Gravitation · Lesson 7 of 7
Chapter Summary and Practice
“Connect the PDF’s gravitational laws, assumptions and numerical applications in one reference and practice lesson.”
• Choose a force, field, energy or period relation for the question. • Keep centre distance, height and depth distinct. • State symmetry, density and reference assumptions. • Practise the PDF’s connected gravitational applications.
Choose the right quantity
| Question | Relation | Condition |
|---|---|---|
| Force between masses | F=Gm1m2/r² | Point masses or justified spherical replacement |
| Field or acceleration | g=F/m | Vector quantity; test mass cancels |
| Potential energy | U=−GMm/r | Exterior point-source model; zero at infinity |
| Potential | φ=−GM/r | Energy per unit test mass |
| Circular orbital motion | v²=GM/r; T²=4π²r³/(GM) | Dominant source, circular radius |
| Escape threshold | ve²=2GM/r0 | Ideal coasting escape, no other sources or losses |
| Inside a uniform Earth | g=g0r/RE | Uniform density is essential |
Formula reference and dimensions
| Quantity | SI unit | Dimensions |
|---|---|---|
| G | N m²/kg² | M⁻¹ L³ T⁻² |
| Force | N | M L T⁻² |
| Gravitational potential energy | J | M L² T⁻² |
| Gravitational potential | J/kg | L² T⁻² |
| Gravitational intensity | N/kg or m/s² | L T⁻² |
Practice from the PDF
Practice Problems
- Use the equal-area law to compare perihelion and aphelion speeds and the two traversal times in source Example 7.1.
- Find the centroid force in the source’s equilateral triangle before and after doubling the mass at A.
- Explain why a spherical shell is not a gravitational shield, and derive the hemisphere field directions c and e.
- Show why the Moon can have a larger tidal effect than the Sun even though its total pull on Earth is smaller.
- Compare g above and below Earth using the exact height expression and uniform-density depth expression; state where each model applies.
- Count the six square-pair energies and distinguish system energy from potential at the centre.
- Find the limiting escape speed between the M and 4M spheres; explain the finite-time qualification at the neutral point.
- Use Phobos data to weigh Mars, lunar data to weigh Earth, Io data to weigh Jupiter and Earth’s orbit to weigh the Sun.
- For the galaxy model in Exercise 7.5, specify the assumed enclosed mass before applying the circular-period expression.
- Check every sign in the 2RE-to-4RE satellite energy transfer and calculate escape energy for the 400 km orbit.
- Solve the two-star head-on approach in Exercise 7.20, distinguishing each star’s speed from relative closing speed.
- Calculate force and potential between the equal 100 kg spheres and test stability along the joining line.
A worked two-body energy application
Problem
Source Exercise 7.20 gives two equal stars of mass M=2×10³⁰ kg and radius Rs=10⁴ km, initially almost at rest with centre separation d0=10⁹ km. Find the speed when they first touch, assuming no distortion.
- 1.At contact their centre separation is 2Rs=2×10⁷ m; initially d0=10¹² m. Equal masses and zero total initial momentum give equal opposite speeds v in the centre-of-mass frame.
- 2.Initial energy is −GM²/d0. Final energy is 2(½Mv²)−GM²/(2Rs).
- 3.Therefore v²=GM[1/(2Rs)−1/d0], giving v≈2.58×10⁶ m/s for each star.
- 4.The relative closing speed is 2v≈5.17×10⁶ m/s. The source answer of about 2.6×10⁶ m/s refers to each star’s speed, not their relative speed.
- 5.This is the exercise’s ideal undeformed-sphere model. Count the interaction potential once for the pair and both stars’ kinetic energies.
Distinctions that prevent common errors
G is a universal constant; g is a local field or acceleration. Gravitational force and field are vectors; potential and potential energy are scalars. A body’s zero net field does not require zero potential. The surface value mgh is not the general inverse-distance energy with its zero at infinity. A satellite is not outside gravity merely because it is weightless. A particle’s linear momentum need not be constant while it follows a central-force orbit; its angular momentum and mechanical energy can be conserved. For the full isolated source-plus-particle system, total linear momentum is conserved as well.
The area law applies to any central force, but the particular period–size relation follows from inverse-square gravity for a shared dominant source. An ellipse’s semi-major axis is not the same as its changing Sun–planet distance. Energy and angular momentum determine different aspects of orbital motion. When a question changes source body, density assumptions, reference level or distance definition, identify that change before substituting numbers.
Check the physical model and the answer
Write the system, source masses and relevant symmetry. Convert kilometres and days to SI when using SI G. Replace altitude by r=R+h and depth by r=R−d. Use the uniform-density assumption only for the inside-Earth formula. Add gravitational force components with signs but add scalar potentials directly. Verify dimensions, energy signs and limiting cases: g should fall at large distance, U should tend to zero from below, and escape speed should decline as launch radius grows. Recompute a printed answer if substitution fails; the audit records the sign and rounding corrections found in the supplied source.
Check your understanding
Quiz
For a spherical source, orbit height h corresponds to centre distance
Add the source radius to altitude.
A zero-field point between two equal masses has potential
Field cancellation does not cancel scalar potential.
For a comet in an ideal elliptical orbit, which pair remains constant?
Gravity is central and conservative.
If identical stars collide head-on with speeds v and −v, their relative closing speed is
The relative velocity magnitude is |v−(−v)|=2v.
Which potential-energy change is correct for a satellite raised from 2RE to 4RE?
−GMm/r becomes less negative as r increases.
Key Takeaways
• Select the relation only after stating the geometry and assumptions. • Preserve vector directions, scalar signs and a common energy reference. • Check numerical answers against the defining equations rather than copying printed signs.
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Satellites: Orbits, Energy and Weightlessness
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