Class 11 Physics Gravitation Notes

Universal Law of Gravitation

Newton’s Law of Gravitation:

Every particle of matter attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

$$F = G \frac{m_1 m_2}{r^2}$$F=Gr2m1​m2​​

Where:

• $F$F = gravitational force
• $G = 6.674 \times 10^{-11} \, \text{N·m²/kg²}$G=6.674×10−11N\cdotpm²/kg² = universal gravitational constant
• $m_1, m_2$m1​,m2​ = masses of the objects
• $r$r = distance between the masses

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🔹 Acceleration Due to Gravity (g)

• Near Earth, every object experiences acceleration $g$g due to gravity:

$$g = \frac{GM}{R^2}$$g=R2GM​

Where:

• $M$M = mass of Earth
• $R$R = radius of Earth
• Weight of a body:

$$W = mg$$W=mg

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🔹 Variation of g with Height and Depth

1. At height h above Earth’s surface:

$$g_h = g \left(1 – \frac{2h}{R}\right)$$gh​=g(1−R2h​)

2. At depth d below Earth’s surface:

$$g_d = g \left(1 – \frac{d}{R}\right)$$gd​=g(1−Rd​)

• $R$R = radius of Earth

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🔹 Gravitational Potential Energy (U)

$$U = – \frac{G m_1 m_2}{r}$$U=−rGm1​m2​​

• Negative because work is done against gravitational attraction.
• Near Earth surface: $U = mgh$U=mgh

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🔹 Escape Velocity

Definition:
Minimum velocity required for an object to escape Earth’s gravity without further propulsion.$$v_{\text{esc}} = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$$vesc​=R2GM​​=2gR​

• Independent of mass of object
• Example: rockets, satellites

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🔹 Orbital Velocity

Definition:
Velocity required for a satellite to stay in a circular orbit:$$v_{\text{orb}} = \sqrt{\frac{GM}{r}}$$vorb​=rGM​​

Where $r$r is distance from Earth’s center.

• For low Earth orbit (LEO) $r \approx R + h$r≈R+h

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🔹 Kepler’s Laws of Planetary Motion

1. First Law (Law of Orbits):
   Every planet moves in an elliptical orbit with Sun at one focus.
2. Second Law (Law of Areas):
   Line joining planet and Sun sweeps equal areas in equal times.
3. Third Law (Harmonic Law):

$$\frac{T^2}{r^3} = \text{constant}$$r3T2​=constant

• $T$T = time period of revolution
• $r$r = mean distance from Sun

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🔹 Gravitational Field and Intensity

• Gravitational field (g-field):

$$\vec{g} = \frac{\vec{F}}{m} = \frac{GM}{r^2} \hat{r}$$g​=mF​=r2GM​r^

• Gravitational potential (V):

$$V = – \frac{GM}{r}$$V=−rGM​

• Relationship: $\vec{g} = – \nabla V$g​=−∇V

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🔹 Important Formulas

Quantity	Formula
Gravitational Force	$F = G \frac{m_1 m_2}{r^2}$F=Gr2m1​m2​​
Acceleration due to gravity	$g = \frac{GM}{R^2}$g=R2GM​
Weight	$W = mg$W=mg
Escape Velocity	$v_{\text{esc}} = \sqrt{2gR}$vesc​=2gR​
Orbital Velocity	$v_{\text{orb}} = \sqrt{\frac{GM}{r}}$vorb​=rGM​​
Gravitational Potential Energy	$U = -\frac{G m_1 m_2}{r}$U=−rGm1​m2​​
Kepler’s Third Law	$\frac{T^2}{r^3} = \text{constant}$r3T2​=constant