For points located outside the Earth, the acceleration due to gravity decreases with distance. It is inversely proportional to the square of the distance from the center of the Earth, meaning as one moves further away, gravity becomes weaker. Specifically, this means that the gravitational force felRead more
For points located outside the Earth, the acceleration due to gravity decreases with distance. It is inversely proportional to the square of the distance from the center of the Earth, meaning as one moves further away, gravity becomes weaker. Specifically, this means that the gravitational force felt at a height above Earth’s surface diminishes with increasing distance from the center.
In contrast, for points inside the Earth, gravity behaves differently. The acceleration due to gravity decreases linearly as one moves closer to the center. This indicates that the gravitational pull inside the Earth is directly related to the distance from the center.
For points lying onside the earth (r > R)
gₕ/g = R²/(R + h)² = R²/r² or gₕ = gR²/r²
gₕ ∝ 1/r²
For points lying inside the earth (r < R)
gₔ = g(R – d)/R = gʳ/R or gₔ = gʳ/R
gₔ ∝ r
The acceleration due to gravity g is slightly altered by Earth's rotational motion, depending on latitude. At the equator, Earth's centrifugal force, due to its maximum velocity of rotation, acts outward and opposes gravity, reducing the effective value of g slightly. As latitude increases toward thRead more
The acceleration due to gravity g is slightly altered by Earth’s rotational motion, depending on latitude. At the equator, Earth’s centrifugal force, due to its maximum velocity of rotation, acts outward and opposes gravity, reducing the effective value of g slightly. As latitude increases toward the poles, the rotational velocity decreases, and the centrifugal force decreases.
At the poles, centrifugal force is negligible, and g is at its maximum value. Therefore, g is lowest at the equator and increases gradually as one moves toward the poles.
Latitude at a place measures how far north or south it is from the Earth’s equator. It’s expressed in degrees, starting at 0° at the equator and going up to 90° at the North and South Poles. Locations in the Northern Hemisphere have positive latitudes, while those in the Southern Hemisphere have negRead more
Latitude at a place measures how far north or south it is from the Earth’s equator. It’s expressed in degrees, starting at 0° at the equator and going up to 90° at the North and South Poles. Locations in the Northern Hemisphere have positive latitudes, while those in the Southern Hemisphere have negative latitudes.
These imaginary lines, called parallels, run horizontally around the globe. Latitude helps pinpoint a location on Earth and also influences the climate and sunlight a region gets throughout the year, playing a key role in geography and navigation.
The orbital period of a satellite depends on the radius of its orbit and the mass of the central body, such as Earth, but it is independent of the satellite's mass. This means that whether the satellite is small or large, its orbital period remains unchanged as long as the orbital radius and the cenRead more
The orbital period of a satellite depends on the radius of its orbit and the mass of the central body, such as Earth, but it is independent of the satellite’s mass. This means that whether the satellite is small or large, its orbital period remains unchanged as long as the orbital radius and the central body’s mass remain constant. This principle highlights that the motion of a satellite is governed by gravitational forces and does not rely on the satellite’s own mass, making it a fundamental aspect of orbital mechanics.
T = 2π√((R + h)³ / GM),
Clearly, T does not depend on the mass m of the satellite.
Johannes Kepler, a renowned 17th-century astronomer, discovered the fundamental laws of planetary motion, which revolutionized our understanding of the solar system. His first law states that planets orbit the Sun in elliptical paths with the Sun at one focus. The second law, or the law of equal areRead more
Johannes Kepler, a renowned 17th-century astronomer, discovered the fundamental laws of planetary motion, which revolutionized our understanding of the solar system. His first law states that planets orbit the Sun in elliptical paths with the Sun at one focus.
The second law, or the law of equal areas, explains that a planet sweeps out equal areas in its orbit in equal times, indicating varying orbital speeds. His third law establishes a relationship between the orbital period and the distance of a planet from the Sun, revealing a consistent mathematical pattern. These laws laid the foundation for modern celestial mechanics.
