Chapter 10: Gravitation

 

Gravitation and Related Concepts

Key Points:

1. Motion and Force

   • Force is needed to change the speed or direction of motion.
   • Objects fall towards the Earth when dropped, indicating a force acting on them.

2. Observations in Nature

   • Planets revolve around the Sun.
   • The Moon revolves around the Earth.

3. Gravitational Force

   • Isaac Newton identified a universal force responsible for these phenomena.
   • This force is termed the gravitational force.

4. Topics Covered in the Chapter

   • Gravitation and the universal law of gravitation.
   • Motion of objects under the influence of Earth's gravitational force.
   • Variation of the weight of a body from place to place.
   • Conditions for objects to float in liquids.

These notes summarize the key ideas related to gravitation and its significance in various natural phenomena.

 Gravitation

Key Concepts and Observations:

1. Moon’s Motion Around Earth:

   • The moon revolves around the Earth instead of moving in a straight line.
   • This indicates the presence of a force pulling the moon towards the Earth.

2. Newton’s Apple Observation:

   • Newton reasoned that the Earth's force attracting the apple also attracts the moon.
   • Concluded that the same type of force acts in both cases.

3. Centripetal Force:

   • In circular motion, an object requires a force directed towards the center, called centripetal force.
   • 
     
     
   • Activity 10.1 demonstrates that:
     • A stone tied to a string moves in a circular path when whirled.
     • Releasing the string causes the stone to fly off tangentially.
   • The moon's circular motion around the Earth is due to centripetal force provided by Earth's gravitational pull.

4. Tangent to a Circle:

   • A straight line touching a circle at only one point is called a tangent.
   • Without centripetal force, the moon would move along a tangent to its orbit.

1. Earth-Apple Interaction:

   • The Earth attracts the apple, and the apple attracts the Earth (Newton’s Third Law).
   • The Earth's movement towards the apple is negligible due to its massive size (Newton’s Second Law).
   • 

2. Universal Gravitation:

   • Newton extended the idea to conclude that all objects in the universe attract each other.
   • This universal force of attraction is termed gravitational force.

Applications:

• Explains why planets orbit the Sun.
• Highlights the role of gravitational force in maintaining celestial motion.

 Isaac Newton, Kepler’s Laws, and Gravitational Force

Isaac Newton (1642–1727)

• Background:

  • Born in Woolsthorpe, England, in a poor farming family.
  • Sent to Cambridge University in 1661.
  • During a plague in 1665, he developed ideas about gravity after observing an apple fall.

• Major Contributions:

  • Formulated the Universal Law of Gravitation.
  • Developed the Laws of Motion.
  • Worked on theories of light and color.
  • Invented the astronomical telescope.
  • Introduced calculus, used to prove that a sphere behaves as if all its mass is concentrated at its center.
  • Synthesized ideas from Copernicus, Kepler, and Galileo, revolutionizing physical science.

• Qualities of Newton’s Work:

  • Based on sound reasoning and mathematics.
  • His theories were simple, elegant, and foundational for modern science.

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Kepler’s Laws of Planetary Motion

1. First Law (Law of Ellipses):

   • The orbit of a planet is an ellipse, with the Sun at one of the foci.

2. Second Law (Equal Areas in Equal Time):

   • A line joining a planet to the Sun sweeps equal areas in equal time intervals.
   • If time from A to B = time from C to D, then area OAB = area OCD.

3. Third Law (Harmonic Law):

   • The cube of the mean distance of a planet from the Sun (r³) is proportional to the square of its orbital period (T²):
     • $r³/T² = \text{constant}$.

• Kepler described planetary motion but couldn’t explain its cause.

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Newton and Gravitational Force

• Newton used Kepler’s third law to calculate the gravitational force between the Sun and planets.
• He showed that the Sun’s gravitational force governs planetary motion.

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Derivation of the Inverse-Square Law

1. Force on an orbiting planet:

   • Force, $F \propto v^2 / r$.

