Gravitation
Short Questions
1. What is the physical significance of the universal gravitational constant G?
Answer : It measures the strength of the gravitational force between two unit masses separated by a unit distance.
2. How does the centripetal force relate to the gravitational force in planetary motion?
Answer : The gravitational force provides the centripetal force required for a planet’s circular orbit around the Sun.
3. Why is the orbit of a planet elliptical according to Kepler’s first law?
Answer : The Sun’s gravitational pull causes planets to follow an elliptical path with the Sun at one focus.
4. What does Kepler’s second law imply about a planet’s speed in its orbit?
Answer : A planet moves faster when closer to the Sun and slower when farther away.
5. How is Kepler’s third law mathematically expressed?
Answer : T² ∝ r³, where T is the period of revolution and r is the mean distance from the Sun.
6. Why does the value of g vary with latitude on Earth?
Answer : Earth’s rotation and oblate shape cause g to be higher at the poles and lower at the equator.
7. What happens to the value of g at the center of the Earth?
Answer : The value of g becomes zero as the effective mass contributing to gravity is zero.
8. How is the escape velocity of a planet derived from energy conservation?
Answer : It is derived by equating initial kinetic energy to the gravitational potential energy at infinity.
9. Why do satellites remain in orbit without falling to Earth?
Answer : Their orbital velocity provides the centripetal force to balance Earth’s gravitational pull.
10. What is the effect of doubling the distance between two objects on their gravitational force?
Answer : The gravitational force reduces to one-fourth of its original value.
11. How does the gravitational force between two objects depend on their masses?
Answer : It is directly proportional to the product of their masses.
12. Why is true free fall only possible in a vacuum?
Answer : In a vacuum, no air resistance or buoyant forces oppose the gravitational force.
13. What is the significance of negative gravitational potential energy?
Answer : It indicates that work is required to move an object to infinity, where potential energy is zero.
14. Why does a feather fall slower than a stone in air but not in a vacuum?
Answer : Air resistance and buoyant forces slow the feather’s fall, absent in a vacuum.
15. How does the Moon’s gravitational force cause high tides on Earth?
Answer : It pulls water closer to it, raising the water level directly beneath it.
16. What is the role of the center of mass in Newton’s law of gravitation?
Answer : The gravitational force acts along the line joining the centers of mass of two objects.
17. Why is the weight of an object on the Moon 1/6th of its weight on Earth?
Answer : The Moon’s gravitational acceleration is 1/6th that of Earth’s due to its smaller mass and radius.
18. How does the value of g change with increasing height above Earth’s surface?
Answer : It decreases as the distance from Earth’s center increases, following g = GM/(R+h)².
19. What is the relationship between escape velocity and a planet’s radius?
Answer : Escape velocity is inversely proportional to the square root of the planet’s radius.
20. Why do astronauts appear weightless in a spacecraft?
Answer : They are in free fall, moving at the same velocity as the spacecraft under Earth’s gravity.
Long Questions
1. Explain Newton’s universal law of gravitation.
Answer : Newton’s law states that every object attracts every other object with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. The formula is F = G (m₁m₂/r²), where G is the universal gravitational constant. This force acts along the line joining the centers of the two objects.
2. Describe Kepler’s three laws of planetary motion.
Answer : Kepler’s first law states planets orbit the Sun in ellipses with the Sun at one focus. The second law says the line joining a planet and the Sun sweeps equal areas in equal times. The third law states the square of a planet’s orbital period is proportional to the cube of its average distance from the Sun (T² ∝ r³).
3. How does the value of g vary on Earth’s surface and why?
Answer : The value of g varies due to Earth’s non-spherical shape and rotation. It is highest at the poles (9.832 m/s²) because the radius is smaller, and lowest at the equator (9.78 m/s²) due to the larger radius and rotational effects. The formula g = GM/R² shows g depends on Earth’s radius.
4. What is centripetal force, and how is it related to planetary motion?
Answer : Centripetal force is the force that keeps an object moving in a circular path, directed toward the center. In planetary motion, the gravitational force between a planet and the Sun acts as the centripetal force, keeping the planet in its orbit. The formula is F = mv²/r, where m is the planet’s mass, v is its speed, and r is the orbital radius.
5. Why do objects in free fall reach the ground simultaneously regardless of mass?
Answer : In free fall, objects move under gravity alone, with acceleration g independent of mass. According to Newton’s second law, F = mg, the mass cancels out, resulting in the same acceleration for all objects. This was demonstrated by Galileo, showing that in a vacuum, all objects fall at the same rate.
6. Explain the concept of escape velocity and its derivation.
Answer : Escape velocity is the minimum speed needed for an object to escape Earth’s gravitational pull without further propulsion. It is derived using conservation of energy, where initial kinetic energy (½mv²) equals the gravitational potential energy (GMm/R) at infinity, giving v = √(2GM/R). For Earth, this is approximately 11.2 km/s.
7. What is the difference between mass and weight, and how do they vary on different planets?
Answer : Mass is the amount of matter in an object, constant everywhere, measured in kg. Weight is the force due to gravity (W = mg), varying with the planet’s gravitational acceleration. On the Moon, weight is 1/6th of Earth’s due to lower g, but mass remains unchanged.
8. How does gravitational potential energy change with height?
Answer : Gravitational potential energy is given by PE = -GMm/(R+h), where h is height above Earth’s surface. For small h, it approximates to mgh, increasing with height. At infinite distance, PE is zero, and it becomes more negative closer to Earth due to stronger gravitational attraction.
9. Why do astronauts experience weightlessness in space?
Answer : Astronauts feel weightless because they are in free fall toward Earth while orbiting. The spacecraft and astronauts fall at the same rate under gravity, so no relative force is felt. This is not due to zero gravity but the absence of a reaction force from a surface.
10. How does the gravitational force cause tides on Earth?
Answer : Tides result from the Moon’s gravitational pull on Earth’s oceans. Water directly under the Moon is pulled more strongly, causing high tides, while areas at 90° experience low tides. The Sun also contributes, but the Moon’s effect is dominant due to its proximity.
11. Explain why the value of g decreases with depth inside Earth.
Answer : As we go deeper inside Earth, only the mass within the radius contributes to gravity, reducing effective mass M. The distance r from the center also decreases, but the reduction in M dominates. Thus, g decreases linearly with depth, becoming zero at the center.
12. How did Newton use Kepler’s laws to formulate the inverse square law?
Answer : Newton analyzed Kepler’s third law (T² ∝ r³) for planets in circular orbits. He derived that the centripetal force (F = mv²/r) must be inversely proportional to r², matching the gravitational force. This led to the formulation of F = G (m₁m₂/r²), confirming the inverse square law.
13. What happens to an object’s weight when it is taken to a height of 2R from Earth’s center?
Answer : Weight is given by W = GMm/r², where r is the distance from Earth’s center. At r = 2R, the weight becomes W = GMm/(2R)² = mg/4, where g is surface gravity. Thus, the weight is one-fourth of its value on Earth’s surface.
14. Why does a stone thrown upwards return to Earth?
Answer : The Earth’s gravitational force acts downward, reducing the stone’s upward velocity to zero at its peak. After reaching maximum height, the stone accelerates downward due to gravity. The initial velocity determines the maximum height, but gravity ensures its return.
15. What are gravitational waves, and why are they significant?
Answer : Gravitational waves are ripples in space-time caused by massive accelerating objects, predicted by Einstein. They carry information about cosmic events like black hole mergers, detected by sensitive instruments like LIGO. Their discovery in 2016 opened new ways to study the universe.
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