Chapter 8: Gravitation
Class 11 Physics | NCERT
Newton's Law of Gravitation
\( F = G \frac{m_1 m_2}{r^2} \)
What it means: The force of gravity between two objects depends on their masses (\( m_1 \), \( m_2 \)) and the distance (\( r \)) between their centers. \( G \) is the gravitational constant (\( 6.67 \times 10^{-11} \, \text{N·m}^2/\text{kg}^2 \)).
Example: Imagine Earth and the Moon. The force pulling them together keeps the Moon orbiting Earth, not flying off into space!
Tip: Bigger masses or closer objects mean stronger gravity. That's why you feel heavier on Earth than you would on the Moon!
Gravity Variations
\( g = \frac{G M}{R^2} \)
Acceleration Due to Gravity: The acceleration due to gravity (\( g \approx 9.8 \, \text{m/s}^2 \)) depends on Earth's mass (\( M \)) and radius (\( R \)).
Example: Drop a pencil. It falls because Earth's gravity pulls it down at \( 9.8 \, \text{m/s}^2 \). On the Moon, it'd fall slower because \( g \) is smaller there!
\( g_h = g \left( 1 - \frac{2h}{R} \right) \)
With Height: Gravity (\( g_h \)) decreases as you go higher above Earth's surface (\( h \) is height, \( R \) is Earth's radius).
For small heights, gravity doesn't change much, but it's why astronauts feel lighter in space!
\( g_d = g \left( 1 - \frac{d}{R} \right) \)
With Depth: Gravity (\( g_d \)) decreases as you go deeper below Earth's surface (\( d \) is depth).
Example: If you're in a deep mine, gravity is a bit weaker because some of Earth's mass is above you. At Earth's core (\( d = R \)), \( g = 0 \)!
Orbital Motion
\( v_e = \sqrt{\frac{2 G M}{R}} = \sqrt{2 g R} \)
Escape Velocity: The speed needed to escape a planet's gravity (e.g., \( 11.2 \, \text{km/s} \) for Earth).
Example: Rockets need to hit escape velocity to leave Earth's atmosphere and explore space, like the Apollo missions to the Moon!
\( v_o = \sqrt{\frac{G M}{r}} \)
Orbital Velocity: The speed a satellite needs to stay in a circular orbit at distance \( r \) from the planet's center.
Closer orbits need faster speeds—low-orbit satellites zip around faster!
Kepler's Laws
\( T^2 \propto r^3 \)
3rd Law (Period Law): The square of the orbital period is proportional to the cube of the orbit's radius.
Example: Mars takes longer to orbit the Sun than Earth because it's farther away.
Bigger orbits, much longer trips!