Write the expression for the total kinetic energy of a ball rolling without slipping, and state the relationship between its linear and angular speed.

The total kinetic energy is ½mv² + ½Iω², where m is the mass in kg, v the speed of the centre in m s⁻¹, I the moment of inertia in kg m² and ω the angular velocity in rad s⁻¹. Rolling without slipping means v = rω, where r is the radius in m.

A solid ball rolls from rest down a slope of vertical height 1.2 m without slipping. Calculate its speed at the bottom, taking g as 9.81 m s⁻².

Energy conservation gives mgh = 0.7mv², so v = √(9.81 × 1.2 ÷ 0.7) = √16.8 = 4.10 m s⁻¹. Substituting back, 0.7 × 4.10² = 11.8 J per kilogram. That is equal to gh, since 9.81 × 1.2 is also 11.8.

A hollow ball behaving as a thin spherical shell rolls from rest down a slope of height 1.2 m. Calculate its speed at the bottom, taking g as 9.81 m s⁻².

Energy conservation gives mgh = ⅚mv², so v = √(9.81 × 1.2 × 6 ÷ 5) = √14.1 = 3.76 m s⁻¹. Substituting back, ⅚ × 3.76² = 11.8 J per kilogram. That is equal to gh for this slope.

A solid ball and a hollow ball of the same mass and radius are released together at the top of a slope. Which reaches the bottom first, and why?

The solid ball wins, because it has the smaller moment of inertia and so gives up a smaller share of the released potential energy to spinning. More of that energy is left as translational kinetic energy, so its centre moves faster all the way down.

WJEC A-Level Physics 1420QS

Basic Physics

Topic overview Units and quantities Scalars, vectors and density Moments, centre of gravity and equilibrium Prefixes, derived units and unit conversion Adding coplanar vectors and the triangle of forces Resolving forces on an inclined plane Taking moments: beams, ladders and stability

Kinematics

Topic overview Describing motion Equations of motion and free fall Projectile motion Non-uniform acceleration and curved motion graphs Projectiles launched at an angle Measuring speed, acceleration and g in the laboratory

Dynamics

Topic overview Newton's second law and momentum Force and Newton's laws Impulse and the force–time graph Collisions and explosions in one dimension Free body diagrams and connected bodies Newton's third law pairs and everyday forces

Energy Concepts

Topic overview Work and power Conservation of energy and efficiency The work–energy relationship Gravitational potential energy and energy interchange Elastic potential energy and the force–extension graph Power, velocity and the efficiency of machines

Solids Under Stress

Topic overview Stress, strain and the Young modulus Stress–strain graphs and the behaviour of materials Measuring the Young modulus, and material properties Hooke's law and elastic strain energy Crystalline, amorphous and polymeric solids Dislocations, ductile and brittle fracture

Using Radiation to Investigate Stars

Topic overview Stellar spectra and black bodies Wien's law, Stefan's law and stellar properties The inverse square law and stellar distance Finding the radius and temperature of a star Multiwavelength astronomy

Particles and Nuclear Structure

Topic overview Quarks, leptons and hadrons The four interactions and conservation laws The first generation of quarks and leptons Antiparticles and annihilation Baryons, antibaryons and mesons Deducing quark compositions from charge Beta decay and quark flavour change Testing a reaction against the conservation rules

Conduction of Electricity

Topic overview Current, charge and the ampere Drift velocity and the equation I = nAvq The coulomb and the charge on an electron Conductors, insulators and other charge carriers The free-electron model and the speed of conduction

Resistance

Topic overview Calculating resistance from resistivity Potential difference, Ohm's law and power Resistivity, temperature and superconductivity The volt: energy transferred per unit charge Current–voltage characteristics of non-ohmic components Measuring resistance and resistivity in the laboratory The heating effect of a current and choosing a power equation Thermistors and how a semiconductor's resistance falls with temperature Conductance, conductivity and comparing materials

D.C. Circuits

Topic overview Kirchhoff's laws, and resistors in series and parallel Potential dividers, emf and internal resistance Cells, resistivity, power and energy Working out currents and potential differences in resistor networks Sensor circuits with thermistors and LDRs Internal resistance, power transfer and short-circuit current Ammeters, voltmeters and their effect on the circuit

The Nature of Waves

Topic overview What a wave is, and the wave equation Describing a wave: peaks, compressions and phase Polarisation and plane-polarised waves Displacement–time and displacement–position graphs Wavefronts and rays Phase difference between two points on a wave

Wave Properties

Topic overview Energy transfer and the wave equation Transverse and longitudinal waves, path difference and intensity Diffraction and superposition Two-source interference and the diffraction grating Stationary waves How much a wave spreads at a gap or an obstacle Coherence and the conditions for observable interference Young's double slit and λ = ay/D Measuring wavelength with a diffraction grating Measuring the speed of sound with stationary waves

