Overview
This lecture covers the CBSE Class 9 Physics chapter "Gravitation." It explains the universal law of gravitation, forces in circular motion, gravity, mass, weight, acceleration due to gravity, free fall, pressure, thrust, fluids, Archimedes' principle, buoyancy, and applications of flotation. The lecture also includes solved examples and important numericals.
Universal Law of Gravitation
- Every object in the universe attracts every other object with a force.
- The force is:
- Directly proportional to the product of their masses.
- Inversely proportional to the square of the distance between their centers.
- Formula:
[
F = G \frac{M_1 M_2}{r^2}
]
where:
- ( F ) = gravitational force
- ( G ) = universal gravitational constant (( 6.67 \times 10^{-11} ) N·m²/kg²)
- ( M_1, M_2 ) = masses of the two objects
- ( r ) = distance between the centers of the two objects
- The force is always attractive and acts along the line joining the centers.
- This law is universal: it applies to all objects, regardless of size or location (planets, satellites, subatomic particles).
Example Numerical:
Find the gravitational force between Earth and Moon.
Given:
- Mass of Earth (( M_1 )) = ( 6 \times 10^{24} ) kg
- Mass of Moon (( M_2 )) = ( 7.4 \times 10^{22} ) kg
- Distance (( r )) = ( 3.84 \times 10^8 ) m
- ( G = 6.7 \times 10^{-11} ) N·m²/kg²
[
F = \frac{6.7 \times 10^{-11} \times 6 \times 10^{24} \times 7.4 \times 10^{22}}{(3.84 \times 10^8)^2}
]
[
F \approx 2.01 \times 10^{20} \text{ N}
]
Forces in Circular Motion
- Centripetal Force: The force directed toward the center that keeps an object moving in a circular path.
- Formula:
[
F_c = \frac{mv^2}{r}
]
where ( m ) = mass, ( v ) = speed, ( r ) = radius.
- Centrifugal Force: Apparent force acting outward, away from the center, in a rotating frame; equal in magnitude to centripetal force but opposite in direction.
- Examples:
- The Moon revolves around the Earth due to Earth's centripetal force.
- A stone tied to a string and whirled in a circle moves tangentially if the string breaks.
Gravity, Mass, and Weight
- Gravitation: The attractive force between any two objects with mass.
- Gravity: The force of attraction between Earth and any object.
- Mass:
- Amount of matter in a body.
- Scalar quantity; measured in kilograms (kg).
- Constant everywhere (does not change with location).
- Weight:
- The force with which gravity attracts a body.
- Formula:
[
W = mg
]
where ( m ) = mass, ( g ) = acceleration due to gravity.
- Vector quantity; measured in newtons (N).
- Changes with the value of ( g ) at different locations.
- Weight on the Moon:
- ( g_{\text{moon}} = \frac{1}{6}g_{\text{earth}} )
- Weight on Moon = ( \frac{1}{6} ) of weight on Earth.
Example Numerical:
If an object weighs 10 N on Earth, what is its weight on the Moon?
[
W_{\text{moon}} = \frac{1}{6} \times 10 = 1.66 \text{ N}
]
Acceleration Due to Gravity (g)
- The acceleration produced in a freely falling body due to Earth's gravity.
- Standard value: ( g = 9.8 ) m/s² (use 10 m/s² for easy calculations).
- Formula:
[
g = G \frac{M}{r^2}
]
where ( M ) = mass of the planet, ( r ) = radius of the planet.
- ( g ) is independent of the mass of the falling body but depends on the planet's mass and radius.
Example Numerical:
Calculate the value of ( g ) on Earth using:
- ( G = 6.7 \times 10^{-11} ) N·m²/kg²
- ( M = 6 \times 10^{24} ) kg
- ( r = 6.4 \times 10^6 ) m
[
g = \frac{6.7 \times 10^{-11} \times 6 \times 10^{24}}{(6.4 \times 10^6)^2} \approx 9.8 \text{ m/s}^2
]
Free Fall and Equations of Motion
- Free Fall: Motion of a body only under the influence of gravity (no other forces).
- Initial velocity (( u )) = 0 (if simply dropped).
