The Cosmic Glue: Mastering the Laws of Gravitation

Gravitation is the weakest of the four fundamental forces, yet it’s the one that rules the universe. It’s the reason the Earth stays in orbit, the reason stars are born, and the reason time itself slows down near a Black Hole.

In this chapter, we move beyond “g = 9.8” and explore how mass warps the space around it. We treat the Earth not as a flat floor, but as a massive sphere in a vacuum. To master Gravitation, you must understand that every mass in the universe is “talking” to every other mass.

The Core Pillars of Gravitation

1. The Inverse Square Law

Newton’s Universal Law of Gravitation tells us that the force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them.

• F = G (m₁m₂) / r²If you double the distance, the gravity doesn’t just halve; it drops to one-fourth.

2. The Variation of ‘g’

The acceleration due to gravity (g) isn’t a constant. It changes based on:

• Altitude: It decreases as you go up.
• Depth: It decreases as you go down (at the center of the Earth, you are weightless!).
• Rotation: The Earth’s spin makes you slightly lighter at the equator than at the poles.

3. Gravitational Potential Energy (V)

Unlike mgh (which only works near the surface), the universal formula for potential energy is U = -GMm / r. The negative sign is crucial—it signifies that the object is “bound” to the planet. You have to “pay” energy to escape to infinity.

4. Kepler’s Laws: The Planetary Dance

1. Orbits are Ellipses: The Sun is at one focus.
2. Equal Areas in Equal Time: Planets move faster when closer to the Sun.
3. The Harmonic Law: The square of the orbital period is proportional to the cube of the mean distance (T² ∝ R³).

The Gauntlet: 10 Challenging Aptitude Questions

Question 1: The Tunnel Through Earth

A straight tunnel is dug through the center of the Earth. A ball is dropped into the tunnel. Describe the motion of the ball and calculate its time period. (Assume Earth is a uniform sphere of constant density).

Question 2: The Energy of Orbit Shift

A satellite is orbiting Earth in a circular path of radius R. How much additional energy is required to move the satellite to a higher circular orbit of radius 2R?

Question 3: The Weightless Equator

At what angular velocity would the Earth need to rotate so that a person standing at the equator feels weightless? What would be the length of a “day” in this scenario?

Question 4: Binary Star Dynamics

Two stars of masses m₁ and m₂ form a binary system. They rotate about their common center of mass under their mutual gravitational attraction. Find the orbital period T of the system if the distance between them is L.

Question 5: Escape Velocity from a Moving Planet

The escape velocity from Earth is vₑ. If a planet has the same density as Earth but its radius is double that of Earth, what would be the escape velocity from that planet?

Question 6: The Neutral Point

The Moon is at a distance D from the Earth. The mass of Earth is approximately 81 times the mass of the Moon. At what distance from the Earth’s center is the net gravitational force on a spaceship zero?

Question 7: Kepler’s Area Constant

A planet moves around the Sun in an elliptical orbit. If the ratio of its maximum distance (aphelion) to its minimum distance (perihelion) from the Sun is k, what is the ratio of its maximum speed to its minimum speed?

Question 8: The Falling Meteorite

A meteorite is at rest at an infinite distance from Earth. It begins to fall toward Earth due to gravity. With what velocity will it strike the Earth’s surface? (Ignore air resistance).

Question 9: Gravitational Field vs. Potential

The gravitational potential in a region is given by V = (5x + 12y) J/kg. Find the magnitude of the gravitational field intensity at the origin (0,0).

Question 10: Geostationary Height

Why must a geostationary satellite be placed at a specific height (~36,000 km)? What would happen if we tried to place a geostationary satellite directly over the North Pole?

Detailed Explanations & Solutions

1. The Tunnel Through Earth

Gravity inside the Earth varies linearly with distance: F = -(GmM/R³)r. This is the condition for Simple Harmonic Motion.

Result: The ball will oscillate with a period T = 2π√(R/g), which is approximately 84.6 minutes.

2. Orbit Shift Energy

Total Energy in orbit E = -GMm / 2r.

Initial Energy E₁ = -GMm / 2R. Final Energy E₂ = -GMm / 4R.

Energy required = E₂ – E₁ = GMm / 4R.

3. Weightless Equator

For weightlessness, centripetal acceleration must equal gravity: g = ω²R.

ω = √(g/R).

Result: The day would be about 1.4 hours long.

4. Binary Star System

Force F = Gm₁m₂ / L². This force provides centripetal acceleration for both.

For mass m₁: Gm₁m₂ / L² = m₁ω²r₁ (where r₁ is distance to COM).

Using r₁ = m₂L / (m₁+m₂).

Result: T = 2π √[L³ / G(m₁+m₂)].

5. Scaling Escape Velocity

vₑ = √(2GM/R). Since mass M ∝ ρR³, then vₑ ∝ R√(ρ).

If density is constant and R doubles, vₑ doubles.

Result: 2vₑ.

6. The Neutral Point

Let distance from Earth be x.

G(81M) / x² = GM / (D-x)².

Taking the square root: 9 / x = 1 / (D-x) → 9D – 9x = x → 10x = 9D.

Result: x = 0.9 D.

7. Kepler’s Speed Ratio

By conservation of angular momentum: m v_max r_min = m v_min r_max.

v_max / v_min = r_max / r_min.

Result: k.

8. The Falling Meteorite

Cons. of Energy: (PE + KE)_infinity = (PE + KE)_surface.

0 + 0 = -GMm/R + ½mv².

v² = 2GM/R.

Result: v = Escape Velocity (approx. 11.2 km/s).

9. Field Intensity

E_x = -dV/dx = -5. E_y = -dV/dy = -12.

Magnitude E = √(5² + 12²) = 13 N/kg.

10. Polar Geostationary?

A geostationary satellite must have an orbital period of 24 hours AND its orbital plane must contain the center of the Earth. A satellite over the North Pole would require a constant force pushing it “sideways” to stay there, which gravity cannot provide.

Result: It is physically impossible; geostationary satellites must be in the Equatorial plane.

Pro-Tip: The “Negative” Sign

Always remember that Gravitational Potential Energy is negative. If an object’s total energy (KE + PE) is:

• Negative: It is trapped in an orbit (Bound).
• Zero: It is just barely escaping (Parabolic path).
• Positive: It is zooming away (Hyperbolic path).