Gravitation – Newton’s Law, Orbits & Escape Velocity | Entrance Exam Physics Notes

GRAVITATION — SHORT NOTES (Entrance Exam Focus)

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1. Newton’s Law of Gravitation

Every mass attracts every other mass with a force:

$$F=G\frac{m_1{m}_2}{r^{\mathrm{2}}}$$

• $G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2$
  

• Force acts along the line joining masses.

• Universal constant → same everywhere.

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2. Gravitational Field (g)

Field strength at a point:

$$g=\frac{F}{m}=G\frac{M}{r^{\mathrm{2}}}$$

Vector quantity → directed toward the mass.

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3. Variation of g

(a) With Height

$$g_h = g\left(1 - \frac{2h}{R}\right) \quad (h \ll R)$$

(b) With Depth

$$g_d = g\left(1 - \frac{d}{R}\right)$$

(c) With Latitude

$$g=g_0(1+\frac{{\omega }^{2}R}{g_0}{\cos }^2\varphi )$$

Due to Earth’s rotation → minimum at equator, maximum at poles.

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4. Gravitational Potential (V)

$$V=-\frac{GM}{r}$$

Scalar quantity → always negative.

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5. Gravitational Potential Energy (U)

$$U=-\frac{GMm}{r}$$

Max (least negative) at infinity → 0.

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6. Escape Velocity

Minimum speed needed to escape Earth’s gravitational field.

$$v_e=\sqrt{2gR}=\sqrt{\frac{2GM}{R}}$$

On Earth:

$$v_e\approx 11.2\text{km/s}$$

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7. Orbital Velocity

Velocity for a satellite in circular orbit:

$$v_o=\sqrt{\frac{GM}{r}}$$

For low Earth orbit (LEO) ≈ 7.9 km/s

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8. Orbital Time Period

$$T=2\pi \sqrt{\frac{r^3}{GM}}$$

Important: Independent of satellite’s mass.

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9. Kepler’s Laws

1. Law of Orbits

Planets move in elliptical orbits with Sun at one focus.

2. Law of Areas

$$\frac{dA}{dt}=\text{constant}$$

Planet sweeps equal areas in equal times → conservation of angular momentum.

3. Law of Periods

$$T^2\propto a^3$$

a = semi-major axis.

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10. Satellite Energy

Total mechanical energy

$$E=-\frac{GMm}{2\mathrm{r}}$$

Kinetic energy

$$K=\frac{GMm}{2\mathrm{r}}$$

Potential energy

$$U=-\frac{GMm}{r}$$

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11. Geostationary Satellite

• Period: 24 hours

• Orbits above equator

• Height ≈ 36,000 km

• Same angular velocity as Earth.

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12. Weightlessness

Occurs when acceleration = g (free fall):

• Astronauts in ISS

• Lift falling freely

• Satellite orbiting Earth

Apparent weight = 0.

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13. Most Expected Numericals

1. Escape velocity from Earth/Moon

2. Orbital speed & orbital period

3. Variation of g with height/depth

4. Energy of satellite (U, K, total energy)

5. Kepler’s third law ratio calculations

6. Comparing gravitational forces between bodies