2. Orbital velocity:

   • $v = 2\pi r / T$, so $v^2 \propto r^2 / T^2$.

3. Kepler’s Third Law:

   • $r^3 / T^2 = \text{constant}$, implies $v^2 \propto 1/r$.

4. Combining relations:

   • $F \propto v^2 / r$, and $v^2 \propto 1/r$, so $F \propto 1 / r^2$.

This shows that gravitational force decreases with the square of the distance between two objects

  Universal Law of Gravitation

Statement of the Law

• Every object in the universe attracts every other object.
• The force is:
  • Proportional to the product of their masses ($M \times m$).
  • Inversely proportional to the square of the distance ($d^2$) between their centers.

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Mathematical Expression

1. Force Proportionality:

   • $F \propto M \times m$
   • $F \propto \frac{1}{d^2}$

2. Combined Expression:

   • $F \propto \frac{M \times m}{d^2}$
   • Introducing the gravitational constant $G$: $$F = G \frac{M \times m}{d^2}$$

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Gravitational Constant ($G$)

• $G = 6.673 \times 10^{-11} \, \text{N m}^2 \text{kg}^{-2}$.
• Determined experimentally by Henry Cavendish using a sensitive balance.

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Properties of the Law

• Universal: Applicable to all objects, whether celestial or terrestrial, big or small.
• Inverse-Square Law:
  • If $d$ increases by a factor of 6, $F$ decreases by $\frac{1}{6^2} = \frac{1}{36}$.

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Force Between Two Objects

• Calculate $F$ using $F = G \frac{M \times m}{d^2}$.
• Example: Force between you and a nearby friend is negligible due to the small masses involved compared to Earth.

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Key Takeaways

• Gravitational force acts along the line joining the centers of two objects.
• Even though the force exists between all objects, it is perceptible only when at least one object has a large mass (e.g., Earth and Sun).

 Importance of the Universal Law of Gravitation and Free Fall

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Importance of the Universal Law of Gravitation

The law explains the following phenomena:

1. Binding Force: The force that binds us to the Earth.
2. Motion of the Moon: The moon's motion around the Earth.
3. Planetary Motion: The motion of planets around the Sun.
4. Tides: The occurrence of tides due to the Moon and the Sun.

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Free Fall

• Definition: Objects falling solely under the influence of Earth's gravitational force are in free fall.
• Effect of Gravity:
  • Causes a change in velocity (magnitude) while falling.
  • The resulting acceleration is called acceleration due to gravity (g).
• Unit of $g$: $\text{m/s}^2$.

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Formula for Gravitational Force During Free Fall

1. From the Second Law of Motion:

   $$F = m g$$

   ($m$ = mass of the object, $g$ = acceleration due to gravity)

2. From the Universal Law of Gravitation:

   $$F = G \frac{M \times m}{d^2}$$

3. Equating the two:

   $$g = G \frac{M}{d^2}$$

   Where:

   • $M$ = Mass of the Earth
   • $d$ = Distance between the object and the Earth's center

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Special Case: On or Near the Earth's Surface

• Distance ($d$) = Earth's radius ($R$)
• Formula: $$g = G \frac{M}{R^2}$$

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Calculation of $g$

• Given values:

  • $G = 6.7 \times 10^{-11} \, \text{N m}^2 \text{kg}^{-2}$
  • $M = 6 \times 10^{24} \, \text{kg}$
  • $R = 6.4 \times 10^{6} \, \text{m}$

• Substituting into the formula:

  $$g = \frac{6.7 \times 10^{-11} \times 6 \times 10^{24}}{(6.4 \times 10^6)^2}$$
  • Result: $g = 9.8 \, \text{m/s}^2$.

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Variation of $g$

1. $g$ is slightly greater at the poles than at the equator due to the Earth's shape.
2. For objects far from Earth, use: $$g = G \frac{M}{d^2}$$

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Key Takeaway

• The universal law of gravitation unifies diverse phenomena and provides the foundation for understanding the motion of celestial and terrestrial bodies.

