further mechanics

• one radian (rad) is defined as the angle subtended at the centre of a circle radius r by an arc length of length r
• $360°=$$2π$rad
• the time taken for an object moving in a circle to complete one full rotation is know as the period T
• the (rotational) frequency f of an object moving in a circle is the number of complete revolutions completed per second
• linear velocity v is the rate of change of linear displacement. linear velocity is measured in metres per second

  $v=$$2πr/t$ or $v=$$2πrf$
• angular displacement θ is the change in the angle of a body with respect to its initial angular position. angular displacement is a vector quantity
• angular speed ω is the angle through which the radius to a point in the circle turns in one second. angular speed is measured in radians per second

  $ω=$$θ/t$
• motion in a circular path at constant speed implies there is an acceleration and requires a centripetal force, which is always directed towards the centre of rotation
• the amplitude of oscillations is the maximum displacement of the oscillating object from its equilibrium position
• the time period (T) of the oscillating motion is defined as the time taken for one complete cycle of oscillation
• the frequency (f) of oscillations is the number of complete cycles that the oscillating object completes per second
• the phase difference between two objects oscillating at the same frequency is the fraction of a cycle between their oscillations (in fractions of cycles or radians)
• phases difference (in radians) $=$$2πΔt/T$
• simple harmonic motion (SHM) is defined as the motion in which the acceleration:

  is directly proportional to the displacement from the equilibrium posotion

  is always in the opposite direction to the displacement

  $a∝-x$
• in the case of SHM, the period is independent of the amplitude (isochronic)
• the v-t graph is derived from the gradient of the x-t graph
• the a-t graph is derived from the gradient of the v-t graph
• x-t graph

  
• v-t graph

  
• a-t graph

  
• in accordance with N2L

  -kx=ma

  -kx=-(2πf)^2mx

  f=1/2π*√k/m or $T=$$2π*$$√m/k$
• the equations associated with the simple harmonic motion of a simple pendulum are only valid for small angles (about 10° or less)
• a free oscillation is one in which there are no forces acting other than internal forces

  there is no periodic (driving) force acting on the system

  resistive forces are negligible

  the total energy of the system remains constant

  the amplitude remains constant over time
• an object undergoing SHM transfers energy between its potential energy and kinetic energy stores as it oscillates

  
• damped oscillations are those in which the amplitude of the oscillation is reduced due to the dissipation of energy to the surroundings
• forced vibrations are situations in which the vibrations are forced when periodic (driving) force is applied. the frequency of these vibrations is determined by the frequency of the driving force
• resonance occurs when the frequency of the applied force is equal to the natural frequency, producing vibrations of a large amplitude
• maximum energy is transferred at resonance
• damping reduces the sharpness of resonance and reduces the amplitude at the resonant (or natural) frequency