Simple Harmonic Motion

Simple Harmonic Motion

1. Measuring oscillations

   • Where an object oscillates around an equilibrium
   • Displacement from equilibrium continually changes during the motion
     • Decreases as it returns to equilibrium
     • Reverses then increases as it moves away from equilibrium
   • Amplitude is the mazimum displacement from equilibrium
   • If the amplitude is constant and no frictional forces are present, then the oscillations are described as free oscillations
   • Time period is the time for one complete cycle of oscillations
   • Frequency is number of cycles per second, measured in Hz

2. Phase difference

   • For two objects oscillating at the same frequence, phase difference in radians = 2πΔt/T
   • Δt is the time between successive instants when the two objects are at maximum displacement in the same direction

3. Principles of simple harmonic motion

   • Velocity increases to a maximum at equilibrium and decreases to a minimum at maximum displacement
   • The variation of vvelocity with time is given by the gradient of a displacement/time graph
   • Velocity is greatest when the gradient is greatest
   • The variation of acceleration with time is given by the gradient of the velocity/time graph
   • The acceleration is greatest when the gradient of the velocity time graph is greatest
   • This is when velocity is zero and occurs at maximum displacement in the opposite direction
   • Acceleration is always in the opposite direciton to displacement
   • Simple harmonic motion is a regularly repeating (oscillating) motion in which the acceleration is proportional to displacement and always in the opposite direction to displacement
   • Acceleration therefore, a = - constant x displacement
     • a = -(2πf)^2x

4. Sine waves

   • To directly see the link between simple harmonic motion and circular motion and sine curves
   • Use a projecor to cast the shadow of a ball in uniform circular motion onto a screen
   • Also project the shadow of an oscillating bob onto the same screen
   • When their time periods are matched, the two have the same horizontal motion, so acceleration of the bob is the same as the acceleration
   • The ball in simple harmonic motion has acceleration equal to -v^2/r
     • Since speed = 2πrf, then a = -(2πf)^2r
     • The component of acceleration of the ball parallel to the screen = acosθ
     • Therefore the acceleration of the ball's shadow = -(2πf)^2rcosθ = -(2πf)^2x

5. Applications

   • Frequency of an oscillating mass-spring system can be reduced
     • Add extra mass, as the extra mass increases the inertia of the system
     • Use weaker springs, as the restoring force on the trolley at any given displacement would be less so the trolley's acceleration and speed would be less
   • Mass-spring
     • Spring tension = kΔL
     • Tension =-kΔx, as the change of tension always tries to restore the object to equilibrium position
     • Fince F = ma, then acceleration = restoring force over mass. In this case, the restoring force is -kx
     • Therefore, a = -kx/m
     • Combining with the equation for acceleration, and rearranging for 1/f (or T), you get the equation on the data sheet
   • Simple pendulum
     • The weight, mg, has components mgcosθ perpendicular to the path of the bob and mgsinθ along the path towards equilibrium
     • Restoring force = -mgsinθ, so the acceleration = -mgsinθ/m, or -gsinθ
     • If θ doesnt exceed about 10 degrees, then sinθ = s/L, so a = -(g/L) s = -(2πf)^2s

6. Energy

   • A freely oscillating object oscillates with a constant amplitude because there is no friction acting upon it
   • At maximum displacement, all of the energy is potential energy, and at equilibrium all energy is kinetic
   • If no friction is present, the total energy in the system is constant
   • Damped oscillations
     • Oscillations of a simple pendulum gradually die away because dissipative forces
     • Motion is said to be damped if dissipative forces are present
     • Light damping
       • Time period is independent of amplitude so each cycle takes the same aount of time as oscillations die away
     • Critical damping
       • Just enough to stop the system oscillating after it has been displaced and released from equilibrium
       • Oscillating object returns to equilibrium in the shhortest time possible without overshooting
       • Important in vehicle suspension
     • Heavy damping
       • Damping is so strong that the displaced object returns to equilibrium much more slowly than if the system is critically damped
       • No more oscillating occurs

7. Forced oscillations and resonance

   • When a system oscillates without the application of a periodic force (one applied at regular intervals), its frequency is referred to as its natural frequency
   • When a periodic force is applied to an oscillating system, the response depends on the frequency of the periodic force
     • System undergoes forced oscillations when a periodic force is applied
   • As the applied frequency approaches the natural frequency of a mass-spring system, the amplitude increases and the phase difference between the displacement and periodic force increases from zero to 1/2π at the natural frequency
   • When applied frequency is equal to natural frequency of a mass spring system
     • Amplitude becomes very large - lighter damping, larger amplitude
     • System is in resonance
     • The phase difference between the displacement and the periodic force is 1/2π a tresonance - the periodic force is then exactly in phase with the velocity of the oscillating object
   • As applied frequency passes natural frequency, amplitude decreases and phase difference increases until the displacement is π radians out of phase with the periodic force