NEET Gravitation — Set 17

Question 1

Three masses of $ 12 \, \text{kg} $ each form an equilateral triangle with side $ 9 \, \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^2}{r^2} = 6.67 \times 10^{-11} \frac{12 \times 12}{9^2} = 1.185 \times 10^{-10} \, \text{N} $. Two forces 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} $.

Question 2

A satellite of mass $ 500 \, \text{kg} $ orbits Earth at $ 2 R_E $ from the center. What is its kinetic 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 $)

Accepted Answer

$ K = \frac{G M_E m}{2 r} $. $ r = 2 R_E = 2 \times 6.4 \times 10^6 = 1.28 \times 10^7 \, \text{m} $. $ K = \frac{6.67 \times 10^{-11} \times 6 \times 10^{24} \times 500}{2 \times 1.28 \times 10^7} $. $ K = \frac{2.001 \times 10^{17}}{2.56 \times 10^7} \approx 7.82 \times 10^9 \, \text{J} $.

Question 3

The time period of a satellite near Earth’s surface is 84 minutes. What is its time period at a height $ h = R_E $? ($ R_E = 6.4 \times 10^6 \, \text{m} $)

Accepted Answer

$ T^2 = k (R_E + h)^3 $, where $ k = \frac{4\pi^2}{G M_E} $. For $ h = 0 $, $ T_0 = 84 \, \text{min} $, $ T_0^2 = k R_E^3 $. For $ h = R_E $, $ r = 2 R_E $, $ T^2 = k (2 R_E)^3 = 8 k R_E^3 $. $ T = \sqrt{8} T_0 = 2.828 \times 84 \approx 237 \, \text{min} $.

NEET Physics: Gravitation — Practice Set 17

Q1. Three masses of \( 12 \, \text{kg} \) each form an equilateral triangle with side \( 9 \, \text{m} \). What is the net force on one mass? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q2. A satellite of mass \( 500 \, \text{kg} \) orbits Earth at \( 2 R_E \) from the center. What is its kinetic energy? (\( M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q3. The time period of a satellite near Earth’s surface is 84 minutes. What is its time period at a height \( h = R_E \)? (\( R_E = 6.4 \times 10^6 \, \text{m} \))

Q4. Why is the gravitational force on a point mass outside a hollow spherical shell equivalent to that of a point mass at its center?

Q5. Why does the gravitational potential energy approach zero as the distance between two masses increases?

Q6. How much energy is required to move a \( 100 \, \text{kg} \) satellite from \( 4 R_E \) to \( 8 R_E \) 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. A \( 900 \, \text{kg} \) satellite orbits Earth at \( 14 R_E \) from the center. What is its total energy? (\( M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q8. A \( 7 \, \text{kg} \) mass is moved from \( 7 R_E \) to \( 14 R_E \) 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. What is the escape speed from a planet with mass \( 3.6 \times 10^{24} \, \text{kg} \) and radius \( 5 \times 10^6 \, \text{m} \)? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q10. Energy required to move a \( 300 \, \text{kg} \) satellite from \( 3 R_E \) to \( 6 R_E \) is? (\( M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))