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M. Manser
A2 Level Physics
Newton’s laws of motion
 Newton’s first law: A body remains at rest or continues
in motion with constant velocity unless an externally
applied force acts on it.
 Newton’s second law: Force is proportional to the rate
of change of momentum. Force is defined as rate of
change of momentum: F = Δ(mv)/Δt
 When the mass m of a body is constant (at speeds
much less than the speed of light), Newton’s second law
becomes F = ma where a is acceleration.
 Newton’s third law: If a force acts on a body then an
equal and opposite force of the same type acts on
another body.
Circular motion
 Linear momentum (a vector) = mass × velocity: p = mv
 Impulse of a force = change in momentum = F Δt
 Area under a force–time graph = impulse
 Principle of conservation of momentum: In any
collision or explosion, momentum is conserved,
provided that no external force acts.
 In an elastic (or perfectly elastic) collision there is no
loss of kinetic energy. Most collisions are inelastic,
with some loss of kinetic energy.
 Intermolecular collisions are perfectly elastic, with no
loss of kinetic energy.
 Angle in radians = arc length/radius. 180° = π rad
 Angular velocity (or angular frequency) ω is the rate of
change of angle. ω = Δθ/Δt = 2π/T = 2πf where T is
the time period and f is the frequency.
 A body moving in a circle at constant speed is
changing direction, and therefore changing velocity. It
is thus accelerating. The centripetal acceleration and
force are directed towards the centre of the circle.
 For a body moving at speed v in a circle of radius r
with period T:
v = 2πr/T a = v2/r F = ma = mv2/r
Gravitational fields (1)
 Gravitational field strength g is force per unit mass.
g = F/m
 On the Earth’s surface, the gravitational field strength
g = 9.81 N kg−1 is approximately equal to the
acceleration of free fall: g = 9.81 m s−2
 A gravitational field may be represented by field lines.
In a uniform field, the lines are parallel; around the
Earth, the lines are radial.
 Newton’s law of gravitation: The force of attraction F
between point masses m1 and m2 at a separation r is
F = −Gm1m2/r2 where G = 6.67 × 10−11 N m2 kg−2 is the
universal constant of gravitation.
Gravitational fields (2)
 At a distance r from a mass M, the gravitational field
strength g = −GM/r2
 Equating GMm/r2 = mv2/r for the force on a mass m in
orbit round a mass M, and using v = 2πr/T leads to
GMT2 = 4π2r3 and if the radius of orbit r and period T
are known, M may be calculated.
 Johannes Kepler stated this result as his third law for
planets in orbit round the Sun: T 2 ∝ r 3
 A satellite in a geostationary orbit round the Earth has
a period of 24 h, so that it remains above one point on
the equator.
Simple harmonic motion (1)
 Examples of SHM are the vertical oscillations of a
mass on a spiral spring, and the simple pendulum.
The horizontal oscillation of a trolley between
stretched springs is damped harmonic motion.
 In SHM the period T of oscillation is independent of
amplitude A.
 The displacement x of a body in SHM from the central
position is: x = A cos ωt = A cos 2πft if time t = 0 at an
extremity of motion; x = A sin ωt = A sin 2πft if time
t = 0 at the centre of the motion.
 The maximum speed of an oscillator is:
vmax = ωA = (2πf)A
Simple harmonic motion (2)
 Definition: SHM occurs when the acceleration a of a
body is directed to a fixed point and is proportional to
the displacement x from that point. Acceleration
a = −ω2x = −(2πf)2x amax = −(2πf)2A
 The displacement–time, velocity–time and
acceleration–time graphs for SHM are all sinusoidal,
with ¼ cycle, i.e. π/2 rad, phase differences.
 The total energy of a body in SHM is constant. The
energy–displacement graph for SHM shows zero KE
at x = A, maximum KE at x = 0, and maximum PE
(gravitational and/or elastic) at x = A
Damping and forced oscillations
 Damping does not change the period of motion, but
leads to exponentially decreasing amplitude.
 Resonance is the maximum amplitude of a forced
oscillation when the driving frequency equals the
natural frequency of the driven oscillator. When this
happens there is maximum energy transfer.
