Gravitational Field | Jamb(UTME)

Jamb(utme) key points on gravitational field; newton law of universal gravitation; gravitational potential; conservative and non-conservative field

Gravitational Field

1. A gravitational field is a region in space where a mass experiences a gravitational force.
2. It is represented by field lines pointing toward the center of the mass creating the field.
3. The strength of the gravitational field $(g)$ is the force per unit mass:
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   $g = \frac{F}{m}$
4. The SI unit of $g$ is $N/kg$ or $m/s$.
5. Near the Earth's surface, the gravitational field strength is approximately $9.8{m/s}^2$.
6. Gravitational field strength decreases with distance from the mass creating the field.
7. For a point mass $M$, the gravitational field strength is:
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   $g = \frac{GM}{r^2}$ where $G$ is the gravitational constant, $M$ is the mass, and $r$ is the distance from the mass.
8. Gravitational fields are vector fields, meaning they have both magnitude and direction.
9. The Earth’s gravitational field is strongest at its surface and weaker as you move farther away.
10. In a uniform gravitational field, $g$ is constant, such as near the Earth's surface.
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Newton's Law of Universal Gravitation

11. Newton’s law of universal gravitation states that every two masses attract each other with a force: $F = G \frac{m_1 m_2}{r^2}$
12. $F$ is the gravitational force, $m_1$ and $m_2$ are the masses, $r$ is the distance between their centers, and $G$ is the gravitational constant.
13. The gravitational constant $G$ is $6.674 \times 10^{-11}{Nm}^2/kg^2$.
14. The force is always attractive, pulling the masses toward each other.
15. Gravitational force decreases with the square of the distance $(r^2)$, making it an inverse-square law.
16. Larger masses produce stronger gravitational forces.
17. Gravitational force acts along the line joining the centers of the two masses.
18. Newton’s law explains phenomena like planetary orbits and the motion of moons and comets.
19. Gravitational force is extremely weak compared to other fundamental forces but dominates at large scales.
20. The law of gravitation applies to all objects with mass, regardless of size.
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Gravitational Potential

21. Gravitational potential $(V)$ at a point is the work done to bring a unit mass from infinity to that point.
22. The formula for gravitational potential is:
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    $V = -\frac{GM}{r}$
23. Gravitational potential is negative because the force is attractive.
24. The SI unit of gravitational potential is $J/kg$.
25. Gravitational potential is a scalar quantity, having magnitude but no direction.
26. At infinity, gravitational potential is zero.
27. The closer you are to a massive object, the more negative the gravitational potential.
28. The difference in gravitational potential between two points determines the work done to move a mass between them.
29. For a spherical mass like Earth, $V$ depends only on the distance from the center of the mass.
30. Gravitational potential energy $(U)$ for a mass $m$ is:
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    $U = mV = -\frac{GMm}{r}$

Conservative Fields

31. A conservative field is one where the work done to move an object between two points is independent of the path taken.
32. The gravitational field is a conservative field.
33. In a conservative field, the total mechanical energy (kinetic + potential) remains constant in the absence of external forces.
34. Work done in a conservative field depends only on the initial and final positions, not the route.
35. Potential energy is well-defined in a conservative field.
36. For example, lifting a mass in Earth’s gravitational field stores potential energy, which can be fully recovered when the mass falls.
37. In a conservative field, energy can be transformed between potential and kinetic without loss.
38. Gravitational potential energy near Earth's surface is approximated as:
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    $U = mgh$ where $h$ is the height above a reference point.
39. The gravitational field is conservative because the force is derived from a potential function $(V)$.
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Non-Conservative Fields

40. A non-conservative field is one where the work done depends on the path taken.
41. Frictional forces are an example of non-conservative forces.
42. Energy is dissipated (e.g., as heat) in non-conservative fields, making it impossible to fully recover.
43. Non-conservative fields do not have a well-defined potential energy.
44. Work done in a non-conservative field is converted into non-recoverable forms of energy.
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Applications of Gravitational Field and Potential

45. Gravitational fields explain the motion of planets, moons, and artificial satellites.
46. Orbits of planets are maintained by the balance between gravitational attraction and inertial motion.
47. Gravitational field strength determines the weight of objects on a planet.
48. Gravitational potential energy is used in hydroelectric dams, where falling water converts potential energy into electricity.
49. Understanding gravitational potential helps in launching rockets and calculating escape velocity.
50. Gravitational forces and potentials are critical in astrophysics, from studying black holes to modeling galaxies.
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Jamb(utme) key points on acceleration due to gravity; variation of g on the earth surface; distinction between mass and weight

Acceleration Due to Gravity (g)

1. Acceleration due to gravity $(g)$ is the rate at which objects accelerate toward Earth when in free fall.
2. On Earth's surface, $g$ has an approximate value of $9.8{m/s}^2$.
3. The value of $g$ is independent of the mass of the falling object, assuming no air resistance.
4. The formula for ( g ) is derived from Newton’s law of gravitation:
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   $g = \frac{GM}{R^2}$ where $G$ is the gravitational constant, $M$ is the mass of Earth, and $R$ is the radius of Earth.
5. $g$ causes all objects, regardless of their mass, to fall at the same rate in a vacuum.
6. Acceleration due to gravity is a vector quantity, directed toward the center of Earth.
7. $g$ decreases with altitude because the distance $(r)$ from Earth’s center increases:
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   $g' = g \left( 1 - \frac{2h}{R} \right)$ for small heights $h$ above the surface.
8. $g$ also decreases with depth below the surface because the effective mass causing gravitational attraction reduces.
9. At the center of Earth, $g = 0$ because there is no net gravitational pull.
10. $g$ is highest at the poles and lowest at the equator due to Earth's shape and rotation.
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Variation of g on Earth’s Surface

