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.

🔹 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.

🔹 Inertial & Gravitational Mass

🔸 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​

🚀 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.

🔸 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.

✳ Differential Equation of Motion

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

🌟 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

⚡ 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

🌞 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.

🛰 Satellite Motion

Orbital Velocity:

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

Escape Velocity:

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

⭐ Geosynchronous Orbit

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

Altitude approx: 36,000 km

⭐ Weightlessness in Orbit

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

🌐 Global Positioning System (GPS)

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

👨‍🚀 Physiological Effects on Astronauts

Solution: Exercise, nutritional support, artificial gravity research.

🔍 Quick Summary