# Oscillations & Fields

## Contents

# PH4.1 VIBRATIONS

Content

- Circular motion
- Physical and mathematical treatment of undamped simple harmonic

motion.

- Energy interchanges during simple harmonic motion.
- Damping of oscillations.
- Free oscillations, forced oscillations and resonance.

AMPLIFICATION OF CONTENT Candidates should be able to:

4.1 (a) understand and use period of rotation, frequency, the radian measure of angle,

4.1 (b) define and use angular velocity ω ,

4.1 (c) recall and use v =ωr, and hence a =ω 2r,

4.1 (d) define simple harmonic motion as a statement in words,

4.1 (e) recall, recognise and use a = −ω 2 x as a mathematical defining equation of simple harmonic motion,

4.1 (f) illustrate, and interpret graphically, the variation of acceleration with displacement during simple harmonic motion,

4.1 (g) recall and use x = Asin(ω t +ε ) as a solution to a = −ω 2 x ,

4.1 (h) explain the terms frequency, period, amplitude and phase (ω t +ε ) ,

4.1 (i) recall and use the period as 1/f or 2π/ω,

4.1 (j) recall and use v = Aω cos (ω t +ε ) for the velocity during simple harmonic motion,

4.1 (k) illustrate, and interpret graphically, the changes in displacement and velocity with time during simple harmonic motion,

4.1 (l) recall and use the equation T = 2π √m/k for the period of a system having stiffness (force per unit extension) k and mass m,

4.1 (m) illustrate, and interpret graphically, the interchange between kinetic energy and potential energy during undamped simple harmonic motion, and perform simple calculations on energy changes,

4.1 (n) explain what is meant by free oscillations and understand the effect of damping in real systems,

4.1 (o) describe practical examples of damped oscillations, and the importance of critical damping in appropriate cases such as vehicle suspensions,

4.1 (p) explain what is meant by forced oscillations and resonance, and describe practical examples,

4.1 (q) sketch the variation of the amplitude of a forced oscillation with driving frequency and know that increased damping broadens the resonance curve,

4.1 (r) appreciate that there are circumstances when resonance is useful e.g. circuit tuning, microwave cooking and other circumstances in which it should be avoided e.g. bridge design.

# PH4.2 MOMENTUM CONCEPTS

Content

- Linear momentum.
- Newton's laws of motion.
- Conservation of linear momentum; particle collision.
- The momentum of a photon.

AMPLIFICATION OF CONTENT Candidates should be able to:

4.2 (a) define linear momentum as the product of mass and velocity,

4.2 (b) recall Newton's laws of motion and know that force is rate of change of momentum, applying this in situations where mass is constant,

4.2 (c) state the principle of conservation of momentum and use it to solve problems in one dimension involving elastic collisions (where there is no loss of kinetic energy) and inelastic collisions (where there is loss of kinetic energy).

4.2 (d) use the formula for the momentum of a photon:p = h/λ = hc/f;

4.2 (e) appreciate that the absorption or reflection of photons gives rise to radiation pressure.

# PH4.3 THERMAL PHYSICS

Content

- Ideal gas laws and the equation of state.
- Kinetic theory of gases.
- The kinetic theory of pressure of a perfect gas
- Internal energy.
- The internal energy of an ideal gas
- Energy transfer.
- First law of thermodynamics.

AMPLIFICATION OF CONTENT Candidates should be able to:

4.3 (a) recall and use Boyles law for an ideal gas,

4.3 (b) recall and use the equation of state for an ideal gas expressed as pV = nRT where R is the molar gas constant, and understand that this equation can be used to define the Kelvin scale of temperature and the absolute zero of temperature,

4.3 (c) recall the assumptions of the kinetic theory of gases which includes the random distribution of energy among the molecules,

4.3 (d) explain how molecular movement causes the pressure exerted by a gas, and understand and use p=1/3ρ<c^2>=1/3N/Vm<c^2> where N is the number of molecules,

4.3 (e) define the Avogadro constant NA and hence the mole;

4.3 (f) understand that the molar mass M is related to the relative molecular mass Mr by M/kg = Mr/1000, and that the number of moles n is given by Molar mass/ Total mass;