A body orbits the Earth at a mean radius that is twice the distance of the parking orbit of a geostationary satellite. According to Kepler's third law, the orbital period of a body increases as the radius of its orbit becomes larger. In this case, the greater radius results in a longer orbital perioRead more
A body orbits the Earth at a mean radius that is twice the distance of the parking orbit of a geostationary satellite. According to Kepler’s third law, the orbital period of a body increases as the radius of its orbit becomes larger. In this case, the greater radius results in a longer orbital period. The body’s period is determined to be approximately 2√2 days, highlighting how the distance from the central body influences the time taken for one complete revolution around the Earth. This demonstrates the proportional relationship between orbital radius and time period.
According to Kepler's law of periods, the ratio of the orbital periods of two planets is related to the ratio of their semi-major axes. Specifically, the ratio of the periods (T₂/T₁) equals the ratio of their radii raised to the power of three-halves. In this case, if the radius of the second planetRead more
According to Kepler’s law of periods, the ratio of the orbital periods of two planets is related to the ratio of their semi-major axes. Specifically, the ratio of the periods (T₂/T₁) equals the ratio of their radii raised to the power of three-halves. In this case, if the radius of the second planet is four times that of the first, the calculation shows that T₂/T₁ equals eight.
Therefore, if the orbital period of the first planet is 1 day, the orbital period of the second planet would be 8 days, demonstrating the significant impact of radius on orbital time.
According to Kepler’s law of periods,
T₂/T₁ = (r₂/r₁)³/² = (4/1)³/² = 8
T₂ = 8T₁= 8 x 1 day = 8 days
A satellite orbiting Earth with a specific orbital radius and time period exhibits a consistent relationship between its time period and orbital radius. According to Kepler's third law, the square of the satellite's orbital period is directly proportional to the cube of its orbital radius. This meanRead more
A satellite orbiting Earth with a specific orbital radius and time period exhibits a consistent relationship between its time period and orbital radius. According to Kepler’s third law, the square of the satellite’s orbital period is directly proportional to the cube of its orbital radius. This means that the ratio of the square of the time period to the cube of the radius remains constant for any satellite orbiting the same central body, such as Earth. This principle reflects the uniformity of gravitational influence and orbital mechanics in determining the motion of satellites around a planet.
When the central gravitational force decreases, it does not produce any torque on the orbiting body because the force acts along the radius vector. As a result, the angular momentum of the body remains conserved. Since angular momentum is directly related to areal velocity (the area swept per unit tRead more
When the central gravitational force decreases, it does not produce any torque on the orbiting body because the force acts along the radius vector. As a result, the angular momentum of the body remains conserved. Since angular momentum is directly related to areal velocity (the area swept per unit time), the areal velocity also remains unchanged. This conservation of angular momentum and areal velocity aligns with Kepler’s second law, which states that a planet sweeps out equal areas in equal time intervals, irrespective of changes in the central force magnitude, as long as no external torque is applied.
A satellite in a circular orbit of radius R has an orbital period of 4 hours. For another satellite orbiting the same planet with a radius of 3R, its orbital period can be determined using Kepler's law of periods. The ratio of the orbital periods is proportional to the ratio of their orbital radii rRead more
A satellite in a circular orbit of radius R has an orbital period of 4 hours. For another satellite orbiting the same planet with a radius of 3R, its orbital period can be determined using Kepler’s law of periods. The ratio of the orbital periods is proportional to the ratio of their orbital radii raised to the power of three-halves. Substituting the values, the period ratio is (3 R/R)³/² = √27. Therefore, the period of the second satellite is √27 x 4, or approximately 4√27 hours.
Draw a graph showing the variation of acceleration dur to gravity g with distance r from the centre of the earth.
For points located outside the Earth, the acceleration due to gravity decreases with distance. It is inversely proportional to the square of the distance from the center of the Earth, meaning as one moves further away, gravity becomes weaker. Specifically, this means that the gravitational force felRead more
For points located outside the Earth, the acceleration due to gravity decreases with distance. It is inversely proportional to the square of the distance from the center of the Earth, meaning as one moves further away, gravity becomes weaker. Specifically, this means that the gravitational force felt at a height above Earth’s surface diminishes with increasing distance from the center.
