Gravitation

Gravitation

Introduction to Gravitation

• Motion and Force: Objects move because of forces. A force can change the speed or direction of an object.
• Everyday Observations:
  • Objects fall towards the earth when dropped.
  • Planets orbit the Sun.
  • The moon orbits the Earth.
• Newton’s Insight: Isaac Newton realized that the same force is responsible for all these motions. This force is called gravitational force.

Newton’s Apple and Gravitation

• Newton’s Thought: An apple falling from a tree made Newton think if the Earth can attract an apple, can it also attract the moon?
• Centripetal Force: The force that keeps an object moving in a circular path. For the moon, this force is provided by Earth’s gravity.

Universal Law of Gravitation

• Statement: Every object in the universe attracts every other object with a force.
• Formula:
  • ​
  • G is the universal gravitational constant.
• Understanding the Formula:
  • Direct Proportion: Force increases with the masses of the objects.
  • Inverse Proportion: Force decreases with the square of the distance between the objects.
• Example Calculation: Calculate the gravitational force between the Earth and the Moon using the given masses and distance.

Importance of the Universal Law of Gravitation

• Explains Many Phenomena:
  • Why we stay on the ground (gravity binds us to Earth).
  • The moon’s orbit around Earth.
  • Planets’ orbits around the Sun.
  • Tides caused by the gravitational pull of the moon and the Sun.

Additional Notes

• Tangent to a Circle: A straight line that touches a circle at only one point.
• Inverse-Square Law: If the distance increases by a factor, the force decreases by the square of that factor (e.g., distance x6, force ÷36).

By understanding these principles, we can see how gravity affects everything from falling apples to orbiting planets!

Free Fall

Understanding Free Fall

Free Fall:

• When objects fall towards Earth due to gravity alone, they are in free fall.
• During free fall, the velocity changes due to gravitational acceleration, called acceleration due to gravity (g).
• Value of g: 9.8 m/s².

Calculating Gravitational Force

• Gravitational Force Formula:
  • 𝐹 = 𝑚×𝑔
  • Where 𝑚 is the mass of the object.
• Gravitational Force and Distance:
  • 

Calculation of value of g

• Given Values:
  • Universal Gravitational Constant 𝐺 = 6.7 × 10⁻¹Nm²/kg²
  • Mass of Earth 𝑀 = 6 × 10²⁴ kg
  • Radius of Earth 𝑅 = 6.4 × 10⁶ m
• Calculation:
  • 

Motion of Objects Under Earth’s Gravity

• Galileo’s Experiment:
  • Objects of different masses fall at the same rate in the absence of air resistance.

Equations of Motion with Gravity:

• 
• Where 𝑎a is replaced by g (9.8 m/s²).

Examples

• Example 9.2: Car Falling from a Ledge
  • Given: Time t = 0.5s, Initial velocity 𝑢=0
  • Find: Final speed, average speed, and height.
  • Solutions:
    • Speed on striking ground: 𝑣 = 𝑔 × 𝑡 = 10 × 0.5 = 5 m/s
    • 
• Example 9.3: Object Thrown Upwards
  • Given: Height 𝑠=10 m, Final velocity 𝑣=0v=0
  • Find: Initial velocity and time to reach the highest point.
  • Solutions:
    • 

Understanding free fall and gravitational force helps us see how gravity affects different objects, whether they are heavy or light.

Mass

• Definition: Mass measures an object’s inertia.
• Key Points:
  • Greater mass means greater inertia.
  • Mass is constant everywhere (Earth, Moon, space).

Weight

• Definition: Weight is the force with which Earth attracts an object.
• Formula: 𝑊=𝑚×𝑔
  • 𝑊: Weight
  • 𝑚: Mass
  • 𝑔: Acceleration due to gravity (9.8 m/s² on Earth)
• Unit: Newton (N)
• Key Points:
  • Weight varies with location because 𝑔 varies.
  • On Earth, weight is directly proportional to mass.

