In 1798, the English scientist Henry Cavendish experimentally determined the value of the gravitational constant 𝐺. The apparatus used is depicted in the figure. It consists of two small identical lead spheres, each of mass 𝑚, attached to the ends of a lightweight rod, forming a dumbbell. This rod iRead more
In 1798, the English scientist Henry Cavendish experimentally determined the value of the gravitational constant 𝐺. The apparatus used is depicted in the figure. It consists of two small identical lead spheres, each of mass 𝑚, attached to the ends of a lightweight rod, forming a dumbbell. This rod is suspended vertically by a thin fiber. Two larger lead spheres, each of mass 𝑀, are placed near the smaller spheres, ensuring all four spheres lie in a horizontal plane. The small spheres are attracted to the larger spheres by the gravitational force, given by:F = G Mm/r²
where 𝑟 is the distance between the centers of a large sphere and its neighboring small sphere.
Torque and Equilibrium:
The forces on the two small spheres form a couple that exerts a torque on the dumbbell. This torque causes the rod to rotate and twist the suspension fiber until the restoring torque of the fiber balances the gravitational torque. The angle of deflection (𝜃) is measured using a lamp and scale arrangement, which detects the deflection of a light beam.
Deflecting Torque: tau_deflecting = F . L = G Mm/r² . L,
where 𝐿 is the length of the rod.
Restoring Torque:
tau_restoring = k𝜃,
where 𝑘 is the torsion constant of the fiber (restoring torque per unit angle of twist).
In rotational equilibrium, the two torques are equal and opposite:
G Mm/r² . L = k𝜃,
Rearranging, the value of 𝐺 is:
𝐺 = k𝜃r² / MmL
Determination of 𝐺
By measuring all the quantities on the right-hand side during the experiment, the value of 𝐺 can be calculated. Cavendish’s experiment laid the foundation for precise determination of
𝐺, and modern methods have refined its measurement. The currently accepted value is:
G = 6.67 × 10⁻¹¹ Nm² kg⁻².
Numerous observations validate the law of gravitation. Key examples include: 1. Planetary and Lunar Motion: The Earth's orbit around the Sun and the Moon's revolution around the Earth are well-explained by this law. 2. Tides: Gravitational attraction between the Moon and seawater causes ocean tides.Read more
Numerous observations validate the law of gravitation. Key examples include:
1. Planetary and Lunar Motion:
The Earth’s orbit around the Sun and the Moon’s revolution around the Earth are well-explained by this law.
2. Tides:
Gravitational attraction between the Moon and seawater causes ocean tides.
3. Eclipses:
Predictions of solar and lunar eclipse timings using this law are highly accurate.
4. Artificial Satellites:
The law enables precise calculation of satellite orbits and periods of revolution.
5. Variation in g:
Changes in the value of g across Earth’s surface align with gravitational principles.
In 1687, Newton published the universal law of gravitation in his book Principia. The law states: Every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. ThisRead more
In 1687, Newton published the universal law of gravitation in his book Principia. The law states: Every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This force acts along the line joining the two particles.
Consider two bodies with masses m ₁ and m₂ separated by a distance r.
According to the law:
F ∝ m ₁ m₂ and F ∝ 1/r²
F ∝ m ₁m₂ /r²
or F = 𝐺 m ₁ m₂ / r²
where 𝐺 is the universal gravitational constant.
Definition of 𝐺 :
If m ₁ = m₂ = 1 and r = 1, then F = G. The universal gravitational constant is defined as the force of attraction between two bodies, each of unit mass, placed 1 unit distance apart.
Units of 𝐺 :
SI Unit: 𝐺 = 6.67 x 10⁻¹¹ Nm² kg⁻²
Cgs Unit, 𝐺 = 6.67 x 10⁸ dyn cm²g⁻².
Dimensions of 𝐺 :
From the formula 𝐺 = F r² / m ₁ m₂ , the dimensions of 𝐺 are:
[G] = MLT⁻² x L² / M x M = [M⁻¹ L³T⁻²]
Properties of 𝐺 :
The value of 𝐺 is constant and does not depend on the nature, size, or composition of the interacting bodies, making it a universal constant.
In 1665, Sir Isaac Newton observed an apple falling from a tree, inspiring him to formulate the law of gravitation. He hypothesized that the same force pulling the apple towards the Earth also kept the Moon in its orbit. By comparing the acceleration due to gravity experienced by the Moon and objectRead more
In 1665, Sir Isaac Newton observed an apple falling from a tree, inspiring him to formulate the law of gravitation. He hypothesized that the same force pulling the apple towards the Earth also kept the Moon in its orbit. By comparing the acceleration due to gravity experienced by the Moon and objects near the Earth’s surface, Newton developed his theory of gravitation.