In contrast, for points inside the Earth, gravity behaves differently. The acceleration due to gravity decreases linearly as one moves closer to the center. This indicates that the gravitational pull inside the Earth is directly related to the distance from the center.
For points lying onside the earth (r > R)
gₕ/g = R²/(R + h)² = R²/r² or gₕ = gR²/r²
gₕ ∝ 1/r²
For points lying inside the earth (r < R)
See lessgₔ = g(R – d)/R = gʳ/R or gₔ = gʳ/R
gₔ ∝ r
Explain how is the acceleration due to gravity affected at a latitude due to the rotational motion of the earth.
The acceleration due to gravity g is slightly altered by Earth's rotational motion, depending on latitude. At the equator, Earth's centrifugal force, due to its maximum velocity of rotation, acts outward and opposes gravity, reducing the effective value of g slightly. As latitude increases toward thRead more
The acceleration due to gravity g is slightly altered by Earth’s rotational motion, depending on latitude. At the equator, Earth’s centrifugal force, due to its maximum velocity of rotation, acts outward and opposes gravity, reducing the effective value of g slightly. As latitude increases toward the poles, the rotational velocity decreases, and the centrifugal force decreases.
See lessAt the poles, centrifugal force is negligible, and g is at its maximum value. Therefore, g is lowest at the equator and increases gradually as one moves toward the poles.
Define latitude at a place.
Latitude at a place measures how far north or south it is from the Earth’s equator. It’s expressed in degrees, starting at 0° at the equator and going up to 90° at the North and South Poles. Locations in the Northern Hemisphere have positive latitudes, while those in the Southern Hemisphere have negRead more
Latitude at a place measures how far north or south it is from the Earth’s equator. It’s expressed in degrees, starting at 0° at the equator and going up to 90° at the North and South Poles. Locations in the Northern Hemisphere have positive latitudes, while those in the Southern Hemisphere have negative latitudes.
See lessThese imaginary lines, called parallels, run horizontally around the globe. Latitude helps pinpoint a location on Earth and also influences the climate and sunlight a region gets throughout the year, playing a key role in geography and navigation.
A body is projected from earth’s surface to become its satellite. Its time period of revolution will not depend upon
The orbital period of a satellite depends on the radius of its orbit and the mass of the central body, such as Earth, but it is independent of the satellite's mass. This means that whether the satellite is small or large, its orbital period remains unchanged as long as the orbital radius and the cenRead more
The orbital period of a satellite depends on the radius of its orbit and the mass of the central body, such as Earth, but it is independent of the satellite’s mass. This means that whether the satellite is small or large, its orbital period remains unchanged as long as the orbital radius and the central body’s mass remain constant. This principle highlights that the motion of a satellite is governed by gravitational forces and does not rely on the satellite’s own mass, making it a fundamental aspect of orbital mechanics.
T = 2π√((R + h)³ / GM),
See lessClearly, T does not depend on the mass m of the satellite.
Kepler discovered
Johannes Kepler, a renowned 17th-century astronomer, discovered the fundamental laws of planetary motion, which revolutionized our understanding of the solar system. His first law states that planets orbit the Sun in elliptical paths with the Sun at one focus. The second law, or the law of equal areRead more
Johannes Kepler, a renowned 17th-century astronomer, discovered the fundamental laws of planetary motion, which revolutionized our understanding of the solar system. His first law states that planets orbit the Sun in elliptical paths with the Sun at one focus.
See lessThe second law, or the law of equal areas, explains that a planet sweeps out equal areas in its orbit in equal times, indicating varying orbital speeds. His third law establishes a relationship between the orbital period and the distance of a planet from the Sun, revealing a consistent mathematical pattern. These laws laid the foundation for modern celestial mechanics.
