Class 9

   

•  Gravitation
• Newton's Law of Gravitation
•  Centripetal Force
• Acceleration Due to Gravity (g)
• Variation of g
• Mass vs Weight
• Kepler laws
• Archimedes' Principle

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Gravitation 

Introduction

• Gravitation is a force of attraction between any two bodies.
• The story began with Newton observing an apple falling from a tree.
• Unlike simple attraction, gravitation is a force that both bodies exert on each other.

📖 Newton's Law of Gravitation

• Statement: Every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
• Consider two bodies with masses ${�}_1$ and ${�}_2$, separated by a distance $�$. The gravitational force $�$ between them is given by:$$�=�\frac{{�}_1{�}_2}{{�}^2}$$where $�$ is the universal gravitational constant.
• $�=6.67\times 10^{-11}{\text{N m}}^2\mathrm{/}{\text{kg}}^2$
• Deriving the Unit of G:
  • $�=�\frac{{�}_1{�}_2}{{�}^2}$
  • $�=\frac{�{�}^2}{{�}_1{�}_2}$
  • Units: $\frac{\text{N}\cdot {\text{m}}^2}{{\text{kg}}^2}$

Centripetal Force

• When an object moves in a circular path, a centripetal force acts on it, directed towards the center of the circle.
• Example: Planets orbiting the Sun. The gravitational force provides the necessary centripetal force.

Acceleration Due to Gravity (g)

• When an object falls towards the Earth, it accelerates due to gravity.
• The acceleration due to gravity ($�$) can be derived from Newton's Law of Gravitation:
  • $�=�\frac{{�}_1{�}_2}{{�}^2}$
  • $�={�}_1�$
  • Equating the two: ${�}_1�=�\frac{{�}_1{�}_2}{{�}^2}$
  • $�=�=�\frac{{�}_2}{{�}^2}$
  • Here, ${�}_2$ is the mass of the Earth and $�$ is the radius of the AEarth.
• The standard value of $�$ is approximately $9.8{\text{m/s}}^2$ or $10{\text{m/s}}^2$.

Variation of g

• The value of $�$ is not constant everywhere. It varies with:
  • Altitude: $�$ decreases as altitude increases.
  • Depth: $�$ decreases as depth increases.
  • Shape of the Earth: $�$ is highest at the poles and lowest at the equator.
    • The Earth is not a perfect sphere; it is slightly flattened at the poles.

Mass vs. Weight 

• Mass:
  • Scalar quantity.
  • Remains constant everywhere.
  • Measured using a balance.
  • Unit: kilogram (kg).
  • Can never be zero.
  • Does not change based on location.
• Weight:
  • Vector quantity (has direction).
  • Varies depending on the value of $�$.
  • Measured using a spring balance.
  • Unit: Newton (N).
  • Can be zero (in the absence of gravity).
  • Changes based on location.
• Weight $�$ is given by:$$\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">�</span>}$$where $�$ is mass and $�$ is the acceleration due to gravity.
• Weight on the Moon: The acceleration due to gravity on the Moon is approximately 1/6th of that on Earth. Therefore, an object's weight on the Moon will be 1/6th of its weight on Earth, but its mass will remain the same.

Example Problem

What is the weight of an 80 kg man on the moon? Calculate his mass on Earth and Moon.

Solution Mass on Earth = 80 kg Mass on Moon = 80 kg (mass remains constant) Weight on Moon = $�\cdot �$, where ${�}_{\text{moon}}=\frac{{�}_{\text{earth}}}{6}\approx \frac{10}{6}{\text{m/s}}^2$ Weight on Moon = $80\cdot \frac{10}{6}=\frac{800}{6}=\frac{400}{3}\approx 133.33\text{N}$

Free Fall

• Free fall is the motion of an object under the influence of gravity alone (no other forces acting on it).
• The equations of motion are modified:
  • $�=�+��$
  • $�=��+\frac{1}{2}�{�}^2$
  • ${�}^2={�}^2+2��$

Sign Conventions

• Downward direction: negative.
• Upward direction: positive.
• $�$ is almost always negative.

Example Problem

A stone is released from the top of a tower of 20 m. Calculate its velocity just before touching the ground.

Solution s = -20 m (downward) u = 0 m/s (released from rest) g = -10 m/s${}^2$ Using ${�}^2={�}^2+2��$: ${�}^2=0+2\cdot (-10)\cdot (-20)=400$ $�=\sqrt{400}=20\text{m/s}$ The negative sign is implicit because the velocity will be in the downward direction.

Kepler's Laws

• First Law (Law of Orbits): Planets move in elliptical orbits with the Sun at one of the foci.
• Second Law (Law of Areas): A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
• Third Law (Law of Periods): The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.$${\mathrm{<span\; style="background-color:\; white;">�</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\propto </span>{\mathrm{<span\; style="background-color:\; white;">�</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}$$

Thrust and Pressure

• Thrust: Force acting on an object perpendicular to the surface. Unit: Newton (N).
• Pressure: Thrust acting per unit area.$$\mathrm{<span\; style="background-color:\; white;">�</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">�</span>}}{\mathrm{<span\; style="background-color:\; white;">�</span>}}$$Unit: Pascal (Pa) or ${\text{N/m}}^2$.
• 1 Pascal is the pressure exerted when a force of 1 Newton acts on an area of 1 square meter.
• Applications:
  • Sharp knives cut better due to smaller area, resulting in higher pressure.
  • Wide straps on bags reduce pressure on shoulders.
  • Camels have broad feet to reduce pressure on sand.

Pressure in Fluids

• Fluids (liquids and gases) exert pressure.
• Pressure in a fluid depends on:
  • Depth: Pressure increases with depth.
  • Density: Pressure increases with density.

Buoyancy and Buoyant Force

• When an object is immersed in a fluid, it experiences an upward force called the buoyant force.
• Buoyancy: The tendency of a fluid to exert an upward force on an object placed in it.
• Conditions for Floatation:
  • If ${�}_{�}>{�}_{�}$ (gravitational force > buoyant force): Object sinks.
  • If ${�}_{�}<{�}_{�}$ (gravitational force < buoyant force): Object moves upward.
  • If ${�}_{�}={�}_{�}$ (gravitational force = buoyant force): Object floats.
• Density and Floatation:
• Object denser than fluid: sinks.
• Object less dense than fluid: floats.

Archimedes' Principle

• Statement: When an object is partially or fully immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced by the object.$$\text{<span style="background-color: white;">Buoyant Force</span>}<span\; style="background-color:\; white;">=</span>\text{<span style="background-color: white;">Weight of fluid displaced</span>}$$
• Applications:
  • Submarines: Control buoyancy by changing the amount of water in their tanks.
  • Hot air balloons: Heated air is less dense, causing the balloon to rise.
  • Hydro meters and Lactometers: Used to measure the density of liquids.
  • Ships: Designed to displace a large volume of water, creating a large buoyant force.