Bihar Board - Class 11 physics - Chapter 8: Gravitation Handwritten Notes

The chapter "Gravitation" explores the universal force of attraction between any two objects with mass, as described by Newton's law of gravitation. This chapter covers the concepts of gravitational force, acceleration due to gravity, planetary motion, and gravitational potential energy. It also delves into the application of gravitation in celestial mechanics, such as satellite motion and Kepler's laws.

Key Points

1. Newton's Law of Gravitation

   • Every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them: $$F = G \frac{m_1 m_2}{r^2}$$where $G$ is the universal gravitational constant ($6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2$).

2. Gravitational Field and Gravitational Force

   • The gravitational field at a point is the gravitational force experienced by a unit mass placed at that point: $$g = \frac{F}{m} = \frac{GM}{r^2}$$ where $g$ is the acceleration due to gravity, $M$ is the mass of the object, and $r$ is the distance from the center of the mass.

3. Acceleration Due to Gravity (g)

   • Near the surface of the Earth, $g$ is approximately $9.8 \, \text{m/s}^2$. Its value decreases with altitude, depth, and latitude.

4. Variation of g with Height and Depth

   • Height: $$g_h = g \left( \frac{R}{R + h} \right)^2$$ where $h$ is the height above the Earth's surface, and $R$ is the Earth's radius.
   • Depth: $$g_d = g \left( 1 - \frac{d}{R} \right)$$where $d$ is the depth below the Earth's surface.

5. Gravitational Potential Energy

   • The gravitational potential energy of a system of masses is given by: $$U = -\frac{G M m}{r}$$where $U$ is negative because the gravitational force is attractive.

6. Escape Velocity

   • The minimum velocity required to escape the gravitational pull of a celestial body: $$v_{\text{escape}} = \sqrt{\frac{2GM}{R}}$$ For Earth, the escape velocity is approximately $11.2 \, \text{km/s}$.

7. Kepler's Laws of Planetary Motion

   • First Law (Law of Ellipses): Planets move in elliptical orbits with the Sun at one of the foci.
   • Second Law (Law of Equal Areas): A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.
   • Third Law (Law of Periods): The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit: $$T^2 \propto r^3$$
     

8. Satellites and Orbital Motion

   • Orbital Velocity: The velocity required for a satellite to stay in a stable orbit around a planet: $$v_{\text{orbital}} = \sqrt{\frac{GM}{R + h}}$$
   • Geostationary Satellites: These satellites have an orbital period equal to Earth's rotational period and appear stationary relative to Earth.

9. Weightlessness in Orbit

   • Astronauts experience weightlessness in orbit because they are in a state of free fall, with the gravitational force acting as the centripetal force.

Conclusion

The chapter "Gravitation" provides a detailed understanding of the universal force that governs the motion of celestial bodies and objects on Earth. By exploring concepts like gravitational potential, satellite motion, and Kepler's laws, students gain a foundational understanding of both terrestrial and cosmic phenomena.

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