1. The gravitational force between two masses is unaffected by the medium between them. 2. Gravitational forces between two bodies are equal and opposite, following Newton's third law of motion. 3. The law of gravitation applies accurately to point masses. 4. Gravitational force acts along the lineRead more
1. The gravitational force between two masses is unaffected by the medium between them.
2. Gravitational forces between two bodies are equal and opposite, following Newton’s third law of motion.
3. The law of gravitation applies accurately to point masses.
4. Gravitational force acts along the line joining two point masses, depending only on the distance (r) without any angular dependence, showing spherical symmetry.
5. Gravitational force is conservative, meaning work done depends only on initial and final positions.
6. The gravitational force between two bodies is not influenced by the presence of other bodies.
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.
Impact of the Rebellion of 1857 on British Policies in India 1. End of East India Company Rule: The British Crown assumed direct control, establishing the British Raj in 1858. 2. Abolition of Doctrine of Lapse: Policy changes prevented the annexation of princely states without heirs. 3. Military RefRead more
Impact of the Rebellion of 1857 on British Policies in India
1. End of East India Company Rule: The British Crown assumed direct control, establishing the British Raj in 1858.
2. Abolition of Doctrine of Lapse: Policy changes prevented the annexation of princely states without heirs.
3. Military Reforms: Restructured military reduced reliance on native troops and increased recruitment of British soldiers.
4. Religious and Social Sensitivity: Policies aimed to avoid interference in religious and social customs to prevent antagonism.
5. Administrative Changes: Reforms in governance and judicial systems were introduced, with gradual inclusion of Indians in administrative roles.
6. Propaganda and Education: Emphasis on education and propaganda to promote British viewpoints and loyalty to the Crown.
These policy shifts reflected a more cautious approach by the British to maintain control while addressing grievances following the Rebellion of 1857.
British Methods to Secure Submission of Rebel Landowners in Awadh 1. Military Superiority: Utilized well-equipped and disciplined forces to overpower rebel armies, establishing military dominance. 2. Diplomatic Tactics: Engaged in negotiations offering amnesty to surrendering rebels and assurance ofRead more
British Methods to Secure Submission of Rebel Landowners in Awadh
1. Military Superiority: Utilized well-equipped and disciplined forces to overpower rebel armies, establishing military dominance.
2. Diplomatic Tactics: Engaged in negotiations offering amnesty to surrendering rebels and assurance of protection to non-rebelling populations, fracturing unity.
3. Punitive Measures: Inflicted harsh punitive actions on rebel strongholds, destroying property to deter further resistance.
4. Capture of Leaders: Focused on isolating and capturing rebel leaders, weakening organizational strength.
5. Reinstating Governance: After suppressing the rebellion, reestablished administrative control, employing military occupation and reforms to ensure future stability.
Through a combination of military prowess, strategic diplomacy, punitive actions, and governance restructuring, the British successfully subdued rebel landowners in Awadh, reinstating their authority in the region.
Impact of Bahadur Shah Zafar's Support on People and Ruling Families during the Rebellion of 1857: 1. Symbol of Unity: Zafar's endorsement provided a unifying figure for rebel factions, boosting morale and inspiring resistance against British rule. 2. Popular Support: Widely seen as a symbol of resiRead more
Impact of Bahadur Shah Zafar’s Support on People and Ruling Families during the Rebellion of 1857:
1. Symbol of Unity: Zafar’s endorsement provided a unifying figure for rebel factions, boosting morale and inspiring resistance against British rule.
2. Popular Support: Widely seen as a symbol of resistance, Zafar gained widespread support among the people, especially in North India.
3. Ruling Families’ Response: Influenced other ruling families to either support, oppose, or remain neutral in the rebellion, based on their alignment with Zafar.
4. British Reaction: Zafar’s involvement intensified British efforts to quell the rebellion, leading to the demise of the Mughal Empire.
5. Legacy: Zafar faced exile and the end of the Mughal dynasty, marking the rebellion’s failure and the British assertion of dominance.
Bahadur Shah Zafar’s support became a rallying point for resistance, but the rebellion’s failure resulted in dire consequences for Zafar and his dynasty, ultimately leading to the end of the Mughal Empire.
Reasons for British Confidence in India before May 1857 1. Military Strength: Possessed a well-equipped and disciplined military, giving confidence in suppressing potential uprisings. 2. Political Control: Established alliances with local rulers, ensuring political stability and compliance across reRead more
Reasons for British Confidence in India before May 1857
1. Military Strength: Possessed a well-equipped and disciplined military, giving confidence in suppressing potential uprisings.
2. Political Control: Established alliances with local rulers, ensuring political stability and compliance across regions.
3. Economic Dominance: Implemented policies enabling wealth extraction, fostering the belief in economic control and stability.
4. Communication Networks: Developed efficient communication systems like railways and telegraphs for quick response and governance.
5. Perceived Stability: Believed the political situation was stable, with minimal organized resistance to challenge their rule.
6. Divisions in Society: Assumed social, religious, and regional divisions in Indian society would prevent unified opposition against British authority.
These factors collectively contributed to the British rulers’ confidence in their control over India before the outbreak of the Indian Rebellion of 1857, which drastically challenged their preconceived notions and forced a reevaluation of their governance.
Mention the characteristic features of gravitational force.
