Short on time? This page is the recap for the calculus-based course (separate from Physics 1). For explanations, the field graph and worked examples, use the full study guide.
Recap
- Gravity is attractive, acts along the line of centers, and can be taken to act at each system’s center of mass.
- The two forces in a pair are equal in size; the accelerations are not.
- The field is force per unit mass. With gravity as the only force, a = g.
- Weight = mg. Treat g as constant when r hardly changes (near Earth’s surface, g ≈ 9.8 N/kg).
- Apparent weight is the normal force. Weightless means N = 0: no forces at all, or gravity alone.
- Equivalence principle: inside a closed cabin you cannot tell a gravitational field from an acceleration of the cabin.
- Inertial mass (resists change in motion) and gravitational mass (takes part in attraction) are found by experiment to be equal.
Key relationships
| Relationship | Units | Notes |
|---|---|---|
| F_g = G m₁m₂ / r² | N | G = 6.67 × 10⁻¹¹ N·m²/kg²; r between centers |
| g = F_g/m = GM/r² | N/kg = m/s² | points toward M |
| W = mg | N | near a planet’s surface |
| N = m(g + a_y) | N | +y upward; apparent weight |
| g_surface = (4/3)πGρR | N/kg | uniform planet; same ρ, double R → double g |
| Thin shell: F = 0 inside; GMm/r² outside | N | shell theorem, used without proof |
| Uniform sphere inside: M_partial = Mr³/R³, F = (GMm/R³)r = kr | N | toward center: F = −kr |
Assumptions behind the numbers
- Spheres are uniform (or made of uniform shells); other bodies’ pulls are ignored unless stated.
- Constant g only over heights that are small compared with the planet’s radius (g falls 1% by about 32 km above Earth).
- Air resistance ignored for “free fall”.
Mistakes to avoid
- Using the altitude as r. Add the planet’s radius.
- Forgetting to square r, or squaring the mass instead.
- Saying apparent weight depends on velocity. It depends on acceleration.
- Using the whole mass inside a sphere. Only the partial mass within r counts.
- Thinking the field inside a shell is zero only at the center. It is zero everywhere inside.
- “Weightless means no gravity.” It means nothing pushes back against gravity.
Quick self-check
- Two masses move twice as far apart. What happens to F_g? (It becomes ¼ as large.)
- g at a planet’s surface is 9.8 N/kg. What is it at r = 2R? (2.45 N/kg)
- A 50 kg person in a lift accelerating upward at 2.0 m/s². What does the scale read? (50 × 11.8 = 590 N)
Next: practice questions.