NEET Gravitation — Set 7

Question 1

A satellite orbits a planet at $ 6 \times 10^7 \, \text{m} $ from its center with a period of 15 hours. What is the planet’s mass? ($ G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 $)

Accepted Answer

$ M = \frac{4\pi^2 r^3}{G T^2} $. $ T = 15 \times 3600 = 54000 \, \text{s} $, $ T^2 = 2.916 \times 10^9 \, \text{s}^2 $. $ r^3 = (6 \times 10^7)^3 = 2.16 \times 10^{23} \, \text{m}^3 $. $ M = \frac{4 \times (3.14)^2 \times 2.16 \times 10^{23}}{6.67 \times 10^{-11} \times 2.916 \times 10^9} $. $ M = \frac{8.51 \times 10^{24}}{1.948 \times 10^{-1}} \approx 4.37 \times 10^{25} \, \text{kg} $.

Question 2

A projectile is launched at $ 7 \, \text{km/s} $ from Earth’s surface. What is its maximum distance from the center? (Escape speed = $ 11.2 \, \text{km/s}, R_E = 6.4 \times 10^6 \, \text{m} $)

Accepted Answer

$ \frac{1}{2} v_i^2 - \frac{v_e^2}{2} = -\frac{v_e^2}{2} \frac{R_E}{r} $. $ 24.5 - 62.72 = -62.72 \frac{R_E}{r} $. $ \frac{r}{R_E} = \frac{62.72}{38.22} \approx 1.64 $. $ r = 1.64 \times 6.4 \times 10^6 \approx 1.05 \times 10^7 \, \text{m} $.

Question 3

What is the escape speed from a planet with mass $ 3 \times 10^{24} \, \text{kg} $ and radius $ 4 \times 10^6 \, \text{m} $? ($ G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 $)

Accepted Answer

$ v_e = \sqrt{\frac{2 G M}{R}} $. $ v_e = \sqrt{\frac{2 \times 6.67 \times 10^{-11} \times 3 \times 10^{24}}{4 \times 10^6}} $. $ v_e = \sqrt{\frac{4.002 \times 10^{14}}{4 \times 10^6}} = \sqrt{1.001 \times 10^8} $. $ v_e \approx 1.0 \times 10^4 \, \text{m/s} = 10 \, \text{km/s} $.

NEET Physics: Gravitation — Practice Set 7

Q1. A satellite orbits a planet at \( 6 \times 10^7 \, \text{m} \) from its center with a period of 15 hours. What is the planet’s mass? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q2. A projectile is launched at \( 7 \, \text{km/s} \) from Earth’s surface. What is its maximum distance from the center? (Escape speed = \( 11.2 \, \text{km/s}, R_E = 6.4 \times 10^6 \, \text{m} \))

Q3. What is the escape speed from a planet with mass \( 3 \times 10^{24} \, \text{kg} \) and radius \( 4 \times 10^6 \, \text{m} \)? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q4. Why does the gravitational force on a body inside Earth decrease linearly with depth?

Q5. Which of the following statements is incorrect about Kepler’s second law?

Q6. What is the gravitational force on a \( 8 \, \text{kg} \) mass \( 5 \, \text{m} \) from the center of a spherical shell of mass \( 400 \, \text{kg} \) and radius \( 3 \, \text{m} \)? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q7. A projectile is launched at \( 2 \, \text{km/s} \) from Earth’s surface. What is its maximum distance from the center? (Escape speed = \( 11.2 \, \text{km/s}, R_E = 6.4 \times 10^6 \, \text{m} \))

Q8. A \( 9 \, \text{kg} \) mass is moved from \( 9 R_E \) to \( 18 R_E \) from Earth’s center. What is the change in potential energy? (\( M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q9. At what depth below Earth’s surface is \( g \) reduced to \( 8.33 \, \text{m/s}^2 \)? (\( g_0 = 9.8 \, \text{m/s}^2, R_E = 6.4 \times 10^6 \, \text{m} \))

Q10. Why does Newton’s law of gravitation apply universally to all objects?