Newton assumed the Moon moved in a circular orbit with a radius of R (= 3.84 x 10⁸m) and an orbital period of T = 27.3 days = 27.3 x 86,400s.From this, he calculated the Moon’s speed and centripetal acceleration. The Moon’s centripetal acceleration was much smaller than the acceleration due to gravity on the Earth’s surface (g = 9.8 ms ⁻² ).
Newton proposed that the gravitational force weakens with increasing distance from the Earth’s center, following an inverse-square law. Using the Earth’s radius Rₑ, he demonstrated that:
a꜀ = (Rₑ/R)² x g.
With Rₑ/R = 1/60, the calculated value of a꜀ matched the observed value, confirming the inverse-square relationship. This test, known as Newton’s “Moon Test,” validated his hypothesis.
Newton also concluded that the gravitational force is proportional to the masses of the interacting objects. By applying the third law of motion, he showed that the forces are equal and opposite. These insights led to the formulation of Newton’s Universal Law of Gravitation, describing how all objects in the universe attract one another.
Rani Lakshmibai of Jhansi: An Unusual Woman for Her Times 1. Warrior Queen: Uncommon for her era, she received training in martial arts, horse riding, and warfare skills. 2. Leadership and Governance: After her husband's demise, she took charge of Jhansi's administration, showcasing exceptional leadRead more
Rani Lakshmibai of Jhansi: An Unusual Woman for Her Times
1. Warrior Queen: Uncommon for her era, she received training in martial arts, horse riding, and warfare skills.
2. Leadership and Governance: After her husband’s demise, she took charge of Jhansi’s administration, showcasing exceptional leadership abilities.
3. Fearless and Resilient: Displayed unparalleled bravery during the Indian Rebellion of 1857, leading her troops into battle against the British.
4. Symbol of Rebellion: Defied societal norms by actively participating in the fight against British rule, becoming an iconic figure of resistance.
5. Sacrifice for Freedom: Fought valiantly for India’s independence but tragically lost her life in battle, leaving behind a legacy of courage and patriotism.
Rani Lakshmibai’s remarkable qualities and defiance of gender stereotypes of her time make her an inspirational figure in Indian history, celebrated for her bravery and dedication to India’s freedom struggle.
Briefly explain the Cavendish’ s experiment for the determination of the universal constant G.
In 1798, the English scientist Henry Cavendish experimentally determined the value of the gravitational constant 𝐺. The apparatus used is depicted in the figure. It consists of two small identical lead spheres, each of mass 𝑚, attached to the ends of a lightweight rod, forming a dumbbell. This rod iRead more
In 1798, the English scientist Henry Cavendish experimentally determined the value of the gravitational constant 𝐺. The apparatus used is depicted in the figure. It consists of two small identical lead spheres, each of mass 𝑚, attached to the ends of a lightweight rod, forming a dumbbell. This rod is suspended vertically by a thin fiber. Two larger lead spheres, each of mass 𝑀, are placed near the smaller spheres, ensuring all four spheres lie in a horizontal plane. The small spheres are attracted to the larger spheres by the gravitational force, given by:F = G Mm/r²
See lesswhere 𝑟 is the distance between the centers of a large sphere and its neighboring small sphere.
Torque and Equilibrium:
The forces on the two small spheres form a couple that exerts a torque on the dumbbell. This torque causes the rod to rotate and twist the suspension fiber until the restoring torque of the fiber balances the gravitational torque. The angle of deflection (𝜃) is measured using a lamp and scale arrangement, which detects the deflection of a light beam.
Deflecting Torque: tau_deflecting = F . L = G Mm/r² . L,
where 𝐿 is the length of the rod.
Restoring Torque:
tau_restoring = k𝜃,
where 𝑘 is the torsion constant of the fiber (restoring torque per unit angle of twist).
In rotational equilibrium, the two torques are equal and opposite:
G Mm/r² . L = k𝜃,
Rearranging, the value of 𝐺 is:
𝐺 = k𝜃r² / MmL
Determination of 𝐺
By measuring all the quantities on the right-hand side during the experiment, the value of 𝐺 can be calculated. Cavendish’s experiment laid the foundation for precise determination of
𝐺, and modern methods have refined its measurement. The currently accepted value is:
G = 6.67 × 10⁻¹¹ Nm² kg⁻².
Mention some experimental evidences in support of Newton’s law of gravitation.
Numerous observations validate the law of gravitation. Key examples include: 1. Planetary and Lunar Motion: The Earth's orbit around the Sun and the Moon's revolution around the Earth are well-explained by this law. 2. Tides: Gravitational attraction between the Moon and seawater causes ocean tides.Read more
Numerous observations validate the law of gravitation. Key examples include:
1. Planetary and Lunar Motion:
The Earth’s orbit around the Sun and the Moon’s revolution around the Earth are well-explained by this law.