 Motion of Objects Under the Influence of Gravitational Force

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Activity: Falling Objects

• Objective: To observe if all objects (hollow or solid, big or small) fall at the same rate.
• Steps:
  1. Drop a sheet of paper and a stone simultaneously from a height (e.g., first floor).
  2. Observation: The paper reaches the ground later than the stone due to air resistance, which affects the paper more than the stone.
  3. In a vacuum (air removed), both objects would fall at the same rate.
• Conclusion: Objects fall at the same rate regardless of mass, assuming air resistance is absent.

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Acceleration Due to Gravity (g)

• The acceleration experienced during free fall is independent of the object's mass.
• This means that all objects, whether solid or hollow, big or small, experience the same acceleration due to gravity (g) near the Earth's surface.

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Galileo's Experiment

• Galileo demonstrated this concept by dropping different objects from the Leaning Tower of Pisa.
• The experiment showed that, in the absence of air resistance, objects fall at the same rate.

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Equations of Motion Under Gravity

• When an object falls under the influence of gravity, we use the equations of uniformly accelerated motion with acceleration $a = g$:

1. $v = u + gt$ (Equation 10.10)
   • $v$: final velocity
   • $u$: initial velocity
   • $t$: time
2. $s = ut + \frac{1}{2}gt^2$ (Equation 10.11)
   • $s$: distance covered
3. $v^2 = u^2 + 2gs$ (Equation 10.12)

• Note:
  • Take acceleration $a$ as positive when it is in the direction of motion.
  • Take acceleration $a$ as negative when it opposes the motion.

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Key Takeaway

• All objects fall with the same acceleration under the influence of gravity, provided air resistance is negligible.

 Mass and Weight

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 Mass

• Definition: Mass is the measure of an object's inertia, which means how much it resists changes in motion.
• Key Property: The mass of an object remains constant, regardless of its location (whether on Earth, the Moon, or in outer space).
• Inertia: Greater mass implies greater inertia.

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 Weight

• Definition: Weight is the force with which the Earth attracts an object towards itself.
• Formula: $$F = m \times g$$ where $F$ is the weight, $m$ is the mass of the object, and $g$ is the acceleration due to gravity.
  • This implies: $$W = m \times g$$ where $W$ is the weight of the object.
• Units: The SI unit of weight is the Newton (N), the same as the unit for force.
• Weight vs. Mass:
  • Mass: Constant everywhere.
  • Weight: Depends on the local value of $g$, i.e., the gravitational force acting on the object.
  • Weight is proportional to mass at a given location, so we can use weight as an indirect measure of mass.

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 Weight of an Object on the Moon

• On the Moon: The weight of an object on the Moon is the force with which the Moon attracts the object.

• The Moon's mass is smaller than Earth's, so it exerts less gravitational force.

• Formula for weight on the Moon:

  $$W_m = \frac{G \times M_m \times m}{R_m^2}$$

  where:

  • $W_m$: Weight on the Moon
  • $G$: Gravitational constant
  • $M_m$: Mass of the Moon
  • $m$: Mass of the object
  • $R_m$: Radius of the Moon

• Formula for weight on Earth:

  $$W_e = \frac{G \times M_e \times m}{R_e^2}$$

  where:

  • $W_e$: Weight on Earth
  • $M_e$: Mass of the Earth
  • $R_e$: Radius of the Earth

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Weight Comparison (Earth vs. Moon)

• Given the masses and radii of the Earth and the Moon:

  • Earth: Mass = $5.98 \times 10^{24} \, \text{kg}$, Radius = $6.37 \times 10^6 \, \text{m}$
  • Moon: Mass = $7.36 \times 10^{22} \, \text{kg}$, Radius = $1.74 \times 10^6 \, \text{m}$

• Weight on the Moon is approximately $\frac{1}{6}$ of the weight on Earth:

  $$\frac{W_m}{W_e} \approx \frac{1}{6}$$

  Thus, the weight of an object on the Moon is about 1/6 of its weight on Earth.

