OCR A-Level Physics: Circular Motion, SHM and Gravity — Complete Revision Guide (H556)
OCR A-Level Physics: Circular Motion, SHM and Gravity
Circular motion, simple harmonic motion and gravitation form the conceptual heart of OCR A-Level Physics A (H556) Module 5.3 (oscillations) and Module 5.4 (gravitational fields). The three topics are deeply interconnected: a body in uniform circular motion, projected onto a diameter, executes simple harmonic motion; a planet in an elliptical orbit obeys Kepler's third law, which falls out of equating gravitational force to centripetal force for a circular orbit; and the period of a pendulum, the period of a planet, and the period of a mass on a spring all share the same √(restoring-coefficient/inertia) structure. Mastering this module is what unifies the rest of the H556 specification under the heading of "physics is the study of periodic systems and the forces that cause them".
H556 examiners weight this module very heavily because it is the most synoptic on the specification. A candidate who can derive centripetal acceleration v²/r, set up and solve the SHM defining equation a = −ω²x, identify the resonance condition from a driven-oscillator response curve, apply Newton's law of gravitation to compute the field strength at a planet's surface and the gravitational potential at any distance, and use Kepler's third law to deduce the mass of a central body from the period and orbital radius of a satellite, has the toolkit needed for the unified-physics Paper 3 items on satellite systems, particle accelerators (synchrotron radiation), molecular vibrations and atomic spectra. A candidate who cannot do these things will be exposed on every periodic-system question.
Course 8 of the H556 Physics learning path on LearningBro, Circular Motion, SHM and Gravity, develops the full story. It opens with circular-motion kinematics (angular velocity, period, frequency), develops centripetal acceleration and force, defines SHM via a = −ω²x (with the minus sign as the conceptual centrepiece), supplies the canonical solutions x = A cos(ωt) and x = A sin(ωt), develops the energy interchange between kinetic and potential in SHM, treats damping and resonance with the driven-oscillator response curve, and pivots to gravitation: gravitational field strength, Newton's inverse-square law, gravitational potential and potential energy, Kepler's three laws, the dynamics of circular and elliptical orbits, and geostationary orbits as a special case. It sits at the synoptic centre of the LearningBro OCR A-Level Physics learning path.
Guide Overview
The Circular Motion, SHM and Gravity course is built as a sequence of twelve lessons that move from circular kinematics through SHM to gravitation and orbital mechanics.
- Circular Motion: Angular Quantities
- Centripetal Acceleration and Force
- SHM: Defining Equation
- SHM: Solutions
- Energy in SHM
- Damping and Resonance
- Gravitational Field Strength
- Newton's Law of Gravitation
- Gravitational Potential and Energy
- Kepler's Laws
- Orbits
- Geostationary Orbits
OCR H556 Specification Coverage
This course addresses OCR H556 Module 5.3 (circular motion and oscillations) and Module 5.4 (gravitational fields) in full. The specification organises the topic into circular-motion kinematics, centripetal force, SHM as a defined motion type, energy interchange in SHM, damping and resonance, Newton's law of gravitation, gravitational field and potential, and the orbital mechanics that culminates in Kepler's third law and geostationary orbits (refer to the official OCR specification document for exact wording).
| Sub-topic | Spec area | Primary lesson(s) |
|---|---|---|
| Angular velocity ω, period T, frequency f | OCR H556 Module 5.3.1 | Circular Motion: Angular Quantities |
| Centripetal acceleration v²/r and force mv²/r | OCR H556 Module 5.3.1 | Centripetal Acceleration and Force |
| SHM defining equation a = −ω²x | OCR H556 Module 5.3.2 | SHM: Defining Equation |
| Sinusoidal solutions x = A cos(ωt + φ) | OCR H556 Module 5.3.2 | SHM: Solutions |
| KE-PE interchange and total-energy conservation | OCR H556 Module 5.3.2 | Energy in SHM |
| Light/heavy/critical damping and the resonance peak | OCR H556 Module 5.3.3 | Damping and Resonance |
| Gravitational field strength g | OCR H556 Module 5.4.1 | Gravitational Field Strength |
| F = GMm/r² | OCR H556 Module 5.4.2 | Newton's Law of Gravitation |
| V = −GM/r and E_p = −GMm/r | OCR H556 Module 5.4.3 | Gravitational Potential and Energy |
| Kepler's three laws | OCR H556 Module 5.4.4 | Kepler's Laws |
| Orbital dynamics and orbital speeds | OCR H556 Module 5.4.4 | Orbits |
| Geostationary orbits as a special case | OCR H556 Module 5.4.4 | Geostationary Orbits |
Modules 5.3 and 5.4 are examined across all three H556 papers, with calculation-heavy questions on Paper 1 and Paper 2 and extended-response synoptic items on Paper 3. The geostationary-orbit derivation and the Kepler's-third-law application appear reliably; the SHM energy-interchange interpretation appears reliably; the damping/resonance qualitative description appears reliably.
