NEET Gravitation — Set 4

Question 1

What is the kinetic energy of a $ 400 \, \text{kg} $ satellite at $ 6 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 $)

Accepted Answer

$ K = \frac{G M_E m}{2 r} $. $ r = 6 R_E = 3.84 \times 10^7 \, \text{m} $. $ K = \frac{6.67 \times 10^{-11} \times 6 \times 10^{24} \times 400}{2 \times 3.84 \times 10^7} $. $ K = \frac{1.601 \times 10^{17}}{7.68 \times 10^7} \approx 2.08 \times 10^9 \, \text{J} $.

Question 2

What is the gravitational force on a $ 11 \, \text{kg} $ mass $ 9 \, \text{m} $ from the center of a spherical shell of mass $ 500 \, \text{kg} $ and radius $ 7 \, \text{m} $? ($ G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 $)

Accepted Answer

Outside shell: $ F = \frac{G M m}{r^2} $. $ F = \frac{6.67 \times 10^{-11} \times 500 \times 11}{9^2} $. $ F = \frac{3.6685 \times 10^{-8}}{81} \approx 4.53 \times 10^{-10} \, \text{N} $.

Question 3

A satellite’s orbit changes from $ 2 R_E $ to $ 4 R_E $ from Earth’s center. What is the change in its potential energy? ($ m = 200 \, \text{kg}, 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 $)

Accepted Answer

$ \Delta V = -G M_E m \left(\frac{1}{r_2} - \frac{1}{r_1}\right) $. $ r_1 = 2 R_E = 1.28 \times 10^7 \, \text{m} $, $ r_2 = 4 R_E = 2.56 \times 10^7 \, \text{m} $. $ \Delta V = -6.67 \times 10^{-11} \times 6 \times 10^{24} \times 200 \left(\frac{1}{2.56 \times 10^7} - \frac{1}{1.28 \times 10^7}\right) $. $ \Delta V = -8.004 \times 10^{16} (-3.906 \times 10^{-8}) \approx 3.13 \times 10^9 \, \text{J} $.

NEET Physics: Gravitation — Practice Set 4

Q1. What is the kinetic energy of a \( 400 \, \text{kg} \) satellite at \( 6 R_E \) from Earth’s center? (\( M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q2. What is the gravitational force on a \( 11 \, \text{kg} \) mass \( 9 \, \text{m} \) from the center of a spherical shell of mass \( 500 \, \text{kg} \) and radius \( 7 \, \text{m} \)? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q3. A satellite’s orbit changes from \( 2 R_E \) to \( 4 R_E \) from Earth’s center. What is the change in its potential energy? (\( m = 200 \, \text{kg}, M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q4. Which of the following statements is correct about Kepler’s first law?

Q5. Why is the gravitational potential energy of a system of multiple masses the sum of pairwise interactions?

Q6. A satellite near Earth has a period of 81 minutes. What is its period at \( h = 10 R_E \)? (\( R_E = 6.4 \times 10^6 \, \text{m} \))

Q7. A planet’s orbital period around the Sun is 8 years. If Earth’s orbital radius is \( 1.5 \times 10^{11} \, \text{m} \), what is its semi-major axis?

Q8. What is the minimum speed to escape from \( 4 R_E \) from Earth’s center? (\( g = 9.8 \, \text{m/s}^2, R_E = 6.4 \times 10^6 \, \text{m} \))

Q9. What does the zero gravitational potential at infinity imply about gravitational systems?

Q10. A planet orbits the Sun with a period of 9 years. If Earth’s orbital radius is \( 1.5 \times 10^{11} \, \text{m} \), what is its semi-major axis?

Q11. What is the minimum speed to escape from \( 9 R_E \) from Earth’s center? (\( g = 9.8 \, \text{m/s}^2, R_E = 6.4 \times 10^6 \, \text{m} \))

Q12. What is the gravitational force on a \( 10 \, \text{kg} \) mass outside a spherical shell of mass \( 500 \, \text{kg} \) and radius \( 2 \, \text{m} \), at \( 3 \, \text{m} \) from the center? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q13. What is the escape speed from a planet with mass \( 1.2 \times 10^{24} \, \text{kg} \) and radius \( 3 \times 10^6 \, \text{m} \)? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q14. A body weighs \( 196 \, \text{N} \) on Earth’s surface. What is its weight at a depth \( d = R_E/4 \)? (\( g = 9.8 \, \text{m/s}^2 \))

Q15. A body weighs \( 98 \, \text{N} \) on Earth’s surface. What is its weight at a depth of \( R_E/2 \)? (\( R_E = 6.4 \times 10^6 \, \text{m}, g = 9.8 \, \text{m/s}^2 \))