Why does the Moon never crash into the Earth despite being pulled by a colossal gravitational force, and what minimum speed must a spacecraft reach to permanently break free from Earth's gravity well? Newton's Universal Law and Kepler's planetary laws govern the cosmos.
यह अध्याय क्यों महत्वपूर्ण है
In Class 11 Physics, "Gravitation" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.
अध्ययन से पूर्व (आवश्यक ज्ञान)
Gravitational force from Class 9.
Circular motion and centripetal acceleration.
Potential energy.
इस अध्याय के लक्ष्य
State Kepler's Three Laws of Planetary Motion (Ellipses, Equal Areas, Harmonic Law $T^2 \propto a^3$).
State Newton's Universal Law of Gravitation: $F = G \frac{m_1 m_2}{r^2}$.
Analyze acceleration due to gravity ($g = \frac{GM}{R^2}$) and its variation with Altitude ($h$) and Depth ($d$).
Derive Gravitational Potential ($V = -\frac{GM}{r}$) and Potential Energy ($U = -\frac{GMm}{r}$).
Derive Escape Velocity ($v_e = \sqrt{2gR} \approx 11.2\text{ km/s}$) and Orbital Velocity ($v_o = \sqrt{gR}$) of satellites.
अध्याय रूपरेखा एवं प्रगति
11. Kepler's Laws & Newton's Gravita...
22. Variation of $g$ with Height & D...
33. Escape Velocity & Satellites
सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन
1. Kepler's Laws & Newton's Gravitation
Kepler's 1st Law (Orbits): Planets move in ellipses with the Sun at one focus.
Kepler's 2nd Law (Areas): Radius vector sweeps equal areas in equal intervals ($\frac{dA}{dt} = \frac{L}{2m} = \text{constant}$; consequence of Conservation of Angular Momentum!).
Kepler's 3rd Law (Periods): $$\mathbf{T^2 \propto a^3}$$
उत्तर: Because the acceleration due to gravity on the Moon is only $1/6$th of Earth's, giving a low escape velocity ($2.38\text{ km/s}$). The root-mean-square thermal velocities of gas molecules at lunar daytime temperatures exceed this escape speed, so gas molecules escaped into space. Gas thermal speed exceeds low lunar escape velocity.
3
At what height above the surface of the Earth will the acceleration due to gravity be reduced to $g/4$?
उत्तर एवं व्याख्या देखें
उत्तर: $g_h = \frac{GM}{(R+h)^2} = \frac{g}{(1 + h/R)^2}$. Set $g_h = g/4 \implies (1 + h/R)^2 = 4 \implies 1 + h/R = 2 \implies h = R$. At a height equal to Earth's radius ($6400\text{ km}$), gravity becomes $g/4$. h = R (6400 km).
4
Explain why Kepler's second law is a consequence of conservation of angular momentum.
उत्तर एवं व्याख्या देखें
उत्तर: Gravitational force between the Sun and a planet is a central force directed along the line joining them, producing zero torque ($\vec{\tau} = \vec{r} \times \vec{F} = 0$). Since torque is zero, angular momentum is constant ($L = 2m \frac{dA}{dt} = \text{constant}$), meaning areal velocity $dA/dt$ is constant. Central gravity produces zero torque, conserving angular momentum.
5
What is the weight of a body at the center of the Earth?
उत्तर एवं व्याख्या देखें
उत्तर: At the center of the Earth, depth $d = R$. $g_d = g(1 - d/R) = g(1 - 1) = 0$. Therefore, weight $W = mg = 0\text{ N}$ (weightlessness). Zero (weightless).
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