Gravitation Problems

This section provides 100 problems to test your understanding of gravitation, including Newton’s law of gravitation, gravitational fields and potential, Kepler’s laws, orbital motion, escape velocity, and their applications. Inspired by JEE Main, JEE Advanced, and NEET exam patterns, these problems are tailored for exam preparation, offering a mix of numerical, conceptual, and derivation-based challenges. NEET-style problems (66–100) are formatted as multiple-choice questions (MCQs) to match the exam’s objective format. Problems are organized by type to support progressive learning and build confidence in mastering gravitation, a key topic for JEE/NEET success.

Numerical Problems

1. Two point masses $m_1=5kg$ and $m_2=8kg$ are separated by $0.4m$. Calculate the gravitational force between them ($G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

   • (a) $4.17\times {10}^{-8}N$
   • (b) $4.22\times {10}^{-8}N$
   • (c) $4.27\times {10}^{-8}N$
   • (d) $4.32\times {10}^{-8}N$

2. Calculate the gravitational field strength at $1500\text{km}$ above Earth’s surface ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

   • (a) $6.40{\text{m/s}}^2$
   • (b) $6.45{\text{m/s}}^2$
   • (c) $6.50{\text{m/s}}^2$
   • (d) $6.55{\text{m/s}}^2$

3. A satellite of mass $600\text{kg}$ orbits Earth at $800\text{km}$ above the surface. Calculate the gravitational potential at that height ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

   • (a) $-5.60\times {10}^7\text{J/kg}$
   • (b) $-5.65\times {10}^7\text{J/kg}$
   • (c) $-5.70\times {10}^7\text{J/kg}$
   • (d) $-5.75\times {10}^7\text{J/kg}$

4. A satellite orbits Earth at $300\text{km}$ above the surface. Calculate the orbital velocity ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

   • (a) $7650\text{m/s}$
   • (b) $7700\text{m/s}$
   • (c) $7750\text{m/s}$
   • (d) $7800\text{m/s}$

5. Calculate the escape velocity from a planet with $M=4\times {10}^{23}\text{kg}$ and $R=3\times {10}^6\text{m}$ ($G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

   • (a) $4.50\text{km/s}$
   • (b) $4.55\text{km/s}$
   • (c) $4.60\text{km/s}$
   • (d) $4.65\text{km/s}$

6. A planet has a period $T=3$ years and semi-major axis $a=9\text{AU}$. Calculate $T^2/a^3$.

   • (a) $0.105$
   • (b) $0.110$
   • (c) $0.111$
   • (d) $0.115$

7. A satellite of mass $1500\text{kg}$ orbits at $1000\text{km}$ above Earth. Calculate the total orbital energy ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

   • (a) $-4.30\times {10}^{10}\text{J}$
   • (b) $-4.35\times {10}^{10}\text{J}$
   • (c) $-4.40\times {10}^{10}\text{J}$
   • (d) $-4.45\times {10}^{10}\text{J}$

8. Calculate the gravitational force between two identical spheres of mass $20\text{kg}$ with centers $0.3\text{m}$ apart ($G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

   • (a) $2.95\times {10}^{-8}N$
   • (b) $2.96\times {10}^{-8}N$
   • (c) $2.97\times {10}^{-8}N$
   • (d) $2.98\times {10}^{-8}N$

9. Calculate the gravitational field at the midpoint between two masses $M=10\text{kg}$ separated by $0.6\text{m}$ ($G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

   • (a) $0{\text{m/s}}^2$
   • (b) $1.0\times {10}^{-9}{\text{m/s}}^2$
   • (c) $2.0\times {10}^{-9}{\text{m/s}}^2$
   • (d) $3.0\times {10}^{-9}{\text{m/s}}^2$

10. A satellite orbits at $2000\text{km}$ above Earth. Calculate the gravitational potential energy of a $1000\text{kg}$ satellite ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $-5.50\times {10}^{10}\text{J}$
    • (b) $-5.55\times {10}^{10}\text{J}$
    • (c) $-5.60\times {10}^{10}\text{J}$
    • (d) $-5.65\times {10}^{10}\text{J}$

11. Calculate the escape velocity from the Moon ($M_{\text{Moon}}=7.342\times {10}^{22}\text{kg}$, $R_{\text{Moon}}=1738\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $2.35\text{km/s}$
    • (b) $2.37\text{km/s}$
    • (c) $2.39\text{km/s}$
    • (d) $2.41\text{km/s}$

12. A geostationary satellite has a period $T=24$ hours. Calculate the orbital radius ($M_E=5.972\times {10}^{24}\text{kg}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $42000\text{km}$
    • (b) $42100\text{km}$
    • (c) $42200\text{km}$
    • (d) $42300\text{km}$

