NEET Gravitation — Set 8

Question 1

At what depth below Earth’s surface is $ g $ reduced to $ 6.86 \, \text{m/s}^2 $? ($ g_0 = 9.8 \, \text{m/s}^2, R_E = 6.4 \times 10^6 \, \text{m} $)

Accepted Answer

$ g(d) = g_0 (1 - d/R_E) $. $ 6.86 = 9.8 (1 - d/R_E) $. $ 1 - d/R_E = 0.7 $. $ d/R_E = 0.3 $. $ d = 0.3 \times 6.4 \times 10^6 = 1.92 \times 10^6 \, \text{m} $.

Question 2

Three $ 5 \, \text{kg} $ masses form an equilateral triangle of side $ 4 \, \text{m} $. What is the potential energy of the system? ($ G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 $)

Accepted Answer

3 pairs: $ V = -3 \frac{G m^2}{r} $. $ V = -3 \frac{6.67 \times 10^{-11} \times 5 \times 5}{4} $. $ V = -3 \times 4.169 \times 10^{-11} = -1.25 \times 10^{-10} \, \text{J} $.

Question 3

What does the constant $ k $ in Kepler’s third law ($ T^2 = k r^3 $) depend on for satellites?

Accepted Answer

$ T^2 = \frac{4\pi^2}{G M_E} r^3 $ for satellites, where $ k = \frac{4\pi^2}{G M_E} $. Thus, $ k $ depends on the mass of the central body (Earth) and universal constants, not the satellite’s mass.

NEET Physics: Gravitation — Practice Set 8

Q1. At what depth below Earth’s surface is \( g \) reduced to \( 6.86 \, \text{m/s}^2 \)? (\( g_0 = 9.8 \, \text{m/s}^2, R_E = 6.4 \times 10^6 \, \text{m} \))

Q2. Three \( 5 \, \text{kg} \) masses form an equilateral triangle of side \( 4 \, \text{m} \). What is the potential energy of the system? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q3. What does the constant \( k \) in Kepler’s third law (\( T^2 = k r^3 \)) depend on for satellites?

Q4. Three masses of \( 6 \, \text{kg} \) each form an equilateral triangle with side \( 5 \, \text{m} \). What is the net force on one mass? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q5. Which of the following statements is correct about escape speed?

Q6. How much energy is required to move a \( 200 \, \text{kg} \) satellite from \( 6 R_E \) to \( 12 R_E \) 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 body weighs \( 392 \, \text{N} \) on Earth’s surface. What is its weight at a height \( h = R_E/6 \)? (\( g = 9.8 \, \text{m/s}^2 \))

Q8. How much energy is required to move a \( 400 \, \text{kg} \) satellite from \( 8 R_E \) to \( 16 R_E \) from Earth’s center? (\( M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q9. What is the gravitational potential due to Earth at \( 4.48 \times 10^7 \, \text{m} \) from its center? (\( M_E = 6 \times 10^{24} \, \text{kg}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q10. What is the gravitational force on a \( 5 \, \text{kg} \) mass \( 4 \, \text{m} \) from the center of a spherical shell of mass \( 200 \, \text{kg} \) and radius \( 3 \, \text{m} \)? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))