Further mechanics

• When an object is moving in a circle it has constantly changing velocity (vector) and so must be accelerating, known as centripetal acceleration.
• Newton’s first law states that for an object to be accelerating it must have a resultant force.
• for an object to have circular motion it must have an acceleration and a centripetal force acting towards the centre
• For an object to have simple harmonic motion it must have acceleration proportional to displacement in opposite direction
• in SHM, force is proportional to distance from equilibrium position in opposite direction.
• isochronous is when oscillations always have the same time period no matter the size of amplitude.
• a displacement time graph shows a sin graph with equilibrium position at time T.
• velocity is the derivative of displacement, so a v-t graph shows a cos graph where the peaks from d-t graphs become the intercepts and intercepts become peaks.
• acceleration is the derivative of velocity so the a-t graph shows the gradient function of velocity graph (upside down sin graph) where peaks turn into intercepts and vice versa
• amplitude is the max displacement
• to turn degrees into radians, you multiply by pi/180
• to turn rads into degrees you multiply by 180/pi
• for pendulum systems, have to let go from angle less than 10 degrees as small angle approximation was included in the derivation of the formula.
• for vertical mass-spring systems, kinetic energy is transferred to elastic and gravitational potential and the back to kinetic as it oscillates.
• for horizontal mass spring systems, kinetic energy is transferred to elastic potential and back as it oscillates
• at amplitude (max displacement) a SHM system has max potential energy, as it moves to equilibrium position the PE is converted into KE. KE is max at equilibrium position. total energy in the system is constant.
• Damping is when energy in oscillations is lost to environment, leading to reduced amplitude.
• under damping is when amplitude gradually decreases by small amounts each oscillation
• critical damping reduces amplitude to 0 as quick as possible with no oscillations
• over damping is when amplitude decreases, slower than critical, with no oscillations
• resonance is when amplitude drastically increases due to gaining energy from driving force
• resonance occurs when driving frequency (from forced vibrations) equals the natural frequency.
• in a swing, if driving frequency equals natural frequency then the swing swings higher.
• if you increase damping, resonant frequency decreases and the max amplitude peak becomes wider.
• The red line shows under damping

  
• The green line shows critical damping

  
• The purple line shows over damping

  
• An object on a vibration plate is no longer in contact when the downwards acceleration of the plate is greater than acceleration of sand due to gravity
• Elastic potential energy = ½ k e^2