?