Topic-by-Topic Walkthrough
Circular Motion and Centripetal Force
The angular quantities lesson develops angular velocity ω = 2π/T = 2πf, with linear speed v = rω, period T (the time for one revolution), and frequency f = 1/T. The centripetal acceleration and force lesson derives the centripetal acceleration a = v²/r = ω²r from geometry (the velocity vector changes direction at rate ω even when its magnitude is constant), then applies Newton's second law to give the centripetal force F = mv²/r = mω²r. The canonical worked example is the conical pendulum or a banked-track vehicle: the horizontal component of the net force on the body equals the required centripetal force, while the vertical component balances weight. The top-band discriminator is the explicit identification that "centripetal force" is not a separate physical force but the name given to whatever combination of real forces (tension, normal reaction, gravity, friction) supplies the required mv²/r — a conceptual point that examiners reward heavily.
SHM: Defining Equation and Solutions
The SHM defining equation lesson establishes a = −ω²x as the defining condition for simple harmonic motion: acceleration is proportional to displacement from equilibrium, and is directed opposite to that displacement. The minus sign is the conceptual centrepiece — it expresses the restoring nature of the force and is what causes oscillation rather than runaway divergence. Without the minus sign, displacement would amplify exponentially; with it, displacement oscillates sinusoidally. The SHM solutions lesson gives the general solution x = A cos(ωt + φ), where A is amplitude, ω is angular frequency and φ is phase, with velocity v = −Aω sin(ωt + φ) and acceleration a = −Aω² cos(ωt + φ) = −ω²x as required. The two canonical SHM systems are the mass-on-spring (ω = √(k/m), so T = 2π√(m/k)) and the simple pendulum at small angles (ω = √(g/L), so T = 2π√(L/g)). The top-band discriminator is recognising that the small-angle approximation (sin θ ≈ θ) is what makes the pendulum SHM — at larger angles, the restoring torque is non-linear and the motion is not strictly SHM.
Energy in SHM
The energy in SHM lesson develops the kinetic-potential interchange: at the extremes of displacement (x = ±A), kinetic energy is zero and potential energy is maximum; at the equilibrium position (x = 0), kinetic energy is maximum and potential energy is zero. Total mechanical energy is conserved (in the absence of damping) and equals ½kA² for a spring (or ½mω²A² for any SHM oscillator). The kinetic-energy-versus-position curve is an inverted parabola, KE = ½mω²(A² − x²); the potential-energy-versus-position curve is an upright parabola, PE = ½mω²x². The two sum to the constant total energy. A worked example: a 0.20 kg mass on a spring of constant 80 N m⁻¹ with amplitude 0.10 m has ω = √(80/0.20) = 20 rad s⁻¹, period T = 2π/ω = 0.314 s, maximum kinetic energy = ½ × 80 × 0.01 = 0.40 J, maximum speed v_max = √(2 × 0.40 / 0.20) = 2.0 m s⁻¹. The top-band discriminator is the explicit graphical interpretation of energy: sketching KE and PE as functions of x or t and identifying where they cross.
Damping and Resonance
The damping and resonance lesson treats real oscillators where energy is lost to friction or drag. Light damping reduces amplitude gradually over many cycles, with the envelope decaying exponentially; heavy damping suppresses oscillation entirely; critical damping returns the system to equilibrium in the shortest time without oscillation. When an external periodic driving force is applied to a damped oscillator, the system responds with steady-state oscillation at the driving frequency, with amplitude that peaks sharply at the natural frequency of the system — this is resonance. The resonance peak is high and narrow for lightly damped systems, low and broad for heavily damped systems. The canonical engineering examples are the Tacoma Narrows bridge collapse (wind-driven resonance), tuned-mass dampers in skyscrapers (heavy damping to prevent resonance), and NMR spectroscopy (resonance exploited to identify atomic environments). The top-band discriminator is the explicit qualitative response curve, with amplitude on the y-axis, driving frequency on the x-axis, and the resonance peak at f_natural.
Gravitational Field Strength and Newton's Law of Gravitation
The gravitational field strength lesson defines g as the force per unit mass on a test mass placed in the field: g = F/m, with SI units N kg⁻¹ (numerically equal to acceleration in m s⁻², because F = ma). At Earth's surface g ≈ 9.81 N kg⁻¹. The Newton's law of gravitation lesson introduces F = GMm/r² as the attractive inverse-square force between two point masses, with G = 6.67×10⁻¹¹ N m² kg⁻². Combining the two gives g = GM/r² as the field strength at distance r from a point mass M (or equivalently from the centre of a spherically symmetric body of mass M, by the shell theorem). The top-band discriminator is the explicit shell-theorem justification: a spherically symmetric mass distribution behaves gravitationally (for points outside) exactly as if all its mass were concentrated at its centre — this is what allows treating planets as point masses for orbital calculations.