A body is orbiting around earth at a mean radius which is two times as greater as the parking orbit of a geostationary satellite, the period of body is
A body orbits the Earth at a mean radius that is twice the distance of the parking orbit of a geostationary satellite. According to Kepler's third law, the orbital period of a body increases as the radius of its orbit becomes larger. In this case, the greater radius results in a longer orbital perioRead more
A body orbits the Earth at a mean radius that is twice the distance of the parking orbit of a geostationary satellite. According to Kepler’s third law, the orbital period of a body increases as the radius of its orbit becomes larger. In this case, the greater radius results in a longer orbital period. The body’s period is determined to be approximately 2√2 days, highlighting how the distance from the central body influences the time taken for one complete revolution around the Earth. This demonstrates the proportional relationship between orbital radius and time period.
See lessSatellite is revolving around earth. If its height is increased to four times the height of geostationary satellite, what will become its time period?
According to Kepler's law of periods, the ratio of the orbital periods of two planets is related to the ratio of their semi-major axes. Specifically, the ratio of the periods (T₂/T₁) equals the ratio of their radii raised to the power of three-halves. In this case, if the radius of the second planetRead more
According to Kepler’s law of periods, the ratio of the orbital periods of two planets is related to the ratio of their semi-major axes. Specifically, the ratio of the periods (T₂/T₁) equals the ratio of their radii raised to the power of three-halves. In this case, if the radius of the second planet is four times that of the first, the calculation shows that T₂/T₁ equals eight.
Therefore, if the orbital period of the first planet is 1 day, the orbital period of the second planet would be 8 days, demonstrating the significant impact of radius on orbital time.
According to Kepler’s law of periods,
See lessT₂/T₁ = (r₂/r₁)³/² = (4/1)³/² = 8
T₂ = 8T₁= 8 x 1 day = 8 days
A satellite is orbiting around the earth with orbital radius R and time period T. The quantity which remains constant is
A satellite orbiting Earth with a specific orbital radius and time period exhibits a consistent relationship between its time period and orbital radius. According to Kepler's third law, the square of the satellite's orbital period is directly proportional to the cube of its orbital radius. This meanRead more
A satellite orbiting Earth with a specific orbital radius and time period exhibits a consistent relationship between its time period and orbital radius. According to Kepler’s third law, the square of the satellite’s orbital period is directly proportional to the cube of its orbital radius. This means that the ratio of the square of the time period to the cube of the radius remains constant for any satellite orbiting the same central body, such as Earth. This principle reflects the uniformity of gravitational influence and orbital mechanics in determining the motion of satellites around a planet.
See lessIf gravitational constant is decreasing in time, what will remain unchanged in case of a satellite orbiting around earth?
When the central gravitational force decreases, it does not produce any torque on the orbiting body because the force acts along the radius vector. As a result, the angular momentum of the body remains conserved. Since angular momentum is directly related to areal velocity (the area swept per unit tRead more
When the central gravitational force decreases, it does not produce any torque on the orbiting body because the force acts along the radius vector. As a result, the angular momentum of the body remains conserved. Since angular momentum is directly related to areal velocity (the area swept per unit time), the areal velocity also remains unchanged. This conservation of angular momentum and areal velocity aligns with Kepler’s second law, which states that a planet sweeps out equal areas in equal time intervals, irrespective of changes in the central force magnitude, as long as no external torque is applied.
See lessA satellite in a circular orbit of radius R has a period of 4 h. Another satellite with orbital radius 3 R round the same planet will have a period (in hours)
A satellite in a circular orbit of radius R has an orbital period of 4 hours. For another satellite orbiting the same planet with a radius of 3R, its orbital period can be determined using Kepler's law of periods. The ratio of the orbital periods is proportional to the ratio of their orbital radii rRead more
A satellite in a circular orbit of radius R has an orbital period of 4 hours. For another satellite orbiting the same planet with a radius of 3R, its orbital period can be determined using Kepler’s law of periods. The ratio of the orbital periods is proportional to the ratio of their orbital radii raised to the power of three-halves. Substituting the values, the period ratio is (3 R/R)³/² = √27. Therefore, the period of the second satellite is √27 x 4, or approximately 4√27 hours.
T₂/T₁ = (3 R/R)³/² = √27
See lessT₂ = √27T₁ = 4√27 h.