13. Two point masses $m_1=1\text{kg}$ and $m_2=4\text{kg}$ are separated by $0.2\text{m}$. Calculate the gravitational force ($G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $6.65\times {10}^{-9}N$
    • (b) $6.67\times {10}^{-9}N$
    • (c) $6.69\times {10}^{-9}N$
    • (d) $6.71\times {10}^{-9}N$

14. Calculate the gravitational field strength at $500\text{km}$ above Earth ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $8.40{\text{m/s}}^2$
    • (b) $8.45{\text{m/s}}^2$
    • (c) $8.50{\text{m/s}}^2$
    • (d) $8.55{\text{m/s}}^2$

15. A satellite of mass $2000\text{kg}$ orbits at $1200\text{km}$ above Earth. Calculate the orbital velocity ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $7450\text{m/s}$
    • (b) $7500\text{m/s}$
    • (c) $7550\text{m/s}$
    • (d) $7600\text{m/s}$

16. Calculate the escape velocity from a planet with $M=1\times {10}^{24}\text{kg}$ and $R=4\times {10}^6\text{m}$ ($G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $5.75\text{km/s}$
    • (b) $5.77\text{km/s}$
    • (c) $5.79\text{km/s}$
    • (d) $5.81\text{km/s}$

17. A planet has $T=4$ years and $a=16\text{AU}$. Calculate $T^2/a^3$.

    • (a) $0.060$
    • (b) $0.062$
    • (c) $0.064$
    • (d) $0.066$

18. A satellite of mass $800\text{kg}$ orbits at $1500\text{km}$ above Earth. Calculate the total orbital energy ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $-3.40\times {10}^{10}\text{J}$
    • (b) $-3.45\times {10}^{10}\text{J}$
    • (c) $-3.50\times {10}^{10}\text{J}$
    • (d) $-3.55\times {10}^{10}\text{J}$

19. Calculate the gravitational force between two masses $m_1=3\text{kg}$ and $m_2=6\text{kg}$ separated by $0.5\text{m}$ ($G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $4.80\times {10}^{-9}N$
    • (b) $4.82\times {10}^{-9}N$
    • (c) $4.80\times {10}^{-8}N$
    • (d) $4.82\times {10}^{-8}N$

20. Calculate the gravitational field at $1000\text{km}$ above Earth ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $7.30{\text{m/s}}^2$
    • (b) $7.35{\text{m/s}}^2$
    • (c) $7.40{\text{m/s}}^2$
    • (d) $7.45{\text{m/s}}^2$

21. A satellite of mass $1200\text{kg}$ orbits at $600\text{km}$ above Earth. Calculate the gravitational potential energy ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $-7.00\times {10}^{10}\text{J}$
    • (b) $-7.05\times {10}^{10}\text{J}$
    • (c) $-7.10\times {10}^{10}\text{J}$
    • (d) $-7.15\times {10}^{10}\text{J}$

22. Calculate the orbital velocity of a satellite at $1800\text{km}$ above Earth ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $7250\text{m/s}$
    • (b) $7300\text{m/s}$
    • (c) $7350\text{m/s}$
    • (d) $7400\text{m/s}$

23. Calculate the escape velocity from a planet with $M=2\times {10}^{24}\text{kg}$ and $R=5\times {10}^6\text{m}$ ($G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $6.50\text{km/s}$
    • (b) $6.52\text{km/s}$
    • (c) $6.54\text{km/s}$
    • (d) $6.56\text{km/s}$

24. A planet has $T=5$ years and $a=25\text{AU}$. Calculate $T^2/a^3$.

    • (a) $0.040$
    • (b) $0.042$
    • (c) $0.044$
    • (d) $0.046$

25. A satellite of mass $1000\text{kg}$ orbits at $400\text{km}$ above Earth. Calculate the total orbital energy ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $-3.10\times {10}^{10}\text{J}$
    • (b) $-3.15\times {10}^{10}\text{J}$
    • (c) $-3.20\times {10}^{10}\text{J}$
    • (d) $-3.25\times {10}^{10}\text{J}$

26. Calculate the gravitational force between two masses $m_1=2\text{kg}$ and $m_2=5\text{kg}$ separated by $0.1\text{m}$ ($G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $6.65\times {10}^{-8}N$
    • (b) $6.67\times {10}^{-8}N$
    • (c) $6.69\times {10}^{-8}N$
    • (d) $6.71\times {10}^{-8}N$