Gravitational Potential and Energy
The gravitational potential and energy lesson introduces V = −GM/r as the potential at distance r from a point mass M, with V → 0 at infinity and V → −∞ at r → 0. The negative sign reflects the convention that infinity is taken as the zero of potential and bound states have negative potential. Potential energy of a mass m at distance r is E_p = mV = −GMm/r. Work done against gravity to move m from r₁ to r₂ (with r₂ > r₁) is E_p(r₂) − E_p(r₁) = GMm(1/r₁ − 1/r₂), which is positive. Escape velocity from a planet's surface is obtained by equating ½mv_esc² to GMm/R, giving v_esc = √(2GM/R) — for Earth, about 11.2 km s⁻¹. A worked example: the gravitational potential at the Moon's surface (M = 7.35×10²² kg, R = 1.74×10⁶ m) is V = −6.67×10⁻¹¹ × 7.35×10²² / 1.74×10⁶ = −2.82×10⁶ J kg⁻¹, so a 1500 kg lunar module sitting on the surface has gravitational PE = −4.23×10⁹ J relative to infinity. The top-band discriminator is rigorous sign-convention tracking — potential is always negative for an attractive field with V(∞) = 0; the work done against gravity to escape is the positive quantity −V(R) × m.
Kepler's Laws, Orbits and Geostationary Orbits
The Kepler's laws lesson states the three laws: planetary orbits are ellipses with the Sun at one focus (first law); the line from a planet to the Sun sweeps equal areas in equal times (second law, equivalent to conservation of angular momentum); the square of the orbital period is proportional to the cube of the semi-major axis (third law, T² ∝ a³). The orbits lesson derives the third law for circular orbits by equating gravitational force to centripetal force: GMm/r² = mω²r, giving ω² = GM/r³, and since ω = 2π/T, T² = 4π²r³/(GM). This relates T to r via the central body's mass M. The geostationary orbits lesson applies this to satellites with T = 24 hours (the sidereal day, technically) in the equatorial plane, eastward — giving an orbital radius of about 42,000 km from Earth's centre or 36,000 km above the surface. Geostationary satellites are essential for fixed-pointing communications because they remain over the same point on Earth's surface continuously. The top-band discriminator on geostationary problems is recognising that the orbit must be in the equatorial plane and prograde — a polar geosynchronous orbit (T = 24 h, polar inclination) is not geostationary because the satellite traces a figure-eight ground track.
A Typical H556 Paper 2 Question
A standard Paper 2 prompt gives candidates an orbital scenario — the orbital period and radius of a satellite around an unknown central body — and asks for the mass of the central body via Kepler's third law, then asks for the orbital speed and the escape velocity from the satellite's altitude. The AO1 component covers recall of T² = 4π²r³/(GM); the AO2 component covers the rearrangement M = 4π²r³/(GT²) and substitution; the AO3 component appears when the question asks for the additional energy required to lift the satellite from its current orbit to escape, which is the difference between the escape kinetic energy GMm/r at the new altitude and the current kinetic energy ½mv² in the current orbit, with both gravitational potential energy and kinetic energy bookkept carefully. The discriminator at the top band is the explicit decomposition of total mechanical energy of an orbiting satellite as KE + PE = ½GMm/r − GMm/r = −GMm/(2r), with the negative sign indicating bound state; escape requires raising total energy to zero, requiring additional energy GMm/(2r).
Synoptic Links
Circular motion, SHM and gravity are the most synoptic module on the H556 specification. Circular motion connects forwards into magnetic fields in Module 6.3 — a charged particle in a uniform magnetic field executes uniform circular motion with radius r = mv/(qB), and the same centripetal-equals-magnetic-force balance underlies particle accelerators, mass spectrometers and cyclotron radiation. SHM connects forwards into electromagnetic waves (electric and magnetic fields oscillate as coupled SHMs), into AC circuits (capacitor charge and current oscillate sinusoidally), and into the photon-energy-level transitions of Quantum Physics (electrons in atoms can be approximated as harmonic oscillators near equilibrium). Gravity connects forwards into the astrophysics module via stellar formation, planetary atmospheres and the cosmological expansion.
The SHM-circular-motion relationship (SHM is the projection of uniform circular motion onto a diameter) is itself a powerful synoptic insight — it explains why the SHM solution is sinusoidal and gives a geometric interpretation of phase φ as the angular position on the imaginary circle. The Kepler-third-law derivation reuses Newton's law of gravitation and centripetal force, so it sits at the synoptic intersection of Modules 5.3 and 5.4.