2. Tides:
Gravitational attraction between the Moon and seawater causes ocean tides.
3. Eclipses:
Predictions of solar and lunar eclipse timings using this law are highly accurate.
4. Artificial Satellites:
The law enables precise calculation of satellite orbits and periods of revolution.
5. Variation in g:
See lessChanges in the value of g across Earth’s surface align with gravitational principles.
State Newton’s law of gravitation. Hence define G. What are the units and dimensions of G? Why is G called a universal gravitational constant?
In 1687, Newton published the universal law of gravitation in his book Principia. The law states: Every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. ThisRead more
In 1687, Newton published the universal law of gravitation in his book Principia. The law states: Every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This force acts along the line joining the two particles.
Consider two bodies with masses m ₁ and m₂ separated by a distance r.
See lessAccording to the law:
F ∝ m ₁ m₂ and F ∝ 1/r²
F ∝ m ₁m₂ /r²
or F = 𝐺 m ₁ m₂ / r²
where 𝐺 is the universal gravitational constant.
Definition of 𝐺 :
If m ₁ = m₂ = 1 and r = 1, then F = G. The universal gravitational constant is defined as the force of attraction between two bodies, each of unit mass, placed 1 unit distance apart.
Units of 𝐺 :
SI Unit: 𝐺 = 6.67 x 10⁻¹¹ Nm² kg⁻²
Cgs Unit, 𝐺 = 6.67 x 10⁸ dyn cm²g⁻².
Dimensions of 𝐺 :
From the formula 𝐺 = F r² / m ₁ m₂ , the dimensions of 𝐺 are:
[G] = MLT⁻² x L² / M x M = [M⁻¹ L³T⁻²]
Properties of 𝐺 :
The value of 𝐺 is constant and does not depend on the nature, size, or composition of the interacting bodies, making it a universal constant.
How did Newton discover the universal law of gravitation?
In 1665, Sir Isaac Newton observed an apple falling from a tree, inspiring him to formulate the law of gravitation. He hypothesized that the same force pulling the apple towards the Earth also kept the Moon in its orbit. By comparing the acceleration due to gravity experienced by the Moon and objectRead more
In 1665, Sir Isaac Newton observed an apple falling from a tree, inspiring him to formulate the law of gravitation. He hypothesized that the same force pulling the apple towards the Earth also kept the Moon in its orbit. By comparing the acceleration due to gravity experienced by the Moon and objects near the Earth’s surface, Newton developed his theory of gravitation.
Newton assumed the Moon moved in a circular orbit with a radius of R (= 3.84 x 10⁸m) and an orbital period of T = 27.3 days = 27.3 x 86,400s.From this, he calculated the Moon’s speed and centripetal acceleration. The Moon’s centripetal acceleration was much smaller than the acceleration due to gravity on the Earth’s surface (g = 9.8 ms ⁻² ).
Newton proposed that the gravitational force weakens with increasing distance from the Earth’s center, following an inverse-square law. Using the Earth’s radius Rₑ, he demonstrated that:
a꜀ = (Rₑ/R)² x g.
With Rₑ/R = 1/60, the calculated value of a꜀ matched the observed value, confirming the inverse-square relationship. This test, known as Newton’s “Moon Test,” validated his hypothesis.
Newton also concluded that the gravitational force is proportional to the masses of the interacting objects. By applying the third law of motion, he showed that the forces are equal and opposite. These insights led to the formulation of Newton’s Universal Law of Gravitation, describing how all objects in the universe attract one another.
See lessFind out more about Rani Lakshmibai of Jhansi. In what ways would she have been an unusual woman for her times?
Rani Lakshmibai of Jhansi: An Unusual Woman for Her Times 1. Warrior Queen: Uncommon for her era, she received training in martial arts, horse riding, and warfare skills. 2. Leadership and Governance: After her husband's demise, she took charge of Jhansi's administration, showcasing exceptional leadRead more
Rani Lakshmibai of Jhansi: An Unusual Woman for Her Times
1. Warrior Queen: Uncommon for her era, she received training in martial arts, horse riding, and warfare skills.
2. Leadership and Governance: After her husband’s demise, she took charge of Jhansi’s administration, showcasing exceptional leadership abilities.
3. Fearless and Resilient: Displayed unparalleled bravery during the Indian Rebellion of 1857, leading her troops into battle against the British.
4. Symbol of Rebellion: Defied societal norms by actively participating in the fight against British rule, becoming an iconic figure of resistance.
5. Sacrifice for Freedom: Fought valiantly for India’s independence but tragically lost her life in battle, leaving behind a legacy of courage and patriotism.
Rani Lakshmibai’s remarkable qualities and defiance of gender stereotypes of her time make her an inspirational figure in Indian history, celebrated for her bravery and dedication to India’s freedom struggle.
See less