- Equations for Free Fall:
- ( v = gt )
- ( h = \frac{1}{2}gt^2 )
- ( v^2 = 2gh )
- For upward motion, ( g ) is taken as negative (( a = -g )).
Example Numerical:
A car falls off a ledge and drops to the ground in 0.5 s. Find:
- (a) Speed on striking the ground (( v ))
- (b) Average speed
- (c) Height of the ledge (( h )), using ( g = 10 ) m/s²
(a) ( v = gt = 10 \times 0.5 = 5 ) m/s
(b) Average speed = ( \frac{0 + 5}{2} = 2.5 ) m/s
(c) ( h = \frac{1}{2}gt^2 = \frac{1}{2} \times 10 \times (0.5)^2 = 1.25 ) m
Example Numerical (Upward Throw):
An object is thrown vertically up and rises to 10 m. Find initial velocity and time to reach the highest point (( g = 9.8 ) m/s²).
- At top, ( v = 0 ), ( h = 10 ) m
- ( v^2 = u^2 - 2gh \Rightarrow 0 = u^2 - 2 \times 9.8 \times 10 )
- ( u^2 = 196 \Rightarrow u = 14 ) m/s
- ( v = u - gt \Rightarrow 0 = 14 - 9.8t \Rightarrow t = 1.43 ) s
Pressure, Thrust, and Fluids
- Thrust: Force acting perpendicularly (normally) to a surface.
- Pressure: Thrust per unit area.
- Formula:
[
P = \frac{F}{A}
]
where ( F ) = thrust (N), ( A ) = area (m²).
- SI unit: pascal (Pa), ( 1 \text{ Pa} = 1 \text{ N/m}^2 ).
- Solids: Exert pressure at points of contact.
- Fluids (liquids and gases): Exert pressure in all directions.
Example Numerical:
A wooden block of mass 5 kg and dimensions 40 cm Ă— 20 cm Ă— 10 cm is placed on a table. Find the pressure exerted if it rests on:
- (a) 20 cm Ă— 10 cm face
- (b) 40 cm Ă— 20 cm face
First, weight ( W = mg = 5 \times 9.8 = 49 ) N
(a) Area = ( 20 \times 10 = 200 ) cm² = ( 0.02 ) m²
Pressure = ( \frac{49}{0.02} = 2450 ) Pa
(b) Area = ( 40 \times 20 = 800 ) cm² = ( 0.08 ) m²
Pressure = ( \frac{49}{0.08} = 612.5 ) Pa
Archimedes' Principle & Buoyancy
- Buoyant Force (Upthrust): Upward force exerted by a fluid on a submerged object.
- Formula:
[
F_B = \rho V g
]
where ( \rho ) = density of fluid, ( V ) = volume of displaced fluid, ( g ) = acceleration due to gravity.
- Archimedes' Principle: The buoyant force on a body immersed in a fluid is equal to the weight of the fluid displaced by the body.
- Conditions:
- If ( W > F_B ): Object sinks.
- If ( W = F_B ): Object is submerged (neither sinks nor floats).
- If ( W < F_B ): Object floats.
Example:
- A stone sinks in water because its weight is greater than the buoyant force.
- A wooden log floats because its weight is less than the buoyant force.
Applications of Flotation
- Iron ships: Float because the average density (including air inside) is less than water, so the buoyant force exceeds the ship's weight.
- Humans, icebergs, submarines, balloons: Float or submerge based on the balance between weight and buoyant force.
- Ice floats on water: Density of ice is less than water, so it floats.
Key Terms & Definitions
- Gravitation: Force of attraction between any two masses.
- Gravity: Attraction between Earth and another object.
- Mass: Measure of matter in an object; constant everywhere.
- Weight: Force on mass due to gravity; varies with location.
- Centripetal Force: Force toward the center in circular motion.
- Centrifugal Force: Outward force, apparent in rotating frames.
- Thrust: Perpendicular force on a surface.
- Pressure: Thrust per unit area; measured in pascals.
- Buoyant Force (Upthrust): Upward force by fluid on submerged body.
- Archimedes’ Principle: Upthrust equals weight of displaced liquid.
Action Items / Next Steps
- Review and highlight all key formulas and principles from this lecture.
- Practice solving NCERT and example numericals from the Gravitation chapter.
- Note any doubts and prepare questions for the next class.