Refraction of Light

Topic overview What refraction is and why it happens Refractive index and Snell's law Total internal reflection and optical fibres The wave model of refraction and n₁v₁ = n₂v₂ Snell's law at a boundary between two media Optical fibres: core, cladding and pulse broadening

Photons

Topic overview Photon energy and the electronvolt The photoelectric effect Stopping potential and the Planck constant The photoelectric effect and photon energies Line spectra and energy levels Demonstrating the photoelectric effect The electromagnetic spectrum and typical photon energies Electron diffraction and the de Broglie relationship Radiation pressure

Lasers

Topic overview Stimulated emission and the properties of laser light Population inversion, pumping and the laser cavity Why a two-level system cannot give a population inversion Three-level and four-level laser systems The semiconductor laser and its uses

Circular Motion

Topic overview Angular velocity, period and frequency Centripetal acceleration and centripetal force Angular measure, period and frequency Vertical circular motion Horizontal circular motion in practice

Vibrations

Topic overview Simple harmonic motion and its equations Oscillating systems and energy Damping and resonance The simple pendulum The mass–spring oscillator The displacement and velocity equations Graphs of displacement, velocity and acceleration The energy interchange in simple harmonic motion Free and forced oscillations Light, heavy and critical damping

Kinetic Theory

Topic overview The assumptions of kinetic theory Molecular speeds, temperature and real gases The Avogadro constant and the mole The ideal gas equation of state and molar mass The pressure of a gas and mean molecular kinetic energy

Thermal Physics

Topic overview Internal energy and heat Work, the first law and specific heat capacity Thermal equilibrium, temperature and absolute zero The internal energy of an ideal monatomic gas Work done by a gas at constant pressure The p–V diagram and work as an area Sign conventions and special cases of the first law Measuring specific heat capacity

Nuclear Decay

Topic overview Radioactive decay, activity and half-life Distinguishing the radiations, and background radiation Randomness and spontaneity in radioactive decay Writing and balancing nuclear decay equations The decay constant and activity The exponential decay law in calculations

Nuclear Energy

Topic overview Binding energy, mass defect and E = Δmc² Nuclear reactors and the chain reaction The binding energy per nucleon curve Mass–energy calculations for fission and fusion Nuclear power: the arguments for and against

Capacitance

Topic overview Capacitance and the parallel plate capacitor Energy, combinations and RC circuits Charging at constant current and measuring capacitance Dielectrics and relative permittivity The uniform field between capacitor plates Energy stored in a capacitor and the Q–V graph Exponential discharge, the time constant and log graphs Capacitors in series and parallel: charge and potential difference sharing

Electrostatic and Gravitational Fields of Force

Topic overview Electric and gravitational fields compared Field lines, equipotentials and combining fields Electric potential and the work done on a charge Gravitational potential and potential energy Adding potentials and fields from several sources Equipotential surfaces Comparing the two fields point by point

Orbits and the Wider Universe

Topic overview Kepler's laws and orbital motion The Doppler effect, dark matter and the expanding universe Newton's law of gravitation applied to orbits Deriving Kepler's third law Finding the mass of a central object Galactic rotation curves and dark matter The centre of mass of a two-body system Radial velocity and the Doppler shift of spectral lines Hubble's law and the age of the universe

Magnetic Fields

Topic overview Magnetic fields and field patterns Magnetic flux density and the force on a conductor The force on a moving charge and its circular path The Hall effect and the Hall voltage Magnetic fields due to a current in a wire and in a solenoid Forces between current-carrying conductors Charged particle beams, velocity selectors and accelerators

Electromagnetic Induction

Topic overview Magnetic flux and flux linkage Faraday's law of electromagnetic induction Lenz's law and the direction of the induced current Emf induced in a moving conductor The rotating coil and the a.c. generator

Option A: Alternating CurrentsOption A

Topic overview Alternating current and rms values Rms values, mean power and UK mains Induced emf and the oscilloscope Reactance, impedance and resonance Flux linkage in a rotating coil Mean power and energy in a.c. circuits Phasor addition in series a.c. circuits Resonance curves and the Q factor

Option B: Medical PhysicsOption B

Topic overview Medical physics Ultrasound imaging Radiation and living tissue X-ray production and beam control Attenuation and half-value thickness Radiography, fluoroscopy and CT scanning Acoustic impedance and ultrasound reflection Magnetic resonance imaging Absorbed dose, equivalent dose and effective dose Radionuclide tracers, the gamma camera and PET