 Resonance is used in microwave ovens, in radio
tuners, and in musical instruments. Resonance is a
nuisance in suspension bridges.
M. Manser
A2 Level Physics
 Solids, liquids and gases are distinguished by the
spacing, ordering and motion of their atoms or
 Brownian motion is demonstrated by viewing smoke
particles in air under a microscope. The experiment
shows that there is much empty space in air, but that
the motion of the smoke particles is changed by the
random collisions of the much smaller (air) particles
moving randomly at very high speed.
 Pressure p is force F per unit area A: p = F/A
 The elastic collisions of millions of molecules per
second on a surface cause gas pressure.
Internal energy; temperature
 The internal energy of a system is the sum of the
random distribution of kinetic and potential energies of
the molecules in the system.
Specific heat capacity and latent heat
 The heating of a body leads to an increase in internal
energy and to either an increase in temperature or a
change of state.
 The specific heat capacity of a substance is the
energy provided by heating to raise the temperature
of unit mass by 1 °C or 1 K.
 If a mass m of a substance of s.h.c. c is raised
through a temperature difference Δθ, the energy E
transferred is: E = mc Δθ
 Latent heat (of fusion or of vaporisation) is the energy
provided by heating to change the state without
changing the temperature.
Ideal gases
 Boyle’s law: For a fixed mass of gas at constant
temperature T the pressure p is inversely proportional
to the volume V: pV = constant
 For a fixed mass of gas: pV/T = constant or
p1V1/T1 = p2V2/T2
 Kinetic theory assumptions: Gases consist of large
numbers of indivisible molecules moving at high
speeds. Intermolecular collisions are (perfectly)
elastic. Volume of molecules ≪ space in which they
move. Duration of collisions ≪ time between
collisions. Forces between distant molecules are
Electric fields (1)
 Electric field strength E is defined as force F per unit
positive charge Q: E = F/Q
The equation of state for an ideal gas
1023 mol−1
 The Avogadro constant: NA = 6.02 ×
 One mole of any substance contains 6.02 × 1023
 The equation of state: The pressure p, volume V and
temperature T (in kelvins) of an ideal gas are related
by pV = nRT or pV = NkT where n is the amount of
substance in moles, R is the molar gas constant, N is
the number of particles, and k = R/NA is the
Boltzmann constant.
 The mean translational kinetic energy E of an atom in
an ideal gas is directly proportional to the temperature
of the gas in kelvins: E = 3⁄2kT
Electric fields (2)
 Coulomb’s law: The force between two point charges
Q and q at a separation r in air or space is
F = Qq/4πε0r2 where ε0 is the permittivity of free
 The electric field at distance r from a point charge Q is
E = Q/4πε0r2.
 The gravitational field strength around a point mass
and the electric field strength around a point charge
are both inversely proportional to the square of the
distance from the point mass or point charge.
 Electrostatic forces may be attractive or repulsive, but
gravitational forces are only attractive.
 Absolute zero (zero kelvins, 0 K) is the temperature at
which a substance has minimum internal energy.
 The thermodynamic scale of temperature is an
absolute scale that does not depend on the property
of any particular substance.
 Temperatures in kelvins and in degrees Celsius may
be converted using: T/K = θ/ºC + 273.15
 E is a vector quantity, measured in N C−1 (or V m−1).
 Between charged parallel plates there is uniform
electric field, shown by parallel field lines, of
magnitude E = V/d.
 If a charged particle is moving across a uniform field
E, it will follow a parabolic path (similar to a golf ball
moving in air under gravity).
 The electric field lines around a point charge or a
charged sphere are in a radial pattern.
Magnetic fields
 The magnetic field pattern around a long currentcarrying conductor is of concentric circles.
 The magnetic field pattern around a long solenoid is
similar to that around a bar magnet; within the
solenoid the field is uniform.