27. Calculate the gravitational field at $2500\text{km}$ above Earth ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $5.20{\text{m/s}}^2$
    • (b) $5.25{\text{m/s}}^2$
    • (c) $5.30{\text{m/s}}^2$
    • (d) $5.35{\text{m/s}}^2$

28. A satellite of mass $500\text{kg}$ orbits at $900\text{km}$ above Earth. Calculate the gravitational potential energy ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $-2.80\times {10}^{10}\text{J}$
    • (b) $-2.85\times {10}^{10}\text{J}$
    • (c) $-2.90\times {10}^{10}\text{J}$
    • (d) $-2.95\times {10}^{10}\text{J}$

29. Calculate the orbital velocity of a satellite at $700\text{km}$ above Earth ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $7500\text{m/s}$
    • (b) $7550\text{m/s}$
    • (c) $7600\text{m/s}$
    • (d) $7650\text{m/s}$

30. Calculate the escape velocity from a planet with $M=5\times {10}^{23}\text{kg}$ and $R=2.5\times {10}^6\text{m}$ ($G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $5.15\text{km/s}$
    • (b) $5.17\text{km/s}$
    • (c) $5.19\text{km/s}$
    • (d) $5.21\text{km/s}$

31. A planet has $T=2$ years and $a=8\text{AU}$. Calculate $T^2/a^3$.

    • (a) $0.060$
    • (b) $0.062$
    • (c) $0.064$
    • (d) $0.066$

32. A satellite of mass $2000\text{kg}$ orbits at $1100\text{km}$ above Earth. Calculate the total orbital energy ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $-6.00\times {10}^{10}\text{J}$
    • (b) $-6.05\times {10}^{10}\text{J}$
    • (c) $-6.10\times {10}^{10}\text{J}$
    • (d) $-6.15\times {10}^{10}\text{J}$

33. Calculate the gravitational force between two spheres of mass $15\text{kg}$ with centers $0.5\text{m}$ apart ($G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $6.00\times {10}^{-9}N$
    • (b) $6.01\times {10}^{-9}N$
    • (c) $6.02\times {10}^{-9}N$
    • (d) $6.03\times {10}^{-9}N$

34. Calculate the gravitational field at $3000\text{km}$ above Earth ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $4.80{\text{m/s}}^2$
    • (b) $4.85{\text{m/s}}^2$
    • (c) $4.90{\text{m/s}}^2$
    • (d) $4.95{\text{m/s}}^2$

35. A rocket escapes from a planet with $M=6\times {10}^{23}\text{kg}$ and $R=3.5\times {10}^6\text{m}$. Calculate the escape velocity ($G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

    • (a) $5.50\text{km/s}$
    • (b) $5.52\text{km/s}$
    • (c) $5.54\text{km/s}$
    • (d) $5.56\text{km/s}$

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Conceptual Problems

36. What does Newton’s law of gravitation state?

• (a) Force is proportional to the product of masses and inversely proportional to $r$
• (b) Force is proportional to the product of masses and inversely proportional to $r^2$
• (c) Force is proportional to $r^2$
• (d) Force is independent of distance

37. What is the gravitational field inside a uniform spherical shell?

• (a) Zero
• (b) Proportional to $r$
• (c) Proportional to $1/r$
• (d) Proportional to $1/r^2$

38. What does a negative gravitational potential energy indicate?

• (a) Repulsive force
• (b) Bound system
• (c) Unbound system
• (d) Zero energy

39. What does Kepler’s second law imply?

• (a) Orbits are circular
• (b) Equal areas in equal times
• (c) $T^2\propto r^3$
• (d) Velocity is constant

40. What is the unit of gravitational potential?

• (a) $\text{J/kg}$
• (b) ${\text{m/s}}^2$
• (c) $N$
• (d) $\text{J}$

41. What happens to orbital velocity as the radius increases?

• (a) Increases
• (b) Decreases
• (c) Remains the same
• (d) Becomes zero

42. What does a total orbital energy of zero indicate?

• (a) Circular orbit
• (b) Elliptical orbit
• (c) Object escapes the gravitational field
• (d) Object is at rest

43. What is the physical significance of escape velocity?

• (a) Speed to orbit a planet
• (b) Speed to escape the gravitational field
• (c) Speed to reach the surface
• (d) Speed to enter an elliptical orbit

44. What does Kepler’s third law relate?

• (a) Velocity and radius
• (b) Period and radius
• (c) Energy and radius
• (d) Force and radius

45. What is the dimension of the gravitational constant $G$?

• (a) $[{\text{M}}^{-1}{\text{L}}^3{\text{T}}^{-2}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$
• (d) $[\text{L}{\text{T}}^{-2}]$

