Lesson1: Newton's Law of Gravitation: Gravitational Attraction Between Masses

1. Does Newton's Law of Gravitation apply to everything in the universe, or just big things like planets and stars?

It is universal. Every object with mass attracts every other object with mass. The pencil on your desk is gravitationally attracting you right now. However, because the constant G is incredibly small (6.67 × 10⁻¹¹), the force is negligible unless at least one of the objects is massive (like a planet).

2. In the formula F = Gm₁m₂/r², is 'r' the radius of the planet, the altitude, or the distance from the center?

This is a classic trap. 'r' is always the center-to-center distance.

• If an object is on the surface: r = Radius of Planet.
• If an object is at an altitude h: r = Radius of Planet + h.
• Never use just the altitude as r!

3. Is gravitational force always attractive, or is there any situation where it can be repulsive?

In classical Newtonian physics, gravity is always attractive. Mass is always positive, and gravity pulls masses together. While scientists theorize about "dark energy" causing cosmic expansion (acting like a repulsion), for your physics exams, gravity only pulls, never pushes.

4. If I have two planets with the same density but different sizes, does the larger one have higher surface gravity?

Yes. Here is the trick: Mass depends on Volume (Density × Volume), and Volume grows by r³. Since g = GM/r², if you substitute Mass with Density (ρ) and Volume (4/3 π r³), you get g is proportional to ρ × r. So, if density is constant, a planet with 2x the radius will have 2x the surface gravity.

5. How do I solve "Planet X" problems where the Mass is 2x Earth and the Radius is 3x Earth? Is there a shortcut?

Use the "Factor Method" (Ratio Method). Don't plug in the big numbers.g_new = g_earth * (Mass Factor) / (Radius Factor)²g_new = g_earth * (2) / (3)²g_new = g_earth * (2/9). The new gravity is roughly 0.22 times Earth's gravity.

6. Is there a distance where Earth's gravity actually ends (becomes zero), or does it mathematically go on forever?

Mathematically, it goes on forever because 1/r² never truly reaches zero; it just gets infinitely small (asymptotic). However, practically, at a certain distance, the gravitational pull of other bodies (like the Sun or Jupiter) becomes dominant, and Earth's influence becomes negligible.

Lesson 2:  Applying Superposition Principle to Gravitational Forces

7. How do I handle the vector math when I have to find the net gravitational force on a mass placed in a triangle of stars?

Treat gravity as a vector.

1. Draw the force vectors acting on your object (pulling it toward the other masses).
2. Decompose each diagonal force into x and y components using sine and cosine (e.g., F_x = F * cos(θ)).
3. Sum all x-components and all y-components separately.
4. Use Pythagoras (a² + b² = c²) to find the magnitude of the net force.

8. How do I calculate the exact point between the Earth and the Moon where the net gravitational force is zero?

At this point, the pull from Earth equals the pull from the Moon. Set F_earth = F_moon. G * M_earth * m / x² = G * M_moon * m / (D - x)² Cancel G and m. You are left with a ratio. Take the square root of both sides before solving for x (the distance from Earth) to make the algebra much easier.

Lesson 3: Why does Free Fall Acceleration Differ From Gravitational Acceleration?

9. What is the actual difference between Big 'G' and little 'g'? I keep mixing them up

Big G (Universal Gravitational Constant): A constant number everywhere in the universe (6.67 × 10⁻¹¹). It describes how strong gravity is as a fundamental force.

Little g (Acceleration due to Gravity): This changes depending on where you are. It is the acceleration an object feels on a specific planet. g = G * M / r².

10. When a question asks for "weight," is that just the force of gravity or something else?

True Weight: This is F_g = mg. It is the actual gravitational force on you.

Apparent Weight: This is what a bathroom scale reads (Normal Force). If you are accelerating up in an elevator, your apparent weight increases, but your true weight stays the same.

11. If I drop a heavy ball and a light ball, why do they hit the ground at the same time if the force of gravity is stronger on the heavy one?

The force is stronger on the heavy ball, but the heavy ball also has more inertia (resistance to moving). Newton’s 2nd Law: a = F / m. The heavy ball has more F (gravity), but also more m (mass). The ratio F/m remains constant (g). Nature cancels them out perfectly.

12. How does the Earth's rotation affect the measured value of 'g' at the equator versus the poles?

At the equator, you are spinning fast. Some of the gravitational pull is "used up" providing the centripetal force to keep you moving in a circle. This reduces the measured acceleration downward. g_measured = g_gravity - a_centripetal. At the poles, you aren't spinning, so g is higher.

Lesson 4: How Does Gravity Change Inside The Earth

13. Can you explain the integration steps to prove that the force inside a solid sphere decreases linearly (F is proportional to r)?

When you are inside the Earth at radius r, only the mass inside that radius pulls you down (Shell Theorem). The mass inside scales with volume (r³). The gravity equation divides by distance squared (r²). Force ~ Mass / r² ~ r³ / r² = r. Therefore, gravity decreases linearly to zero as you reach the center.

