Circular motion and gravitational fields (A Level)
13. Gravitational Fields
5 min read 12 parts Lesson 14 of 14 Unit quiz
Lesson summary
13. Gravitational Fields — Cambridge AS & A Level Physics 9702 — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Apply Newton's law of gravitation, field strength, orbital motion, geostationary orbits, gravitational potential and potential energy to planets and satell…
- Gravitational field strength g
- Near the Earth's surface
- Field lines
- Planets and stars
- Separation
- Inverse square law
Section 1 of 12Gravitational field and field strength
1Gravitational field
Gravitational field a region of space where a mass experiences a force.
- The force is always attractive and points towards the centre of the mass producing it.
2Gravitational field strength g
Gravitational field strength g the force per unit mass acting on a small mass at that point, g = F/m, in N kg−1.
3Weight
- Weight a planet with a larger g gives objects a larger weight, since weight = mg.
4Notation
- Notation g (field strength) is not G.
- (Newton's gravitational constant)
Section 2 of 12Field lines
| Field type | Field lines | Field strength |
|---|---|---|
| Radial (around a point mass or sphere) | Directed inwards towards the centre; they spread out with distance | g decreases with distance from the centre |
| Uniform (near a planet's surface) | Equally spaced parallel lines towards the surface | g is the same everywhere |
Near the Earth's surface
- Near the Earth's surface the field is almost uniform and can be treated as uniform.
Arrows
- Arrows field lines carry arrows because gravitational forces are attractive only.
Section 3 of 12Point-mass approximation
Rule
- Rule for a point outside a uniform sphere, all the mass can be treated as concentrated at its centre.
- An object can be a point mass when its mass distribution is even and the distance considered is larger than its size.
Field lines
- Field lines those around a uniform sphere are identical to those around a point mass.
Planets and stars
- Planets and stars they qualify because they are nearly uniform spheres and their separations are much greater than their radii.
Section 4 of 12Newton's law of gravitation
Newton's law
Newton's law the force between two point masses is proportional to the product of the masses and inversely proportional to the square of their separation.
- F = Gm1m2/r2, with G = 6.67 × 10−11 N m2 kg−2.What the symbols mean: F = gravitational force (N) · G = 6.67 × 10−11 N m2 kg−2 · m1, m2 or M, m = masses (kg) · r = distance between their centres (m)
Separation
- Separation r is measured between the centres of the masses.
- For a satellite above a planet r = planet radius + height above the surface.
Inverse square law
- Inverse square law doubling the separation reduces the force to one quarter.
Section 5 of 12Field strength due to a point mass
Derivation
- Derivation the force on a small mass m at distance r from mass M is F = GMm/r2, and the definition gives F = mg.
- Equating and cancelling m gives g = GM/r2.
Vector
- Vector g points towards the centre of the body.
Inverse square
- Inverse square g ∝ 1/r2, so it falls quickly as r increases.
Two equations
- Two equations g = F/m is the definition at a point, and g = GM/r2 is the field of a point mass.
Section 6 of 12The value of g near the Earth
Near the surface
- Near the surface g is about 9.81 N kg−1 and is treated as constant, since heights h in the atmosphere are tiny compared with the Earth's radius R.
- At h = 150 km, R + h = 6.55 × 106 m, only slightly more than R = 6.4 × 106 m.
General form
- General form g = GM/(R + h)2, which reduces to GM/R2 at the surface.
Density form
- Density form for a sphere M = ρ × (4/3)πr3, so g = 4πGρr/3, useful for comparing planets.
Section 7 of 12Circular orbits
Centripetal force
- Centripetal force the gravitational force on an orbiting body provides it: GMm/r2 = mv2/r.
Orbital speed
- Orbital speed the satellite mass cancels, so v = √(GM/r), and all satellites at one radius have the same speed.
Direction
- Direction the velocity is along the tangent and the force towards the centre, so the velocity changes continually.
Kepler's third law
- Kepler's third law substituting v = ωr gives ω2r3 = GM, and with ω = 2π/T this gives T2 ∝ r3.
Binary stars
- Binary stars each star's gravitational force provides its centripetal force, and both stars have the same ω.
Section 8 of 12Geostationary orbits
Properties
- Properties a geostationary satellite stays above the equator at a fixed point on the Earth.
- It moves from west to east, the same way as the Earth turns.
- Its period is 24 hours, equal to the Earth's rotation period.
Orbit radius
- Orbit radius setting T = 24 h in T2 = 4π2r3/GM fixes the radius.
Uses
- Uses telecommunications and television broadcasting.
- A ground station sends a signal up, the satellite amplifies it and sends it to chosen areas.
- Receiver dishes point at a fixed position in the sky and need not move.
Section 9 of 12Gravitational potential
Gravitational potential
Gravitational potential φ
Gravitational potential φ the work done per unit mass in bringing a small test mass from infinity to the point.
- It is zero at infinity and negative everywhere else, because gravity is attractive.
Equation
- φ = −GM/r, in J kg−1, which depends on the mass and distance from its centre.
- It becomes less negative as r increases and decreases as a satellite moves closer.
Change in potential
- Δφ = GM(1/r1 − 1/r2) between distances r1 and r2.
Section 10 of 12Gravitational potential energy
1Work done
- Work done moving a mass m through a potential difference Δφ needs work W = mΔφ.
2Gravitational potential energy
Gravitational potential energy the work done to bring the masses from infinite separation to distance r, Ep = −Gm1m2/r.
3Change
- ΔEp = Gm1m2(1/r1 − 1/r2), the work done against gravity from r1 to r2.
4Limit of mgh
- Ep = mgh holds only in a uniform field near a surface; elsewhere use −GMm/r.
Section 11 of 12Key terms
| Term | Meaning |
|---|---|
| Gravitational field | A region of space in which a mass experiences a force. |
| Gravitational field strength | The force per unit mass at a point, g = F/m. |
| Newton's law of gravitation | The force between two point masses is Gm1m2/r2. |
| Geostationary orbit | An equatorial orbit with a 24-hour period, so the satellite stays above one point on Earth. |
| Gravitational potential | Work done per unit mass bringing a small mass from infinity to the point, φ = −GM/r. |
| Gravitational potential energy | Work done to bring masses from infinite separation, Ep = −Gm1m2/r. |
| Inverse square law | A quantity that falls in proportion to 1/r2. |
Section 12 of 12Exam technique
1
Using the height above the surface as r — add the planet's radius to get the distance from the centre.
2
Confusing g and G — g is the field strength in N kg−1 and G is the universal constant 6.67 × 10−11 N m2 kg−2.
3
Writing gravitational potential as positive — it is negative everywhere, with zero at infinity.
4
Using Ep = mgh for large height changes — use −GMm/r.
5
Thinking orbital speed depends on the satellite's mass — the mass cancels, so v depends on M and r only.