Gravitational fields

• One equation carries the orbits: \(GMm/r^2 = mv^2/r\). One sign carries the potential: minus, because zero is at infinity and gravity only attracts. And one habit prevents most lost answers: \(r\) is measured from the centre.
• \(r\) from the centre. Force and field \(1/r^2\), potential \(1/r\). Gravity is the centripetal force. Zero at infinity, so negative everywhere else.

• 13.1.1 I can understand — understand that a gravitational field is an example of a field of force and define gravitational field as force per unit mass
• 13.1.2 I can — represent a gravitational field by means of field lines
• 13.2.1 I can understand — understand that, for a point outside a uniform sphere, the mass of the sphere may be considered to be a point mass at its centre
• 13.2.2 I can recall — recall and use Newton's law of gravitation F = Gm₁m₂/r² for the force between two point masses
• 13.2.3 I can — analyse circular orbits in gravitational fields by relating the gravitational force to the centripetal acceleration it causes
• 13.2.4 I can understand — understand that a satellite in a geostationary orbit remains at the same point above the Earth's surface, with an orbital period of 24 hours, orbiting from west to east, directly above the Equator
• 13.3.1 I can — derive, from Newton's law of gravitation and the definition of gravitational field, the equation g = GM/r² for the gravitational field strength due to a point mass
• 13.3.2 I can recall — recall and use g = GM/r²
• 13.3.3 I can understand — understand why g is approximately constant for small changes in height near the Earth's surface
• 13.4.1 I can define — define gravitational potential at a point as the work done per unit mass in bringing a small test mass from infinity to the point
• 13.4.2 I can use — use ϕ = −GM/r for the gravitational potential in the field due to a point mass
• 13.4.3 I can understand — understand how the concept of gravitational potential leads to the gravitational potential energy of two point masses and use E_P = −GMm/r

The pattern in all seven. The satellite’s mass always cancels; only the central mass and the radius matter. Speed questions use \(v^2 = GM/r\); period, radius and mass questions use \(T^2 = 4\pi^2r^3/(GM)\); ratio questions need neither number, only \(v \propto r^{-1/2}\) and \(T \propto r^{3/2}\).

• “A satellite 400 km up is at \(r = 400\,\mathrm{km}\).” Correct \(r\) is measured from the centre, not the surface. Outside a uniform sphere its mass acts as if concentrated at the centre, so every \(r\) in \(F = Gm_1m_2/r^2\), \(g = GM/r^2\), \(\phi = -GM/r\) and \(v = \sqrt{GM/r}\) is a centre-to-centre distance: \(r = R + h = 6.37 \times 10^{6} + 4.00 \times 10^{5} = 6.77 \times 10^{6}\,\mathrm{m}\).
• “Gravitational field strength is the force on a mass.” Correct It is the force per unit mass on a small test mass, \(g = F/m\), in N kg−1. Without “per unit mass” the definition describes a force, not a field.
• “Doubling the distance halves the force.” Correct Inverse square: doubling \(r\) quarters \(F\) and \(g\). Only the potential goes as \(1/r\), so doubling \(r\) halves \(\phi\).
• “The satellite stays up because an outward force balances gravity.” Correct Only one force acts on a satellite: gravity, towards the centre. It supplies the centripetal force. The satellite is in free fall; its tangential velocity carries it round the planet rather than into it.
• “Potential energy is \(mgh\), so it is positive and grows with height without limit.” Correct Measured from infinity, \(E_P = -GMm/r\): negative everywhere, rising towards zero as \(r\) increases. \(mg\Delta h\) is a good approximation for a change only when \(h\) is small compared with the radius.
• “A geostationary satellite can be parked above any city.” Correct It must be directly above the Equator. Every orbit is centred on the Earth’s centre, so an orbit that is not in the equatorial plane takes the satellite north and south of the Equator every day.
• “The satellite is 400 km up, so \(r = 4.00 \times 10^{5}\,\mathrm{m}\).” Repair \(r\) is measured from the centre of the planet: \(r = R + h = 6.37 \times 10^{6} + 4.00 \times 10^{5} = 6.77 \times 10^{6}\,\mathrm{m}\). Outside a uniform sphere its mass acts at its centre.
• “Gravitational field strength is the force on a mass.” Repair It is the force per unit mass on a small test mass, \(g = F/m\), in N kg−1.
• “\(g = GM/r^2\), so a heavier object has a bigger \(g\).” Repair \(M\) is the mass creating the field. The test mass cancelled in the derivation, so every object at the same point has the same \(g\).
• “Doubling the distance halves the gravitational force.” Repair It quarters it: the force, and the field, follow an inverse-square law.
• “\(\phi = -GM/r^2\) and \(g = GM/r\).” Repair The other way round: the field goes as \(1/r^2\), the potential as \(1/r\). Units decide it: \(GM/r^2\) is in N kg−1, \(GM/r\) in J kg−1.
• “A satellite stays up because the centrifugal force balances gravity.” Repair There is one force on the satellite, gravity, towards the centre. It is not balanced: it provides the centripetal acceleration that keeps changing the direction of the velocity. The satellite is in free fall.
• “Astronauts float because there is no gravity in orbit.” Repair At 400 km, \(g = 8.69\,\mathrm{N\,kg^{-1}}\), 89% of the surface value. The astronauts and the station fall together with the same acceleration, so nothing presses on them.
• “A heavier satellite has to go faster to stay in the same orbit.” Repair \(GMm/r^2 = mv^2/r\) gives \(v = \sqrt{GM/r}\): the satellite’s mass \(m\) cancels. Every object in the same orbit has the same speed.
• “A geostationary satellite can be placed above London.” Repair It must be directly above the Equator. Gravity points to the Earth’s centre, so every orbit is centred there; a tilted orbit carries the satellite north and south of the Equator each day and it cannot stay above one point.
• “Gravitational potential is positive and increases as you go down towards the planet.” Repair It is negative, zero at infinity, and it decreases (becomes more negative) as you approach the mass.
• “\(\Delta E_P = mg\Delta h\) works for any height.” Repair Only while \(g\) is constant, that is for \(h \ll R\); it overestimates by the fraction \(h/R\). Otherwise use \(\Delta E_P = GMm(1/r_1 - 1/r_2)\).
• “The field lines of a planet point outwards.” Repair They point inwards, the direction of the force on a small mass placed there.
• “The Earth pulls the Moon harder than the Moon pulls the Earth.” Repair The two forces are equal in size and opposite in direction, a Newton’s third law pair on different bodies. The Moon’s acceleration is larger only because its mass is smaller.

• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.
• Sketching the two patterns. Radial: lines on true radii, evenly distributed round the sphere, every arrow pointing inwards, lines stopping at the surface. Uniform: straight, parallel, equally spaced vertical lines with downward arrows, ending on a horizontal surface. Arrows pointing outwards, curved lines, or unequal spacing in a uniform field each misrepresent the field.
• Say “less negative”, not “smaller”. \(-3 \times 10^{7}\,\mathrm{J\,kg^{-1}}\) is a higher potential than \(-6 \times 10^{7}\,\mathrm{J\,kg^{-1}}\), though its magnitude is smaller. Writing “the potential decreases as you move away” because the number’s size falls gets the physics backwards.
• Two habits for every line of this card. Every \(r\) is measured from the centre of the body creating the field, so a height becomes \(r = R + h\) before it goes into any equation. And keep the squares straight: force and field go as \(1/r^2\); potential and potential energy go as \(1/r\).
• Interleave with the chapters that use this one. When you reach Topic 18, rewrite the field-structure card for electric fields step by step, and say at each step why the gravitational version has only one sign. When you reach Topic 25, compare the inverse-square fall of radiant flux intensity, \(F = L/(4\pi d^2)\), with the inverse-square fall of \(g\): the star’s luminosity \(L\) is fixed, and what weakens with distance is the flux received. Recalling a topic inside a new context is worth more than another pass over this chapter on its own; at A Level, Paper 4 assumes the whole of the AS content, so nothing here is ever finished with.

• Cambridge International AS & A Level Physics 9702 has five components. Topic 13 is A Level content, so it is examined in Papers 4 and 5. A Level content: examined in Paper 4 (A Level structured, which also requires the AS content) and, as practical context, Paper 5. AS Level candidates take Papers 1, 2 and 3; A Level candidates take all five, either staged over two years (Papers 1–3 in year one, Papers 4 and 5 in year two) or together in one series. Examinations are available in the June and November series, and in March in India.
• Across both the AS Level and the A Level the assessment objectives are weighted AO1 40% (knowledge and understanding), AO2 40% (handling, applying and evaluating information) and AO3 20% (experimental skills and investigations). AS candidates are graded a–e; A Level candidates A*–E. The Data and formulas sheet is printed as page 2 of Papers 1 and 2 and as pages 2 and 3 of Paper 4: it gives the constants and a short list of formulas. Every other equation in this chapter is one the syllabus says you must recall, and this chapter says which is which.
• Topic 13 is not in Paper 1, so there is no multiple-choice item on it. In Paper 4 you may be asked to define gravitational field strength or gravitational potential, to state Newton’s law of gravitation, to show that \(g = GM/r^2\) or \(T^2 \propto r^3\), and to explain why potential is negative, why a geostationary orbit must be equatorial, or why \(g\) is nearly constant near the surface.
• Field strength at a height, orbital speed and period, the mass of a planet from an orbit, the geostationary radius, a neutral point between two bodies, and an energy change from \(\Delta E_P = GMm(1/r_1 - 1/r_2)\). Graphs of \(g\) and \(\phi\) against \(r\) may be sketched. \(\phi = -GM/r\) and \(E_P = -GMm/r\) are on the formulas sheet; \(F = Gm_1m_2/r^2\), \(g = GM/r^2\) and the circular-motion equations are recall.
• Gravitational fields cannot be investigated in a school laboratory, so this topic reaches Paper 5 as an analysis context: supplied orbital radii and periods, a \(\lg T\) against \(\lg r\) graph with error bars, the gradient testing \(T \propto r^{1.5}\), and a central mass from the intercept with its uncertainty. There is no Paper 3 experiment for this topic.
• Read the command word before you decide how much to write. This syllabus has fifteen of them: calculate, comment, compare, define, describe, determine, explain, give, identify, justify, predict, show (that), sketch, state and suggest. Define wants a precise meaning — for a physical quantity, usually an equation in words with every quantity named. State and give want a fact and nothing more. Describe wants the points or the features. Explain wants the reasons and the relationships — a describe-level answer to an explain question is incomplete however well written it is. Show (that) gives you the result and asks for the structured evidence that leads to it, so every step must appear — and a final value worked to one more significant figure than the one printed makes it plain that you calculated it rather than copied it. Sketch wants a freehand graph with its key features — intercepts, asymptotes, the shape — correct, but no plotted scale.