14. How do I graph Gravitational Field Strength vs. Distance from the center of a planet (including inside the core)?

• Inside (0 to R): A straight line going up from 0 to max surface gravity. (Directly proportional to r).
• Outside (R to infinity): A curve dropping down rapidly toward zero. (Inversely proportional to r²).

15. Why does gravity cancel out perfectly inside a hollow shell? (The Shell Theorem intuition).

Imagine you are inside a hollow shell, floating near the right wall. The right wall is pulling you strongly (close distance), but there is very little mass there. The left wall is far away (weak pull), but there is a huge amount of mass on that side. The math works out such that the "small mass/close distance" pull exactly balances the "huge mass/far distance" pull. Net force = 0.

16. If I dug a tunnel through the Earth and jumped in, would I oscillate back and forth forever? What is the period of that motion?

Yes, assuming no air resistance and uniform density, you would undergo Simple Harmonic Motion. You would fall to the center, speed up, overshoot, reach the other side of Earth, stop, and fall back. The period is approx 84 minutes—interestingly, this is the same time it takes a satellite to orbit Earth at the surface!

Lesson 5: Gravitational Potential Energy

17. If gravity is a force, what is a "gravitational field" and why do we need a separate formula for it?

Think of the field as a "map" of gravity's influence.

Force: Tells you how hard a specific object is pulled (depends on the object's mass).

Field (g): Tells you how strong gravity is at a spot, regardless of whether an object is there or not. It's Force per unit Mass (N/kg).

18. Why is Gravitational Potential Energy negative? How can energy be less than zero?

It is a reference point convention. We define potential energy as Zero at Infinity (when you are completely free of the planet). Since you fall towards a planet (losing energy), your energy drops below zero. Being "negative" means you are in a "gravity well"—you are stuck. To escape to zero (freedom), you must add positive energy.

Lesson 6:  Escape Velocity: Kinetic Energy + Gravitational Potential Energy

19. How do I derive the Escape Velocity formula without just memorizing it? Does it start with Force or Energy?

Start with Conservation of Energy. Total Energy at surface = Total Energy at infinity. At infinity, we want the object to just barely arrive and stop, so KE = 0 and PE = 0. ½mv² - GMm/R = 0½mv² = GMm/R Cancel m and solve for v: v = sqrt(2GM/R).

Lesson 7: Kepler's Three Laws: How Planets Move in Elliptical Orbits

20. When using Kepler's Third Law (T²/a³), which units do I have to use so I don't get a crazy answer?

You have two choices, but never mix them:

1. SI Units (Rigorous): Seconds for Time, Meters for Distance. (Use this if you need an answer in standard units).
2. Solar System Units (Easy): Years for Time, AU (Astronomical Units) for Distance. This works because T²/a³ = 1 for our solar system.

21. Why are planetary orbits elliptical and not perfect circles?

What determines how "squashed" the oval is? Circles are rare; they require the exact perfect velocity and angle. Any slight deviation results in an ellipse. The "squashed-ness" is called Eccentricity (e).

• e = 0: Perfect circle.
• 0 < e < 1: Ellipse.
• e ≥ 1: The object escapes (Parabola/Hyperbola).

Lesson 8:  Kinetic and Potential Energy of Satellites

22. Why are astronauts "weightless" on the ISS? Isn't there still gravity up there?

There is plenty of gravity (about 90% of what we feel on Earth!). They are weightless because they are in Free Fall. The ISS, and everything inside it, is falling toward Earth, but it has enough sideways speed to miss the planet. Since the floor falls at the same rate as the astronaut, the floor cannot push back up on their feet. No Normal Force = Weightlessness.

23. Does a heavier satellite need to move faster than a lighter one to stay in the same orbit?

No. Look at the orbital velocity equation: v = sqrt(GM/r). The mass of the satellite (m) does not appear in the equation. A hammer and a feather would orbit at the exact same speed at the same altitude.

24. The Satellite Paradox: If a satellite encounters air resistance and loses energy, why does it speed up?

1. Friction removes total mechanical energy.
2. To compensate, the satellite drops to a lower orbit (lower r).
3. In a lower orbit, gravity is stronger, so you must move faster to stay in orbit (v increases as r decreases).
4. The Trade-off: The satellite loses Potential Energy (a lot) and gains Kinetic Energy (some). The net result is a loss of Total Energy, but an increase in speed.

25. How do I calculate the total work required to move a satellite from a low orbit to a higher orbit (Hohmann transfer basics)?

You cannot just subtract the energies of the two circular orbits.

1. Calculate energy of the first circular orbit.
2. Calculate energy of the transfer ellipse (Burn 1).
3. Calculate energy of the second circular orbit (Burn 2).
4. The Work is the difference in Kinetic Energy required at the two burn points. (Usually tested as simple E_final - E_initial for conservative fields, but in rocketry, we calculate "Delta V"). For standard physics class: Work = Total Energy (Orbit 2) - Total Energy (Orbit 1).