11. The value of $g$ varies with altitude; it decreases as you move farther from Earth's surface.
12. At a height $h$, $g$ is given by:
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    $g_h = \frac{GM}{(R + h)^2}$
13. The further you are from Earth’s center, the weaker the gravitational pull.
14. At sea level, $g$ is stronger than on a mountain.
15. At great depths below Earth's surface, $g$ decreases linearly with depth because only the mass within the radius of depth contributes to gravitational attraction.
16. The formula for $g$ at depth $d$ is:
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    $g_d = g \left( 1 - \frac{d}{R} \right)$
17. Earth’s rotation causes $g$ to vary between the poles and the equator.
18. At the equator, centrifugal force due to rotation opposes gravity, reducing the effective $g$.
19. At the poles, there is no centrifugal force, so $g$ is stronger.
20. The shape of Earth (oblate spheroid) means the radius is slightly larger at the equator than at the poles, further reducing $g$ at the equator.
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Numerical Variation of g

21. At the poles, $g \approx 9.83{m/s}^2$.
22. At the equator, $g \approx 9.78{m/s}^2$.
23. At a height of $10,000m$ above Earth, $g$ is slightly less than $9.8{m/s}^2$.
24. The difference in $g$ between the equator and poles is due to both centrifugal force and Earth's shape.
25. At geostationary satellite orbits (high altitudes), $g$ is negligible compared to its surface value.
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Distinction Between Mass and Weight

26. Mass is the amount of matter in an object.
27. Weight is the gravitational force acting on an object due to $g$.
28. Mass is a scalar quantity, while weight is a vector quantity.
29. The SI unit of mass is the kilogram $(kg)$.
30. The SI unit of weight is the newton $(N)$.
31. Mass is constant and does not change with location.
32. Weight depends on $g$ and changes with altitude, latitude, and planetary body.
33. The relationship between weight $(W)$ and mass $(m)$ is:
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    $W = mg$
34. For example, an object with a mass of $10kg$ has a weight of $98N$ on Earth’s surface $g = 9.8{m/s}^2$.
35. On the Moon, where $g = 1.6{m/s}^2$, the same object weighs only $16N$ but its mass remains $10kg$.
36. Mass measures inertia (resistance to changes in motion), while weight measures the force of gravity on the object.
37. A person’s weight is lower at the equator than at the poles because $g$ is weaker at the equator.
38. Astronauts experience weightlessness in orbit because the spacecraft and their bodies are in free fall, but their mass remains unchanged.
39. To calculate weight on other planets, multiply the mass by the planet’s specific $g$-value.
40. Understanding the difference between mass and weight is crucial in physics, engineering, and space exploration.
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Jamb(utme) key points on escape velocity; parking orbit and weightlessness

Escape Velocity

1. Escape velocity is the minimum velocity required for an object to leave a planet's gravitational field without further propulsion.
2. It ensures the object reaches a distance where the planet’s gravitational pull becomes negligible.
3. Escape velocity is given by the formula:
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   $v_{\text{escape}} = \sqrt{\frac{2GM}{R}}$ where $G$ is the gravitational constant, $M$ is the planet’s mass, and $R$ is its radius.
4. For Earth, the escape velocity is approximately $11.2km/s$ (40,320 km/h) at the surface.
5. Escape velocity depends on the planet's mass and radius; larger or more massive planets have higher escape velocities.
6. The escape velocity is independent of the mass of the object attempting to escape.
7. Rockets achieve escape velocity by gradually building speed, not in a single instant.
8. Escape velocity decreases if the object starts at a higher altitude since $R$ increases.
9. If an object doesn’t reach escape velocity, it will fall back to the planet or enter an orbit around it.
10. Spacecraft need escape velocity to leave Earth’s gravitational influence and travel to other celestial bodies.
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Parking Orbit

11. A parking orbit is a temporary, stable circular orbit around Earth where a spacecraft waits before continuing its mission.
12. It is commonly used during satellite launches, interplanetary missions, and rendezvous with space stations.
13. Parking orbits are usually located in the low Earth orbit (LEO), about 160–2,000 km above Earth’s surface.
14. The spacecraft remains in the parking orbit while systems are checked and alignments are adjusted.
15. Parking orbits allow for precise timing of maneuvers, such as transferring to a higher orbit or departing for another planet.
16. Geostationary satellites are often launched into parking orbits before being transferred to their final positions.
17. Parking orbits optimize fuel efficiency by enabling controlled transfers using specific maneuvers like the Hohmann transfer.
18. A parking orbit is crucial for ensuring safe and accurate satellite deployment.
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Weightlessness

19. Weightlessness occurs when an object experiences no net gravitational force acting upon it.
20. In orbit, astronauts experience weightlessness because they are in free fall toward Earth, but their tangential velocity keeps them in orbit.
21. Weightlessness does not mean zero gravity; gravitational forces still act, but the object is in a continuous state of free fall.
22. Weightlessness is felt during parabolic flights, which simulate short periods of free fall.
23. Astronauts in space appear to float because the spacecraft and their bodies accelerate toward Earth at the same rate.
24. Weightlessness affects the human body, causing muscle and bone loss over long durations in space.
25. Space stations are designed to simulate a weightless environment for scientific experiments and astronaut training.
26. Weightlessness is used in medical research to study how fluids and biological processes behave without gravity.
27. Understanding weightlessness is essential for designing life-support systems and habitats for long-term space missions.
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