Gravitation

Definition

Every particle in the universe 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.

Explanation

Think of it like this: The heavier the objects, the stronger the "pull." But, the further apart they are, the much weaker that pull becomes. It's called "Universal" because this rule applies to an apple, a human, a planet, or a distant galaxy.

• It is a central force (acts along the line joining the centers).
• It follows the inverse-square law (if you double the distance, the force becomes \(1/4\)th).
• It is always attractive (never repulsive).

The Formula

\[F = G \frac{m_1 m_2}{r^2}\]

Where:

• \(F\) = Gravitational Force
• \(G\) = Universal Gravitational Constant
• \(m_1, m_2\) = Masses of the two objects
• \(r\) = Distance between the centers of the masses

Examples

• The Earth and the Moon: The Earth's massive gravity keeps the Moon in a constant loop around it.
• Two Pens on a Table: Technically, the two pens are pulling each other via gravity, but because their masses are so tiny, the force is too small to notice.

Definition

• Gravitational Constant (G): A tiny, fixed number that tells us how strong the force of gravity is in our universe (\(6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2\)).
• Acceleration due to Gravity (g): The acceleration gained by an object due to the Earth's gravitational pull (approximately \(9.8 \, \text{m/s}^2\)).

Key Points

• \(G\) is a scalar quantity and is universal.
• \(g\) is a vector quantity and varies by location.
• Relation: \(g = \frac{GM}{R^2}\) (where \(M\) is Earth's mass and \(R\) is Earth's radius).

Comparison Table: G vs g

Variation in g

1. Altitude (Height): As you go higher, \(g\) decreases because the distance from Earth's center increases.
2. Depth: As you go deeper into a mine, \(g\) decreases. At the center of the Earth, \(g = 0\).
3. Shape of Earth: Earth is an oblate spheroid. Therefore, \(g\) is maximum at the poles and minimum at the equator.

Definition

The energy possessed by an object due to its position in a gravitational field.

Formula

\[U = -\frac{GMm}{r}\]

Note: The negative sign indicates that the gravitational force is attractive; you have to "add" energy to pull the object away to infinity.

Examples

• A Rock on a Cliff: A rock sitting at the edge of a cliff has high GPE.
• A Satellite: A satellite held in orbit has a specific amount of potential energy relative to Earth.

The Three Laws

1. Law of Orbits: All planets move in elliptical orbits, with the Sun at one of the two foci.
2. Law of Areas: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3. Law of Periods: The square of the orbital period (\(T\)) is directly proportional to the cube of the semi-major axis (\(r\)).
   Formula: \[T^2 \propto r^3\]

Orbital Velocity (\(v_o\))

Velocity required for a satellite to maintain a circular orbit around a planet.

\[v_o = \sqrt{\frac{GM}{r}}\]

Escape Velocity (\(v_e\))

The minimum velocity an object must have to break free from a planet's gravitational pull entirely.

\[v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2} \times v_o\]

An orbiting satellite possesses both Kinetic Energy (\(K\)) and Potential Energy (\(U\)):

• Kinetic Energy (\(K\)): \(\frac{GMm}{2r}\)
• Potential Energy (\(U\)): \(-\frac{GMm}{r}\)
• Total Energy (\(E\)): \(K + U = -\frac{GMm}{2r}\)

Important: The total energy is negative, meaning the satellite is "bound" to the planet.

Problem 1: Force Calculation

Calculate the gravitational force between two objects of mass \(50 \, \text{kg}\) and \(100 \, \text{kg}\) placed \(2 \, \text{m}\) apart. (\(G = 6.67 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2\))

Solution:

1. Given: m1=50kg, m2=100kg, r=2m, G=6.67x10^-11
2. Formula: F = G * (m1 * m2) / r^2
3. Calculation:
   F = (6.67 x 10^-11) * (50 * 100) / (2^2)
   F = (6.67 x 10^-11) * (5000 / 4)
   F = (6.67 x 10^-11) * 1250
   F = 8.3375 x 10^-8 N
Answer: 8.3375 x 10^-8 N
        

Problem 2: Escape Velocity

If the escape velocity from Earth is \(11.2 \, \text{km/s}\), what is the orbital velocity of a satellite orbiting very close to the Earth's surface?

Solution:

1. Given: ve = 11.2 km/s
2. Formula: ve = sqrt(2) * vo  =>  vo = ve / sqrt(2)
3. Calculation:
   vo = 11.2 / 1.414
   vo ≈ 7.92 km/s
Answer: 7.92 km/s
        

1. The value of G is:
   a) Zero, b) Variable, c) Universal constant, d) Depends on mass.
   Ans: c
2. If the distance between two masses is doubled, the gravitational force becomes:
   a) Double, b) Half, c) One-fourth, d) Four times.
   Ans: c
3. Where is the value of 'g' maximum?
   a) Equator, b) Poles, c) Center of Earth, d) Mount Everest.
   Ans: b
4. Kepler's Second Law is a consequence of the conservation of:
   a) Linear momentum, b) Angular momentum, c) Energy, d) Mass.
   Ans: b
5. The relationship between ve and vo is:
   a) ve = vo, b) ve = 2vo, c) ve = sqrt(2)vo, d) ve = vo / sqrt(2).
   Ans: c

1. State Newton's Universal Law of Gravitation and write its mathematical form.
   Ans: State the law (attraction between masses) and write \(F = G \frac{m_1 m_2}{r^2}\).
2. Distinguish between G and g.
   Ans: Use the comparison table (G is constant/scalar, g is variable/vector).
3. Explain why the value of g is less at the equator than at the poles.
   Ans: Earth is an oblate spheroid; the equatorial radius is larger. Since \(g \propto 1/R^2\), a larger \(R\) means a smaller \(g\).
4. What is Escape Velocity? Derive its expression.
   Ans: The minimum velocity to break free from gravity. Derive using \(\frac{1}{2}mv_e^2 - \frac{GMm}{R} = 0\).
5. Calculate the weight of a 10 kg object on the Moon if g_moon = 1.6 m/s².
   Ans: \(W = m \times g = 10 \times 1.6 = 16 \, \text{N}\).

• Newton's Law: \(F \propto m_1m_2\) and \(F \propto 1/r^2\).
• G vs g: G is Global (Universal), g is ground-level (Local).
• Kepler's Rule (E-E-S): Elliptical, Equal Area, Square-Cube Law (\(T^2 \propto r^3\)).
• Escape Velocity: \(\sqrt{2} \times\) Orbital Velocity.

Mnemonic for Kepler: "Every Earthling Studies" (Elliptical, Equal Area, Square-Cube).