Weight of an object on the Moon

• Weight Difference:
  • The Moon’s gravity is weaker than Earth’s.
  • Weight on the Moon is about 1/6​th of weight on Earth.
• Calculation:
  • Weight on Moon = 1/6​ × Weight on Earth

Examples

• Example 1: Weight on Earth
  • Given: Mass 𝑚 = 10 kg
  • Find: Weight on Earth
  • Solution:
    • 𝑊 = 𝑚 × 𝑔
    • 𝑊 = 10 kg × 9.8 m/s²
    • 𝑊 = 98 N
  • Answer: Weight on Earth is 98 N.
• Example 2: Weight on Moon
  • Given: Weight on Earth 𝑊 = 10 N
  • Find: Weight on Moon
  • Solution:
    • Weight on Moon = 1/6 × Weight on Earth
    • Weight on Moon = 10/6
    • Weight on Moon ≈ 1.67 N
  • Answer: Weight on Moon is 1.67 N.

Understanding mass and weight helps us see how gravity affects objects differently based on their location, like on Earth or the Moon.

Thrust and Pressure

Understanding Thrust and Pressure

• Thrust: The force acting on an object perpendicular to the surface.
• Pressure: Thrust per unit area.

Examples to Understand Thrust and Pressure

• Fixing a Poster:
  • Pressing a drawing pin into a board.
  • Force applied on the head of the pin acts on a smaller area at the tip.
• Standing vs. Lying on Sand:
  • Standing: Weight acts on the area of your feet, making them sink deep.
  • Lying down: Weight is spread over a larger area, so you sink less.
  • Conclusion: The same force has different effects based on the area it acts on.

Formula for Pressure

• SI Unit: Pascal (Pa), where 1 Pa = 1 N/m²

Example Problem

• Given: A wooden block of mass 5 kg, dimensions 40 cm×20 cm×10 cm.
• Find: Pressure exerted by the block on the table.
• Solution:
  • 
• Conclusion: Smaller area results in higher pressure.

Practical Applications

• Sharp Tools: Have small areas to exert high pressure (e.g., nails, knives).
• Wide Tyres: Distribute weight over a larger area, reducing pressure.
• Tank Tracks: Spread weight over a large area to avoid sinking.

Pressure in Fluids

• Fluids: Liquids and gases that exert pressure on container walls.
• Key Point: Pressure in a confined fluid is transmitted equally in all directions.

By understanding thrust and pressure, we can explain how objects interact with surfaces and why design choices (like wider tyres or pointed tips) matter in practical applications.

Buoyancy

What is Buoyancy?

• Feeling Lighter in Water: When swimming, you feel lighter due to buoyancy.
• Heavy Bucket: A bucket feels heavier when lifted out of water because buoyancy is lost.
• Ships vs. Iron Sheets: Ships float because of their shape and volume, despite being made of heavy iron and steel.

Activity to Understand Buoyancy

• Buoyant Force: The upward force exerted by a fluid on an immersed object.
• Depends on Fluid Density: The denser the fluid, the greater the buoyant force.

Why Objects Float or Sink

• Density Concept:
  • Objects with density less than water float.
  • Objects with density greater than water sink.

Archimedes’ Principle

Archimedes’ Principle Explained

• Principle Statement: When a body is fully or partially immersed in a fluid, it experiences an upward force equal to the weight of the fluid displaced by it.
• Applications:
  • Used in designing ships and submarines.
  • Basis for lactometers (measure milk purity) and hydrometers (measure liquid density).

Archimedes’ Story

• Discovery: Archimedes discovered this principle when he noticed water displacement in a bathtub.
• Famous Exclamation: He ran through the streets shouting “Eureka!” meaning “I have got it.”
• Legacy: Archimedes’ work in geometry and mechanics helped in various fields, including military applications.

Chapter Summary:

• The law of gravitation states that the force of attraction between any two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them.
• The law applies to objects anywhere in the universe. Such a law is said to be universal.
• Gravitation is a weak force unless large masses are involved.
• The force of gravity decreases with altitude. It also varies on the surface of the earth, decreasing from poles to the equator.
• The weight of a body is the force with which the earth attracts it.
• The weight is equal to the product of mass and acceleration due to gravity.
• The weight may vary from place to place but the mass stays constant.
• All objects experience a force of buoyancy when they are immersed in a fluid.
• Objects having density less than that of the liquid in which they are immersed float on the surface of the liquid.
• If the density of the object is more than the density of the liquid in which it is immersed, then it sinks in the liquid.