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Key Takeaways

• Mass is constant and independent of location.
• Weight depends on the local gravitational pull (which differs on the Earth and the Moon).
• An object's weight on the Moon is roughly 1/6 of its weight on the Earth.

Thrust and Pressure

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Thrust and Pressure

• Thrust: The force exerted perpendicular to a surface.

  • Example: Pressing drawing pins with your thumb to fix a poster, where the force is applied perpendicularly to the surface area of the board and the pin's tip.
  • Example: When you stand or lie on sand, your body’s weight acts on the sand. The force is applied perpendicularly (vertically) to the surface.

• Pressure: The force applied per unit area. It is the effect of thrust on the area.

  • Formula: $$\text{Pressure} = \frac{\text{Thrust}}{\text{Area}} \quad \text{(Eq. 10.20)}$$
  • Units:
    • The SI unit of pressure is Newton per square meter (N/m²), or Pascal (Pa).
    • The effect of pressure depends on the area on which the force acts.
  • Examples:
    • Standing on sand: Pressure is greater when you stand because the force (your weight) acts on a smaller area (feet).
    • Lying on sand: Pressure is lesser because the same force acts on a larger area (your body).

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Pressure in Fluids

• Fluids: Liquids and gases are considered fluids.
  • Pressure in Fluids: Fluids exert pressure on the walls and base of the container they are in due to their weight.
  • Transmission of Pressure: In a confined fluid, pressure is transmitted undiminished in all directions.

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Key Takeaways

• Thrust refers to the force acting perpendicular to a surface.
• Pressure is the thrust (force) per unit area.
• Pressure can vary depending on the area on which a force acts (larger area = lower pressure, smaller area = higher pressure).
• In fluids, pressure is exerted in all directions and is uniform within a confined space.

 Buoyancy

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Buoyancy

• Buoyancy: It is the upward force exerted by a fluid (liquid or gas) on an object submerged in it. This force opposes the weight of the object and is responsible for the sensation of "weightlessness" experienced when floating in water.

• Activity to Understand Buoyancy:

  • Floating Bottle: When an empty plastic bottle is submerged in water, it floats. An upward push (buoyant force) is felt as the bottle is pushed deeper into the water. The force increases as the bottle is pushed deeper until it is fully immersed.
  • Upthrust: The upward force exerted by the fluid is known as upthrust or buoyant force. This force is what makes objects float.
  • Effect of Buoyant Force: The object floats or rises when the buoyant force is greater than its weight. To keep the object fully immersed, an external downward force is needed to balance the buoyant force.

• Why Objects Float or Sink:

  • When the buoyant force equals or exceeds the weight of an object, it floats. If the weight of the object is greater than the buoyant force, the object sinks.

• Factors Affecting Buoyant Force:

  • The magnitude of the buoyant force depends on the density of the fluid.
  • The more dense the fluid, the greater the buoyant force exerted on an object immersed in it.

• Examples:

  • Ship Made of Steel: A ship made of iron and steel floats on water due to its shape, which displaces a large amount of water, creating a significant buoyant force. A sheet of iron, having a small surface area, displaces less water and sinks.

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Key Takeaways

• Buoyant force (upthrust) is the upward force exerted by a fluid on an object submerged in it.
• The force depends on the fluid's density and the volume of fluid displaced.
• Objects float when the buoyant force is equal to or greater than their weight; they sink if the buoyant force is less.
• A ship floats while a sheet of metal sinks due to differences in the amount of displaced water.

Floating and Sinking of Objects

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Why Objects Float or Sink on Water:

• Activity 10.5 (Iron Nail in Water):

  • Observation: When an iron nail is placed on the surface of water, it sinks.
  • Explanation: The downward force due to the gravitational attraction on the nail (its weight) is greater than the upthrust or buoyant force exerted by the water. Hence, the nail sinks.