46. What does a zero gravitational field at a point indicate?

• (a) No mass nearby
• (b) Fields cancel due to symmetry
• (c) Object is at infinity
• (d) Object is inside a mass

47. What is the significance of $-\frac{GM}{r}$?

• (a) Gravitational field
• (b) Gravitational potential
• (c) Orbital velocity
• (d) Escape velocity

48. What happens to gravitational potential as distance increases?

• (a) Becomes more negative
• (b) Becomes less negative
• (c) Remains constant
• (d) Becomes positive

49. What does a geostationary satellite’s period equal?

• (a) 12 hours
• (b) 24 hours
• (c) 1 year
• (d) 1 month

50. How does escape velocity depend on mass of the object escaping?

• (a) Proportional to mass
• (b) Inversely proportional to mass
• (c) Independent of mass
• (d) Proportional to square of mass

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Derivation Problems

51. Derive Newton’s law of gravitation for two point masses.

52. Derive the gravitational field of a point mass.

53. Derive the gravitational potential $V=-\frac{GM}{r}$.

54. Derive Kepler’s third law for circular orbits.

55. Derive the escape velocity $v_{\text{esc}}=\sqrt{\frac{2GM}{r}}$.

56. Derive the total energy in a circular orbit.

57. Derive the gravitational force outside a spherical mass.

58. Derive the orbital velocity $v=\sqrt{\frac{GM}{r}}$.

59. Derive Kepler’s second law using angular momentum conservation.

60. Derive the gravitational potential energy $U=-\frac{GMm}{r}$.

61. Derive the gravitational field inside a spherical shell.

62. Derive the period of a geostationary satellite.

63. Derive the minimum energy required for escape.

64. Derive the gravitational acceleration near Earth’s surface.

65. Derive the gravitational field at the midpoint between two masses.

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NEET-style Conceptual Problems

66. What is the unit of gravitational field strength?

• (a) ${\text{m/s}}^2$
• (b) $\text{J/kg}$
• (c) $N$
• (d) $\text{J}$

67. What does a zero gravitational potential indicate?

• (a) Point is at the surface
• (b) Point is at infinity
• (c) No mass is present
• (d) Field is zero

68. Which law describes the elliptical shape of planetary orbits?

• (a) Kepler’s first law
• (b) Kepler’s second law
• (c) Kepler’s third law
• (d) Newton’s law of gravitation

69. What happens to total energy when a satellite moves to a higher orbit?

• (a) Becomes more negative
• (b) Becomes less negative
• (c) Remains the same
• (d) Becomes positive

70. What is the dimension of gravitational potential?

• (a) $[{\text{L}}^2{\text{T}}^{-2}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$
• (d) $[\text{L}{\text{T}}^{-1}]$

71. What does the inverse-square law in gravitation imply?

• (a) Force increases with distance
• (b) Force decreases as $1/r^2$
• (c) Force decreases as $1/r$
• (d) Force is constant

72. What is the role of gravitational force in orbital motion?

• (a) Provides centripetal force
• (b) Causes linear motion
• (c) Increases angular velocity
• (d) Reduces potential energy

73. What happens to escape velocity as the planet’s mass increases?

• (a) Decreases
• (b) Increases
• (c) Remains the same
• (d) Becomes zero

74. Why does a satellite in a higher orbit have a longer period?

• (a) Kepler’s third law: $T^2\propto r^3$
• (b) Kepler’s second law
• (c) Lower gravitational field
• (d) Higher velocity

75. What is the unit of the gravitational constant $G$?

• (a) $\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$
• (b) ${\text{m/s}}^2$
• (c) $\text{J/kg}$
• (d) $\text{J}$

76. What does a constant $\frac{T^2}{a^3}$ indicate?

• (a) Kepler’s first law
• (b) Kepler’s second law
• (c) Kepler’s third law
• (d) Newton’s law

77. Which type of orbit does Kepler’s first law describe?

• (a) Circular
• (b) Elliptical
• (c) Parabolic
• (d) Hyperbolic

78. What is the direction of the gravitational field?

• (a) Away from the mass
• (b) Toward the mass
• (c) Perpendicular to the mass
• (d) Along the velocity

79. What does a pseudo-force do in a rotating frame for orbits?

• (a) Maintains circular motion
• (b) Affects gravitational calculations
• (c) Provides centripetal force
• (d) Reduces friction

80. What is the dimension of orbital velocity?

• (a) $[\text{L}{\text{T}}^{-1}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{L}{\text{T}}^{-2}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