 The direction of the force on a current-carrying
conductor placed at right angles to a magnetic field is
found using Fleming’s left-hand rule:
first finger = field
second finger = current
thumb = motion
M. Manser
A2 Level Physics
Magnetic flux density
 The force on a conductor of length L carrying current I
at an angle θ to a field of flux density B is
F = BIL sin θ
 Magnetic flux density B is the force per unit length per
unit current on a conductor placed at right angles to
the field: B = F/IL
 A field has flux density of 1 tesla (1 T) if the force on a
conductor of length 1 m carrying a current of 1 A at
right angles to the field is 1 N.
 A particle of charge Q moving at speed v at right
angles to a magnetic field of flux density B
experiences a force F = BQv.
Magnetic flux
 The magnetic flux through an area A inclined at an
angle θ to a magnetic field of flux density B is
Φ = BA sin θ.
 If an area of 1 m2 is perpendicular to a magnetic field
of flux density 1 T then the magnetic flux through the
area is 1 weber (1 Wb).
 The magnetic flux linkage through a coil is the
magnetic flux times the number of turns N on the coil:
magnetic flux linkage = NΦ = BAN sin θ
Motion of charged particles in magnetic fields
 The force F on a charge Q moving at speed v across
a magnetic field B causes it to move in a circular path
of radius R because the force is always perpendicular
to the field and to the motion.
 The force on the moving charged particle is the
centripetal force. Therefore BQv = mv2/R and so:
BQR = mv
 A charged particle moving across electric and
magnetic fields at right angles has an undeflected
path if FB = FE, i.e. if BQv = EQ. This is a velocity
selector for speed v = E/B and is used in a mass
Electromagnetic induction
 Faraday’s law of electromagnetic induction: The
induced e.m.f. is proportional to the rate of change of
flux linkage.
 Lenz’s law: The induced e.m.f. acts in such a direction
as to oppose the change causing it. This follows from
the conservation of energy.
 Faraday’s and Lenz’s laws become: induced e.m.f. =
- (rate of change of flux linkage); e = −Δ(NΦ)/Δt
 Alternating current (a.c.) can be generated using a
coil rotating in a magnetic field or a magnet rotating
inside a coil.
 A transformer is made using two separate coils of wire
wound on a laminated iron ring or core. Transformers
work only on a.c., not d.c.
 In an ideal transformer with a primary coil of NP turns
and a secondary coil of NS turns, the ratio of the
secondary and primary voltages is: VS/VP = NS/NP
 Step-up and step-down transformers are used to
transmit electrical energy over long distances. In a
high-voltage power line the low current wastes little
energy as heat in the resistance of the wires.
 The capacitance C of a capacitor is the charge Q
stored per unit potential difference V: C = Q/V.
Discharging capacitors
 When a capacitor discharges through a resistor R the
p.d. V across it, charge Q on it, and current I through
R all fall exponentially with time.
 In an exponential decay each quantity (V, Q or I) falls
by a constant factor in each successive time interval.
 The time constant for a capacitance C discharging
through a resistance R is T = RC.
 The p.d., charge and current all fall with time t
according to the same pattern:
V = V0e−t/CR Q = Q0e−t/CR I = I0e−t/CR
The nuclear atom (1)
 Rutherford’s alpha-scattering experiment shows that
each atom has a tiny massive positively charged
nucleus and much empty space.
 The nucleus of an atom comprises protons and
neutrons and is surrounded by orbiting electrons.
 A capacitor has a capacitance 1 farad (1 F – a very
large unit) if it stores 1 coulomb of charge when the
p.d. across it is 1 volt.
 For three capacitors in parallel: CP = C1 + C2 + C3
 For three capacitors in series:
1/CS = 1/C1 + 1/C2 + 1/C3
 The energy stored in a capacitor is given by W = ½QV
= ½CV 2 and is represented by the area under a graph
of p.d. against charge.
 Nuclear diameters are measured in fm (10−15 m);
atomic diameters are of the order of 10−10 m.
 The density of a proton of diameter 1.2 fm is of the
order of 1017 kg m−3. Nucleons are tightly packed, thus
all nuclei have similar densities.
M. Manser
A2 Level Physics
The nuclear atom (2)
 Two protons in a nucleus have a strong electrostatic
repulsion (Coulomb’s law) and a weak gravitational
attraction (Newton’s law of gravitation).