• Activity 10.6 (Cork and Iron Nail in Water):

  • Observation: A piece of cork floats, while an iron nail sinks, even though they have equal mass.
  • Explanation:
    • The density of cork is less than the density of water, so the buoyant force (upthrust) on the cork is greater than the weight of the cork, causing it to float.
    • The density of the iron nail is greater than the density of water, meaning the upthrust on the nail is less than its weight, causing it to sink.

• Key Concept: Density:

  • Density is defined as the mass per unit volume of a substance.
  • If an object’s density is less than the density of the fluid (e.g., cork in water), it will float.
  • If an object’s density is greater than the density of the fluid (e.g., iron nail in water), it will sink.

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Conclusion:

• Objects with a density lower than that of the fluid will float on the fluid.
• Objects with a density greater than that of the fluid will sink in the fluid.

 Archimedes' Principle

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Activity 10.7 (Buoyant Force on Stone):

• Procedure:
  • Tie a stone to a rubber string or spring balance and suspend it to measure its weight.
  • Dip the stone slowly into water and observe the change in the string elongation or spring balance reading.
• Observations:
  • As the stone is lowered into the water, the elongation of the string or the reading on the spring balance decreases.
  • Once the stone is fully immersed, no further change in elongation or reading is observed.
• Inference:
  • The decrease in elongation indicates that an upward force acts on the stone when it is immersed in water.
  • This upward force is called buoyant force.
  • The buoyant force reduces the effective weight of the stone, causing the reading on the balance to decrease.

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Archimedes' Principle:

• Statement: When a body is immersed fully or partially in a fluid, it experiences an upward force equal to the weight of the fluid displaced by it.

• Key Points:

  • The magnitude of the buoyant force depends on the volume of fluid displaced by the object.
  • Buoyant force is the reason objects float or sink in fluids.

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Applications of Archimedes' Principle:

1. Design of Ships and Submarines:
   • Used to calculate the buoyant force and design vessels that float on water.
2. Lactometers:
   • Instruments used to determine the purity of milk, based on how much milk displaces water.
3. Hydrometers:
   • Used to determine the density of liquids by measuring how much liquid a submerged object displaces.

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Conclusion:

• Archimedes' principle explains the buoyant force that acts on objects submerged in fluids, helping us understand floating, sinking, and the design of objects like ships and hydrometers.

 Archimedes

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Archimedes' Background:

• Nationality: Greek scientist
• Famous Discovery: Archimedes' Principle
  • Origin of Discovery: Archimedes noticed that when he stepped into a bathtub, the water overflowed. This led him to the realization about the displacement of water by an object.
  • Eureka Moment: He famously shouted "Eureka!" which means "I have got it," after discovering the principle.

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Contributions:

1. Archimedes' Principle:

   • Helped determine the buoyant force and the weight of displaced fluid.
   • Used to determine the purity of gold in a crown made for a king.

2. Works in Geometry and Mechanics:

   • Geometry: Archimedes contributed significantly to the understanding of areas and volumes.
   • Mechanics: He studied the use of levers, pulleys, and wheels-and-axles, which had practical applications in warfare.

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Impact:

• Military Contribution: His understanding of mechanics greatly assisted the Greek army during its war with the Roman army.

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Famous Quote:

• "Eureka!" - Meaning "I have found it"

 Relative Density

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Density:

• Definition: Density of a substance is the mass of a unit volume.
• Unit: Kilogram per meter cubed (kg/m³).
• Characteristic Property: The density of a substance is constant under specified conditions and is different for each substance.

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Relative Density:

• Definition: The relative density of a substance is the ratio of its density to the density of water.
  • Formula: $$\text{Relative Density} = \frac{\text{Density of the substance}}{\text{Density of water}}$$
• Unit: Relative density has no unit since it is a ratio of two similar quantities.
• Purpose: It helps express the density of a substance in comparison to that of water and can be used to determine the purity of substances.

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Example:

• Gold: Density = 19300 kg/m³
• Water: Density = 1000 kg/m³
• Relative Density of Gold: $$\frac{19300}{1000} = 19.3$$

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