81. What is the role of angular momentum in Kepler’s second law?

• (a) Increases velocity
• (b) Ensures equal areas in equal times
• (c) Reduces potential energy
• (d) Increases orbital radius

82. What happens to gravitational field strength inside a spherical shell?

• (a) Increases with radius
• (b) Decreases with radius
• (c) Remains zero
• (d) Becomes infinite

83. Why does escape velocity not depend on the object’s mass?

• (a) Mass cancels out in the equation
• (b) Mass increases with velocity
• (c) Mass affects potential energy
• (d) Mass affects field strength

84. What is the significance of $\sqrt{\frac{GM}{r}}$?

• (a) Escape velocity
• (b) Orbital velocity
• (c) Gravitational field
• (d) Gravitational potential

85. What is the unit of total orbital energy?

• (a) $\text{J}$
• (b) ${\text{m/s}}^2$
• (c) $\text{J/kg}$
• (d) $\text{N}$

86. What does a zero total energy indicate in orbital mechanics?

• (a) Circular orbit
• (b) Elliptical orbit
• (c) Object escapes
• (d) Object is at rest

87. What is the physical significance of $\frac{4{\pi }^2{r}^3}{GM}$?

• (a) Gravitational field
• (b) Gravitational potential
• (c) Orbital period squared
• (d) Escape velocity

88. Why does a geostationary satellite stay above the same point?

• (a) High velocity
• (b) Period matches Earth’s rotation
• (c) Low gravitational field
• (d) High potential energy

89. What is the dimension of escape velocity?

• (a) $[\text{L}{\text{T}}^{-1}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{L}{\text{T}}^{-2}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

90. How does gravitational potential affect a rocket’s trajectory?

• (a) Increases kinetic energy
• (b) Determines potential energy
• (c) Reduces velocity
• (d) Increases field strength

91. What is the role of escape velocity in rocket launches?

• (a) Determines orbital radius
• (b) Ensures the rocket leaves the gravitational field
• (c) Increases period
• (d) Reduces energy

92. What does a negative total energy in an orbit indicate?

• (a) Object escapes
• (b) Bound orbit
• (c) Parabolic trajectory
• (d) Hyperbolic trajectory

93. What is the physical significance of $-\frac{GMm}{2r}$?

• (a) Total energy in a circular orbit
• (b) Potential energy
• (c) Kinetic energy
• (d) Gravitational force

94. What is the dimension of gravitational field strength?

• (a) $[\text{L}{\text{T}}^{-2}]$
• (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
• (c) $[\text{L}{\text{T}}^{-1}]$
• (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

95. Why does gravitational field strength decrease with distance?

• (a) Proportional to $r$
• (b) Proportional to $1/r$
• (c) Proportional to $1/r^2$
• (d) Proportional to $r^2$

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NEET-style Numerical Problems

96. Calculate the gravitational force between two masses $m_1=4\text{kg}$ and $m_2=7\text{kg}$ separated by $0.3\text{m}$ ($G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

• (a) $2.07\times {10}^{-8}N$
• (b) $2.08\times {10}^{-8}N$
• (c) $2.09\times {10}^{-8}N$
• (d) $2.10\times {10}^{-8}N$

97. Calculate the gravitational field at $2000\text{km}$ above Earth ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

• (a) $5.65{\text{m/s}}^2$
• (b) $5.68{\text{m/s}}^2$
• (c) $5.71{\text{m/s}}^2$
• (d) $5.74{\text{m/s}}^2$

98. A satellite orbits at $500\text{km}$ above Earth. What is the orbital velocity ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$)?

• (a) $7550\text{m/s}$
• (b) $7600\text{m/s}$
• (c) $7650\text{m/s}$
• (d) $7700\text{m/s}$

99. Calculate the escape velocity from a planet with $M=8\times {10}^{23}\text{kg}$ and $R=4\times {10}^6\text{m}$ ($G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$).

• (a) $5.80\text{km/s}$
• (b) $5.82\text{km/s}$
• (c) $5.84\text{km/s}$
• (d) $5.86\text{km/s}$

100. A satellite of mass $1000\text{kg}$ orbits at $800\text{km}$ above Earth. What is the total orbital energy ($M_E=5.972\times {10}^{24}\text{kg}$, $R_E=6371\text{km}$, $G=6.67\times {10}^{-11}\text{N}\cdot {\text{m}}^2/{\text{kg}}^2$)?
     - (a) $-2.80\times {10}^{10}\text{J}$
     - (b) $-2.85\times {10}^{10}\text{J}$
     - (c) $-2.90\times {10}^{10}\text{J}$
     - (d) $-2.95\times {10}^{10}\text{J}$

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