NEET Gravitation — Set 5

Question 1

A satellite near Earth has a period of 87 minutes. What is its period at $ h = 6 R_E $? ($ R_E = 6.4 \times 10^6 \, \text{m} $)

Accepted Answer

$ T^2 \propto (R_E + h)^3 $. $ T_0^2 = k R_E^3 $, $ h = 6 R_E $, $ r = 7 R_E $. $ T^2 = k (7 R_E)^3 = 343 k R_E^3 $. $ T = T_0 \sqrt{343} = 87 \times 18.52 \approx 1611 \, \text{min} $.

Question 2

What is the kinetic energy of a $ 600 \, \text{kg} $ satellite at $ 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 $)

Accepted Answer

$ K = \frac{G M_E m}{2 r} $. $ r = 8 R_E = 5.12 \times 10^7 \, \text{m} $. $ K = \frac{6.67 \times 10^{-11} \times 6 \times 10^{24} \times 600}{2 \times 5.12 \times 10^7} $. $ K = \frac{2.401 \times 10^{17}}{1.024 \times 10^8} \approx 2.34 \times 10^9 \, \text{J} $.

Question 3

A satellite’s period is 100 minutes at $ h = 0 $. What is its period at $ h = 2 R_E $? ($ R_E = 6.4 \times 10^6 \, \text{m} $)

Accepted Answer

$ T^2 \propto (R_E + h)^3 $. $ T_0^2 = k R_E^3 $, $ T^2 = k (3 R_E)^3 = 27 k R_E^3 $. $ T = T_0 \sqrt{27} = 100 \times 5.196 \approx 520 \, \text{min} $.

NEET Physics: Gravitation — Practice Set 5

Q1. A satellite near Earth has a period of 87 minutes. What is its period at \( h = 6 R_E \)? (\( R_E = 6.4 \times 10^6 \, \text{m} \))

Q2. What is the kinetic energy of a \( 600 \, \text{kg} \) satellite at \( 8 R_E \) from Earth’s center? (\( M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q3. A satellite’s period is 100 minutes at \( h = 0 \). What is its period at \( h = 2 R_E \)? (\( R_E = 6.4 \times 10^6 \, \text{m} \))

Q4. Why does the gravitational force act as if all mass is at the center for a uniform sphere?

Q5. Two masses \( 11 \, \text{kg} \) and \( 22 \, \text{kg} \) are \( 22 \, \text{m} \) apart. What is the gravitational potential at a point \( 10 \, \text{m} \) from the \( 11 \, \text{kg} \) mass? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q6. A satellite orbits Earth at \( 9 R_E \) from the center. What is its orbital speed? (\( g = 9.8 \, \text{m/s}^2, R_E = 6.4 \times 10^6 \, \text{m} \))

Q7. What is the gravitational force on a \( 9 \, \text{kg} \) mass \( 7 \, \text{m} \) from the center of a spherical shell of mass \( 350 \, \text{kg} \) and radius \( 5 \, \text{m} \)? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q8. What is the gravitational potential due to Earth at \( 3.2 \times 10^7 \, \text{m} \) from its center? (\( M_E = 6 \times 10^{24} \, \text{kg}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q9. Two masses \( 9 \, \text{kg} \) and \( 18 \, \text{kg} \) are \( 18 \, \text{m} \) apart. What is the gravitational potential at a point \( 8 \, \text{m} \) from the \( 9 \, \text{kg} \) mass? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q10. At what depth below Earth’s surface is \( g \) reduced to \( 7.35 \, \text{m/s}^2 \)? (\( g_0 = 9.8 \, \text{m/s}^2, R_E = 6.4 \times 10^6 \, \text{m} \))