 There must therefore be a very strong short-range
attractive force (the strong nuclear interaction)
between nucleons.
 For a nuclide
is the nucleon number or mass
number, the number of nucleons in the nucleus; Z is
the proton number or atomic number of the element,
the number of protons in the nucleus.
 For isotopes of an element, Z is the same, but A is
Fundamental particles
 Baryons, massive particles such as protons and
neutrons, are not fundamental and are made up of
charged particles called quarks.
 The six types of quark are: up, down, strange, charm,
top and bottom.
 The quarks in a proton are u, u, d; in a neutron, u, d, d.
 Weak nuclear interaction: a quark may change its
flavour causing β decay of the nucleus. In β− decay
the emissions are an electron and an antineutrino; in
β+ decay, a positron and a neutrino.
 Electrons and neutrinos are leptons, which are
fundamental particles.
 The activity A of a radioactive source is its rate of
decay in becquerels (Bq).
 Decay constant λ is probability of decay per unit time.
 If there are N undecayed nuclei present, then the
activity A = λN.
 The half-life t1/2 of a decay process is the time for the
number of undecayed nuclei or the activity to fall to
half of its original value.
 Half-life and activity are related: λt1/2 = ln 2 = 0.693.
 The activity A and number of undecayed nuclei N fall
exponentially with time according to: A = A0e−λt
N = N0e−λt
Nuclear fission
 Einstein’s equation: If an object is given energy ΔE, its
mass increases by Δm = ΔE/c2 where.c is the speed
of light in vacuo.
 A thermal neutron may cause the fission of a large,
neutron-rich nucleus such as U-235 into two large
fragments and more neutrons.
 Fission reactors, with fuel rods, control rods and
moderators, are energy sources, but produce longlived radioactive waste.
 The binding energy of a nucleus is the energy needed
to break the nucleus into its separate protons and
Medical diagnosis
 A gamma camera detects γ-rays emitted by the
Tc-99m radioisotope absorbed by parts of the body.
 A positron emission tomography (PET) scanner is
used to scan the brain and map changes in the blood
flow. Carbon-11, absorbed into the blood, decays by
positron emission and produces γ-ray photons.
 In magnetic resonance imaging (MRI) a patient is
placed in a strong magnetic field. A radiofrequency
pulse of e-m radiation produces precession of nuclei
at the Larmor frequency.
(~10−10 m)
 X-rays are short-wavelength
waves produced when high-energy electrons hit a
metal surface.
 When they interact with matter, X-rays may be
scattered by the Compton effect, and may cause the
photoelectric effect or produce an electron–positron
pair and then two photons.
 The intensity of an X-ray beam is the power per unit
cross-sectional area.
 The intensity I of a collimated X-ray beam varies with
the thickness x of the medium according to I = I0e−μx
where μ is the linear attenuation coefficient.
The universe (1)
 1 astronomical unit (AU) = 1.5 ×
1 light-year (ly) = 9.5 × 1015 m
1 parsec (pc) = 3.1 × 1016 m = 2.1 × 105 AU = 3.3 ly
 Doppler effect: For an e-m wave source moving at a
speed v relative to an observer, the apparent change
in wavelength Δλ is given by Δλ = (v/c)λ.
 Hubble’s law: The recession velocity v of a distant
galaxy is proportional to its distance d: v = H0d.
1011 m
 H0 = 70 km s−1 Mpc−1
= 70 000/(106 × 3.1 × 1016) = 2.26 × 10−18 s−1
 The age of the universe = 1/H0 = 1.4 × 1010 y
= 14 billion years
The universe (2)
 Olbers' paradox: An infinite universe full of stars
should mean that the light reaching us from all
directions would give a bright night sky. This theory
fails as the universe is not static: stars are evolving.
 The cosmological principle: The universe is
 The 3 K microwave background radiation is evidence
of radiation from the matter and antimatter annihilation
following the big bang.
 The universe may be ‘open’, ‘flat’ or ‘closed’,
depending on its density. The critical density of the
universe is: ρ0 = 3H02/8πG