1. The gravitational force between two masses is unaffected by the medium between them. 2. Gravitational forces between two bodies are equal and opposite, following Newton's third law of motion. 3. The law of gravitation applies accurately to point masses. 4. Gravitational force acts along the lineRead more
1. The gravitational force between two masses is unaffected by the medium between them.
2. Gravitational forces between two bodies are equal and opposite, following Newton’s third law of motion.
3. The law of gravitation applies accurately to point masses.
4. Gravitational force acts along the line joining two point masses, depending only on the distance (r) without any angular dependence, showing spherical symmetry.
5. Gravitational force is conservative, meaning work done depends only on initial and final positions.
6. The gravitational force between two bodies is not influenced by the presence of other bodies.
See lessBriefly 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 lessIn what ways did the British change their policies as a result of the rebellion of 1857?
Impact of the Rebellion of 1857 on British Policies in India 1. End of East India Company Rule: The British Crown assumed direct control, establishing the British Raj in 1858. 2. Abolition of Doctrine of Lapse: Policy changes prevented the annexation of princely states without heirs. 3. Military RefRead more
Impact of the Rebellion of 1857 on British Policies in India
1. End of East India Company Rule: The British Crown assumed direct control, establishing the British Raj in 1858.
2. Abolition of Doctrine of Lapse: Policy changes prevented the annexation of princely states without heirs.
3. Military Reforms: Restructured military reduced reliance on native troops and increased recruitment of British soldiers.
4. Religious and Social Sensitivity: Policies aimed to avoid interference in religious and social customs to prevent antagonism.
5. Administrative Changes: Reforms in governance and judicial systems were introduced, with gradual inclusion of Indians in administrative roles.
6. Propaganda and Education: Emphasis on education and propaganda to promote British viewpoints and loyalty to the Crown.
These policy shifts reflected a more cautious approach by the British to maintain control while addressing grievances following the Rebellion of 1857.
See lessHow did the British succeed in securing the submission of the rebel landowners of Awadh?
British Methods to Secure Submission of Rebel Landowners in Awadh 1. Military Superiority: Utilized well-equipped and disciplined forces to overpower rebel armies, establishing military dominance. 2. Diplomatic Tactics: Engaged in negotiations offering amnesty to surrendering rebels and assurance ofRead more
British Methods to Secure Submission of Rebel Landowners in Awadh
1. Military Superiority: Utilized well-equipped and disciplined forces to overpower rebel armies, establishing military dominance.
2. Diplomatic Tactics: Engaged in negotiations offering amnesty to surrendering rebels and assurance of protection to non-rebelling populations, fracturing unity.
3. Punitive Measures: Inflicted harsh punitive actions on rebel strongholds, destroying property to deter further resistance.
4. Capture of Leaders: Focused on isolating and capturing rebel leaders, weakening organizational strength.
5. Reinstating Governance: After suppressing the rebellion, reestablished administrative control, employing military occupation and reforms to ensure future stability.
Through a combination of military prowess, strategic diplomacy, punitive actions, and governance restructuring, the British successfully subdued rebel landowners in Awadh, reinstating their authority in the region.
See lessWhat impact did Bahadur Shah Zafar’s support to the rebellion have on the people and the ruling families?
Impact of Bahadur Shah Zafar's Support on People and Ruling Families during the Rebellion of 1857: 1. Symbol of Unity: Zafar's endorsement provided a unifying figure for rebel factions, boosting morale and inspiring resistance against British rule. 2. Popular Support: Widely seen as a symbol of resiRead more
Impact of Bahadur Shah Zafar’s Support on People and Ruling Families during the Rebellion of 1857:
1. Symbol of Unity: Zafar’s endorsement provided a unifying figure for rebel factions, boosting morale and inspiring resistance against British rule.
2. Popular Support: Widely seen as a symbol of resistance, Zafar gained widespread support among the people, especially in North India.
3. Ruling Families’ Response: Influenced other ruling families to either support, oppose, or remain neutral in the rebellion, based on their alignment with Zafar.
4. British Reaction: Zafar’s involvement intensified British efforts to quell the rebellion, leading to the demise of the Mughal Empire.
5. Legacy: Zafar faced exile and the end of the Mughal dynasty, marking the rebellion’s failure and the British assertion of dominance.
Bahadur Shah Zafar’s support became a rallying point for resistance, but the rebellion’s failure resulted in dire consequences for Zafar and his dynasty, ultimately leading to the end of the Mughal Empire.
See lessWhat could be the reasons for the confidence of the British rulers about their position in India before May 1857?
Reasons for British Confidence in India before May 1857 1. Military Strength: Possessed a well-equipped and disciplined military, giving confidence in suppressing potential uprisings. 2. Political Control: Established alliances with local rulers, ensuring political stability and compliance across reRead more
Reasons for British Confidence in India before May 1857
1. Military Strength: Possessed a well-equipped and disciplined military, giving confidence in suppressing potential uprisings.
2. Political Control: Established alliances with local rulers, ensuring political stability and compliance across regions.
3. Economic Dominance: Implemented policies enabling wealth extraction, fostering the belief in economic control and stability.
4. Communication Networks: Developed efficient communication systems like railways and telegraphs for quick response and governance.
5. Perceived Stability: Believed the political situation was stable, with minimal organized resistance to challenge their rule.
6. Divisions in Society: Assumed social, religious, and regional divisions in Indian society would prevent unified opposition against British authority.
These factors collectively contributed to the British rulers’ confidence in their control over India before the outbreak of the Indian Rebellion of 1857, which drastically challenged their preconceived notions and forced a reevaluation of their governance.
See less