Further mechanics and thermal physics (A-level only)

Explanation

• An object moving at constant speed in a circle accelerates because its velocity changes direction. Its instantaneous velocity is tangential, while acceleration and resultant force point towards the centre.
• Angular speed is $\omega=v/r=2\pi f$ in $\text{rad s}^{-1}$, with one complete revolution equal to $2\pi$ radians.
• Therefore $a=v^2/r=\omega^2r$ and $F=mv^2/r=m\omega^2r$.
• “Centripetal force” is not a new force: it is the inward resultant supplied by forces such as tension, friction or gravity.
• The specification does not require a direction for the angular-velocity vector.

Explanation

• Simple harmonic motion has acceleration proportional to displacement and directed towards equilibrium: $a=-\omega^2x$.
• A suitable displacement equation is $x=A\cos\omega t$, and speed at displacement $x$ satisfies $v=\pm\omega\sqrt{A^2-x^2}$; the sign follows the direction of motion.
• Speed is greatest at equilibrium, $v_{\max}=\omega A$, whereas acceleration magnitude is greatest at the extremes, $a_{\max}=\omega^2A$.
• On time graphs, velocity is the gradient of displacement and acceleration is the gradient of velocity, so the three sinusoidal curves have fixed quarter-cycle phase differences.
• Define the positive direction before assigning signs.

Explanation

• A mass-spring oscillator has period $T=2\pi\sqrt{m/k}$, while a simple pendulum at small angle has $T=2\pi\sqrt{l/g}$. Other oscillators may be analysed from supplied information.
• In ideal spring SHM, total energy is $E=\tfrac12kA^2$; elastic potential energy is $E_{\text{p}}=\tfrac12kx^2$ and the remainder is kinetic.
• Thus kinetic energy is greatest at equilibrium, potential energy is greatest at the extremes and total energy is constant.
• Damping transfers energy to the surroundings, reducing amplitude.
• Required practical 7 measures how period depends on mass for a spring and on length for a pendulum.

Explanation

• A free vibration follows an initial disturbance and occurs at the system’s natural frequency. A forced vibration is maintained by a periodic driving force and settles at the driving frequency.
• Resonance occurs when driving frequency equals natural frequency, causing maximum energy transfer and the greatest steady amplitude.
• Increasing damping dissipates more energy per cycle, so the resonance peak becomes lower and broader: the resonance is less sharp.
• Resonance is useful in selected mechanical systems and in producing stationary waves, but it can damage structures or machinery driven close to a natural frequency.
• A complete explanation links frequency matching, energy transfer and amplitude.

Explanation

• Internal energy is the sum of randomly distributed particle kinetic and potential energies. Heating a system or doing work on it increases internal energy; energy transferred from the system or work done by it decreases internal energy.
• For a temperature change, $Q=mc\Delta\theta$.
• During a change of state, $Q=ml$ and temperature stays constant while particle potential energy changes rather than average kinetic energy.
• Continuous-flow heating often uses $P=\dot{m}c\Delta\theta$ when losses are negligible.
• Practical determinations of specific heat capacity or specific latent heat must account for energy transferred to the apparatus and surroundings, electrical measurements and rate or mass uncertainties.

Explanation

• For a fixed gas mass, empirical gas laws relate pressure, volume and absolute temperature: Boyle’s law applies at constant temperature and Charles’s law at constant pressure.
• Absolute zero motivates the kelvin scale, so convert using $T/\text{K}=\theta/^{\circ}\text{C}+273.15$.
• The ideal-gas equation is $pV=nRT$ for $n$ moles or $pV=NkT$ for $N$ molecules; distinguish molar mass from one molecule’s mass.
• At constant pressure, work done by an expanding gas is $W=p\Delta V$.
• Required practical 8 investigates Boyle’s and Charles’s laws, with pressure, volume and temperature uncertainties and control variables stated explicitly.

Explanation

• Brownian motion is random motion of visible particles caused by unequal molecular impacts and supports the molecular model. Kinetic theory assumes many identical point molecules in random motion, negligible intermolecular forces except during perfectly elastic collisions, and negligible collision duration.
• Momentum transfer to container walls gives $pV=\tfrac13Nm(c_{\text{rms}})^2$; random isotropic motion produces the factor $1/3$.
• Combining this with $pV=NkT$ gives $\tfrac12m(c_{\text{rms}})^2=\tfrac32kT=3RT/(2N_{\text{A}})$.
• For an ideal gas, internal energy is molecular kinetic energy because intermolecular potential energy is neglected.
• This theory explains empirical gas laws and illustrates how evidence changes scientific models.