Where:
$G = 6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2$G=6.674×10−11Nm2/kg2

Force acts along the line joining the centers of the masses — a central force.

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🔹 Gravitational Potential Energy

Work done to bring a mass $m$m from infinity to a distance $r$r:$$U(r) = -\frac{GMm}{r}$$U(r)=−rGMm​

Negative sign → Gravity is an attractive force.

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🔹 Inertial & Gravitational Mass

Mass Type	Meaning	Experiment
Inertial Mass	Resists acceleration	Newton’s 2nd Law
Gravitational Mass	Measures gravitational force	Cavendish type
Both are proven to be equivalent → Principle of Equivalence (Einstein).

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🔸 Gravitational Field and Potential of Spherical Objects

👉 Spherical Shell

• Inside shell: $$g = 0$$g=0 Field is zero → hollow cavity has no gravitational effect.
• Outside shell:
  Acts like a point mass at center: $$g = \frac{GM}{r^2}$$g=r2GM​

👉 Solid Sphere

• Outside: Point mass behaviour $$g = \frac{GM}{r^2}$$g=r2GM​
• Inside: Proportional to radius $$g = \frac{GMr}{R^3}$$g=R3GMr​

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🚀 Central Force Motion

A force which always acts along the radius vector and depends only on distance:$$\vec{F}(r) = F(r)\hat{r}$$F(r)=F(r)r^

Examples: Gravitational force, Electrostatic force.

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🔸 Two-Body Problem → One-Body Reduction

Two masses $m_1, m_2$m1​,m2​ interacting through central force are reduced to:

• Motion of center of mass
• Relative motion of reduced mass

$$\mu = \frac{m_1 m_2}{m_1 + m_2}$$μ=m1​+m2​m1​m2​​

The problem becomes a single particle of mass $μ$μ moving under a central potential.

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✳ Differential Equation of Motion

$$\mu \ddot{\vec{r}} = \vec{F}(r)$$μr¨=F(r)

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🌟 First Integrals of Motion

1️⃣ Angular Momentum Conservation$$\vec{L} = \mu r^2 \dot{\theta} = \text{constant}$$L=μr2θ˙=constant

2️⃣ Energy Conservation$$E = \frac{1}{2}\mu\dot{r}^2 + \frac{L^2}{2\mu r^2} + U(r) = \text{constant}$$E=21​μr˙2+2μr2L2​+U(r)=constant

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⚡ Power Law Potentials

$$U(r) \propto r^n$$U(r)∝rn

Examples:

• Inverse square law → $U(r) \propto -1/r$U(r)∝−1/r
• Harmonic oscillator → $U(r) \propto r^2$U(r)∝r2

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🌞 Kepler’s Laws of Planetary Motion

Derived from Newton’s gravitation:

1️⃣ Law of Orbits
Planets move in elliptical orbits, Sun at one focus.

2️⃣ Law of Areas
Equal areas in equal times → conservation of angular momentum.

3️⃣ Law of Periods$$T^2 \propto a^3$$T2∝a3

Where $T$T = time period and $a$a = semi-major axis.

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🛰 Satellite Motion

Orbital Velocity:

$$v_0 = \sqrt{\frac{GM}{R+h}}$$v0​=R+hGM​​

Escape Velocity:

$$v_e = \sqrt{\frac{2GM}{R}}$$ve​=R2GM​​

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⭐ Geosynchronous Orbit

• Period = 24 hours
• Always above same point on Earth
• Used for satellite TV & communication

Altitude approx: 36,000 km

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⭐ Weightlessness in Orbit

Astronauts are in free fall around Earth → no normal reaction force.
Hence, they feel weightless.

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🌐 Global Positioning System (GPS)

• Network of 24 satellites
• Provides accurate position using signal triangulation
• Used in: Navigation, defense, mobile tracking

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👨‍🚀 Physiological Effects on Astronauts

Effect	Reason
Muscle weakening	No muscle load in microgravity
Bone density loss	Calcium release due to low stress
Fluid shift to head	Lack of gravity pull
Orientation problems	No up-down reference

Solution: Exercise, nutritional support, artificial gravity research.

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🔍 Quick Summary

Topic	Key Formula / Idea
Gravitational Force	$F = G \frac{m_1 m_2}{r^2}$F=Gr2m1​m2​​
Potential Energy	$U=-\frac{GMm}{r}$U=−rGMm​
Escape Velocity	$v_e = \sqrt{\frac{2GM}{R}}$ve​=R2GM​​
Kepler’s 3rd Law	$T^2 ∝ a^3$T2∝a3
Two-body → one-body	Reduced mass $\mu$μ