4.3 (g) compare pV = 1/3 Nm<c^2> with pV = nRT and deduce that the total translational kinetic energy of a mole of a monatomic gas is given by 3/2RT 2 and hence the average kinetic energy of a molecule is 3/2kT where k=R/NA is the Boltzmann constant, and deduce that T is proportional to the mean kinetic energy

4.3 (h) understand that the internal energy of a system is the sum of the potential and kinetic energies of its molecules;

4.3 (i) understand that the internal energy of an ideal monatomic gas is wholly kinetic so is given by U = 3/2nRT

4.3 (j) understand that heat enters or leaves a system through its boundary or container wall, according to whether the system's temperature is lower or higher than that of its surroundings, so heat is energy in transit and not contained within the system;

4.3 (k) understand that if no heat flows between systems in contact, then they are said to be in thermal equilibrium, and are at the same temperature;

4.3 (l) understand that energy can also enter or leave a system by means of work, so work is also energy in transit;

4.3 (m) use W = pΔV to calculate the work done by a gas under constant pressure;

4.3 (n) understand and explain that, even if p changes, W is given by the area under the p – V graph;

4.3 (o) recall and use the first law of thermodynamics, in the form ΔU = Q−W , knowing how to interpret negative values of ΔU, Q, and W.

4.3 (p) understand that for a solid (or liquid), W is usually negligible, so Q = ΔU ;

4.3 (q) use the formulaQ = mcΔθ , for a solid or liquid, understanding that this is the defining equation for specific heat capacity, c.

# PH4.4 ELECTROSTATIC AND GRAVITATIONAL FIELDS OF FORCE

Content

- Electrostatic and gravitational fields.
- Field strength (intensity).
- Electrical and gravitational inverse square laws.
- Potential in force fields.
- Relation between force and potential energy gradient.
- Relation between intensity and potential gradient.
- Field lines and equipotential surfaces.
- Vector addition of electric fields.
- Potential energy of a system of charges.

AMPLIFICATION OF CONTENT Candidates should be able to:

4.4 (a) recall the main features of electric and gravitational fields as specified in the table.

4.4 (b) recall that the gravitational field outside spherical bodies such as the earth is essentially the same as if the whole mass were concentrated at the centre;

4.4 (c) understand that field lines (or lines of force) give the direction of the field at a point, thus, for a positive point charge, the field lines are radially outward; and that equipotential surfaces join points of equal potential and are therefore spherical for a point charge;

4.4 (d) calculate the net potential and resultant field strength for a number of point charges and point masses;

4.4 (e) appreciate that ΔUP = mgΔh for distances over which the variation of g is negligible.

# PH4.5 APPLICATION TO ORBITS IN THE SOLAR SYSTEM AND THE WIDER UNIVERSE

Content

- Kepler's Laws of Planetary Motion
- Circular orbits of satellites, planets and stars
- Centre of Mass
- Missing mass in galaxies – Dark Matter
- Objects in mutual orbit
- Doppler shift of spectral lines
- Extra-solar planets

AMPLIFICATION OF CONTENT Candidates should be able to:

4.5 (a) state Kepler's three Laws of Planetary Motion,

4.5 (b) recall and use Newton's law of Gravitation F=Gm1m2/r^2 in simple examples, including the motion of planets and satellites;

4.5 (c) derive Kepler's 3rd Law, for the case of a circular orbit from Newton's Law of Gravity and the formula for centripetal acceleration,

4.5 (d) use data on orbital motion, such as period or orbital speed, to calculate the mass of the central object;

4.5 (e) appreciate that the orbital speeds of objects in spiral galaxies implies the existence of dark matter;

4.5 (f) calculate the position of the centre of mass of two sphericallysymmetric objects, given their masses and separation, and calculate their mutual orbital period in the case of circular orbits,

4.5 (g) use the Doppler relationship in the form Δλ/λ=v/c;

4.5 (h) calculate a star's radial velocity (i.e. the component of its velocity along the line joining it and an observer on the Earth) from data about the Doppler shift of spectral lines,

4.5 (i) use data on the variation of the radial velocities of the bodies in a double system (e.g. a star and orbiting planet) and their orbital period to determine the masses of the bodies for the case of a circular orbit edge on as viewed from the Earth.