Paper 3 'Unified physics' items typically deploy this module against unfamiliar contexts. A satellite-systems scenario might give the period of the International Space Station and ask candidates to compute the altitude and the orbital speed, then explain why ISS crew experience apparent weightlessness despite g being only slightly reduced from surface value. A molecular-vibration scenario might give the frequency of a diatomic molecule's vibrational spectrum and ask candidates to deduce the bond stiffness, treating the molecule as a mass-on-spring SHM oscillator. A binary-star scenario might give the orbital period and separation of a binary system and ask candidates to deduce the total mass via Kepler's third law generalised to two-body orbits.
What Examiners Reward
Top-band marks on this module cluster around the explicit minus sign in a = −ω²x (the restoring-force signature of SHM), rigorous identification of centripetal force as the resultant of real forces rather than a separate force, the explicit shell-theorem justification for treating spherical bodies as point masses, and rigorous sign-tracking of gravitational potential and potential energy. For Kepler's-third-law problems, examiners want the explicit derivation route (gravitational force equals centripetal force, T = 2π/ω) rather than direct substitution. For orbital-energy problems, they want the explicit decomposition KE + PE = −GMm/(2r), with the negative total energy diagnostic of a bound state. For resonance problems, they want a qualitative response curve and the natural-frequency identification.
Common pitfalls cluster around six recurring mistakes. First, drawing centripetal force as an additional arrow on a free-body diagram (it is the resultant of the real forces, not an additional force). Second, omitting the minus sign in a = −ω²x and analysing the resulting motion as exponential rather than oscillatory. Third, using surface gravitational field strength at altitude — g decreases as 1/r² above the surface (by the inverse-square law). Fourth, omitting the negative sign in V = −GM/r and consequently miscomputing escape velocity or potential energy. Fifth, applying Kepler's third law without using consistent units (T in seconds, r in metres, M in kilograms, G in SI units). Sixth, assuming geostationary means polar — geostationary requires equatorial plane and prograde direction. Each of these is a one- or two-mark deduction.
Practical Activity Groups (PAGs)
This course anchors PAG 6 (Mechanics) in the OCR practical scheme, specifically the SHM sub-activities. PAG 6.1 measures the period of a simple pendulum as a function of length, exploiting T = 2π√(L/g) to deduce g from the gradient of a T²-versus-L plot. PAG 6.2 measures the period of a mass on a spring as a function of attached mass, exploiting T = 2π√(m/k) to deduce the spring constant k from the gradient of a T²-versus-m plot. PAG 6.3 measures the damping of an oscillator by recording amplitude decay over successive cycles and fitting an exponential envelope. The error-propagation discussion is what differentiates a Top-band PAG write-up from a Mid-band one — typically the timing uncertainty dominates when only one or two periods are timed, but averaging over many periods (e.g. timing 20 periods and dividing) reduces the timing uncertainty to negligibility.
Going Further
Undergraduate analogues of this material extend in three directions. First, Lagrangian mechanics generalises Newton's laws into a variational principle, with the SHM equations of motion arising naturally from a quadratic Lagrangian and Kepler's orbits arising from a central-potential Lagrangian. Second, general relativity replaces Newton's law of gravitation with the curvature of spacetime, predicting corrections (perihelion precession of Mercury, gravitational waves, frame-dragging) that have all been experimentally confirmed. Third, modern celestial mechanics extends Kepler's two-body problem into the three-body problem and chaos theory, with applications in spacecraft trajectory design (Lagrange points, gravity assists). Suggested reading: Marion and Thornton's Classical Dynamics for the Lagrangian framing, and Hartle's Gravity: An Introduction to Einstein's General Relativity for the GR perspective. Oxbridge-style interview prompts include: "If you were standing on the equator of a rapidly rotating asteroid, could you be flung off — and at what rotation rate?" "Why does a pendulum on the Moon swing more slowly than the same pendulum on Earth?" "How would you measure G in a laboratory?"
Authorship and Sign-off
This guide was authored independently by John Haigh, paraphrasing OCR H556 Modules 5.3 and 5.4 as descriptive use. No verbatim spec text, mark-scheme phrasing, examiner-report quotation, or past-paper question reference appears. The worked examples are original.
Start at the Circular Motion, SHM and Gravity course and work through every lesson in sequence. Once centripetal-force decomposition, SHM defining-equation analysis, gravitational potential sign tracking and Kepler's-third-law derivation are automatic, every later H556 mechanics-flavoured topic — magnetic-field circular motion, AC circuit oscillation, stellar orbital dynamics — becomes a recognition task rather than a fresh problem.