Option C: The Physics of SportsOption C

Topic overview Centre of gravity, stability and moments Restitution, moment of inertia and angular momentum Rotational energy, projectiles and the flight of a ball Moments in the muscles and joints Applying the impulse equation in sport Moment of inertia and angular kinematics Torque, angular momentum and its conservation Rotational energy and energy conservation in sport

Option D: Energy and the EnvironmentOption D

Topic overview Factors affecting the rate at which the temperature of the Earth rises The Earth's thermal equilibrium, Wien's law and Stefan's law Renewable and non-renewable sources Solar energy and photovoltaic cells Wind, tidal and hydroelectric power Nuclear fission and fusion as energy sources U-values and rates of heat loss Fuel cells and how they work Fuel cells: benefits, limitations and the hydrogen economy Thermal conduction, conductivity and insulation

Mathematical Requirements (Appendix C)

Topic overview Significant figures and arithmetic means Uncertainty and how uncertainties combine Algebraic symbols and proportionality Rearranging and solving equations Small angle approximations Tangents and instantaneous rates of change Areas under curves and numerical methods Reading a tangent on a motion graph Straight-line graphs and y = mx + c What a gradient and an intercept represent Linearising a relationship to find a constant Log plots of exponential decay Choosing what to plot to find a constant Logarithmic scales and orders of magnitude Working with base-ten logarithms The shapes of common graphs Inverse and root relationships on a graph Areas, volumes and Pythagoras Geometry applied to physical situations Substituting into physical equations Trigonometry and right-angled triangles Resolving a force into components Radians, arc length and force diagrams Random processes and translating between representations Mean values and periodic functions Symbols for approximation and comparison

How Science Works (Appendix D)

Topic overview Apparatus and taking measurements Working safely with radioactive sources Planning an investigation Analysing and evaluating evidence Uncertainty, repeatability and drawing a conclusion

Option C: The Physics of Sports

Rotational energy and energy conservation in sport

3 things to know
A2 Unit 4 Section B

Overview

Recall questions on Rotational energy and energy conservation in sport.

What you need to know

Answer each one before you reveal it.

Your recallNot started

Key term

Explain why a pole vaulter does not clear a bar at exactly the height their run-up speed would predict.

Key term

Show that a rolling solid sphere carries two sevenths of its kinetic energy as rotation.

Key term

What fraction of the kinetic energy of a rolling thin spherical shell is rotational, given I = ⅔mr²?

Questions from this subtopic

Each one with the model answer underneath.

1
Write the expression for the total kinetic energy of a ball rolling without slipping, and state the relationship between its linear and angular speed.
Model answer
The total kinetic energy is ½mv² + ½Iω², where m is the mass in kg, v the speed of the centre in m s⁻¹, I the moment of inertia in kg m² and ω the angular velocity in rad s⁻¹. Rolling without slipping means v = rω, where r is the radius in m.
2
Show that a rolling solid sphere carries two sevenths of its kinetic energy as rotation.
Model answer
Substituting I = ⅖mr² and ω = v/r into ½Iω² gives ⅕mv², so the total is ½mv² + ⅕mv² = 0.7mv². The rotational share is 0.2 divided by 0.7, giving two sevenths or 28.6%.
3
What fraction of the kinetic energy of a rolling thin spherical shell is rotational, given I = ⅔mr²?
Model answer
The rotational term becomes ⅓mv², so the total is ½mv² + ⅓mv² = ⅚mv². The rotational fraction is ⅓ divided by ⅚, giving 0.40 or 40%.
4
A solid ball rolls from rest down a slope of vertical height 1.2 m without slipping. Calculate its speed at the bottom, taking g as 9.81 m s⁻².
Model answer
Energy conservation gives mgh = 0.7mv², so v = √(9.81 × 1.2 ÷ 0.7) = √16.8 = 4.10 m s⁻¹. Substituting back, 0.7 × 4.10² = 11.8 J per kilogram. That is equal to gh, since 9.81 × 1.2 is also 11.8.
5
A hollow ball behaving as a thin spherical shell rolls from rest down a slope of height 1.2 m. Calculate its speed at the bottom, taking g as 9.81 m s⁻².
Model answer
Energy conservation gives mgh = ⅚mv², so v = √(9.81 × 1.2 × 6 ÷ 5) = √14.1 = 3.76 m s⁻¹. Substituting back, ⅚ × 3.76² = 11.8 J per kilogram. That is equal to gh for this slope.
6
A solid ball and a hollow ball of the same mass and radius are released together at the top of a slope. Which reaches the bottom first, and why?
Model answer
The solid ball wins, because it has the smaller moment of inertia and so gives up a smaller share of the released potential energy to spinning. More of that energy is left as translational kinetic energy, so its centre moves faster all the way down.

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